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The critical behaviour of ethylene and hydrogen De Bruyn, John Roy 1987

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T H E C R I T I C A L B E H A V I O U R OF E T H Y L E N E AND H Y D R O G E N by J O H N R O Y d e B R U Y N B.Sc. Physics and Astronomy, University of British Columbia, 1979 M.Sc. Physics, Queen's University, 1982 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES ( D E P A R T M E N T of PHYSICS) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A October 1987 © John Roy deBruyn, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Phys ics The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ^ f O / . 1987 DE-6G/81) ii A b s t r a c t Optical techniques have been used to study the behaviour of ethylene and hydrogen near their liquid-vapour critical points. From measurements of the co-existence curve of ethylene over the reduced temperature range 1.5 x 1 0 - 6 < t < 4.5 x 10~2, where t — (Tc — T)/Tc and Tc is the critical temperature, we find the criti-cal exponent /? = 0 .327± .002 and the corrections-to-scaling exponent A = 0 .46± .02 . Similar measurements for hydrogen over the range 3.2 x 1 0 - 5 < t < 7.0 x 1 0 - 2 give /? = 0.326 ± .002 and A = 0.46 ± .02. Measurements of the compressibility of hy-drogen give the critical exponent 7 = 1.19 ± .05 and the critical amplitude ratio FQ/TQ = 5.2 ± .4. With the exception of A , which is slightly lower than its pre-dicted value of 0.5, the results for these universal quantities are in agreement with theoretical predictions. The leading coexistence curve amplitude for hydrogen, BQ = 1 .19± .03 , is lower than the corresponding values for ethylene, BQ = 1.56 ± .03, and for other room-temperature fluids. This decrease is in qualitative agreement with the predictions of a theory of quantum effects on critical behaviour. Measurements of the coexistence curve diameter for both fluids show an anomaly near the critical point having a form consistent with the predicted tl~a temperature dependence. These results are in agreement with a recent theory of the effects of many-body forces on the diameter; the hydrogen data indicate that these forces are attractive in that fluid. This suggests that quantum mechanical exchange interactions are important near the critical point of hydrogen. iii T a b l e o f C o n t e n t s Abstract ii List of Tables vi List of Figures vii Acknowledgments x Chapter I Introduction 1 I- l An Introduction to Critical Phenomena in Fluids 1 1-2 Summary of Past Work 4 I- 3 Summary of This Work 9 Chapter II Theory 11 II— 1 The van der Waals Equation in the Critical Region 11 II—2 Homogeneity and Scaling 15 II—3 Renormalization Group 22 II—4 Corrections to Scaling 26 II—5 Universal Amplitude Ratios 28 II— 6 Quantum Mechanical Effects on Critical Behaviour 29 Chapter III The Experiments 32 III— 1 The Focal Plane and Image Plane Experiments 32 III-2 The Ethylene Experiment 40 III—3 The Hydrogen Experiment 46 III-4 The Prism Cell Experiment 56 iv III—5 Experimental Limitations 61 Gravitational Rounding 61 Temperature Gradients 63 III— 6 Data Reduction and Experimental Error 64 Chapter IV Results, Part 1: Ethylene 67 IV- 1 The Prism Experiment 67 The Lorenz-Lorentz Function 67 The Coexistence Curve 70 The Coexistence Curve Diameter 77 IV-2 The Fringe Experiment 82 IV- 3 Combined Results 84 Chapter V Results, Part 2: Hydrogen 91 V - l The Coexistence Curve 91 V-2 The Coexistence Curve Diameter 99 V - 3 The Compressibility 103 Chapter VI Discussion 109 V I - 1 Comparison with Previous Work 109 VI-2 Universal Critical Parameters I l l VI-3 Non-Universal Properties and Quantum Effects 112 The Coexistence Curve and the Compressibility 112 The Coexistence Curve Diameter 118 VI-4 Crossover Effects 119 Chapter VII Conclusions 124 References 126 Appendix A Conversion of Refractive Indices to Densities 132 Appendix B Data Listings 136 B-1 Lorenz-Lorentz Coefficient for Ethylene 137 £{p) at 298.4 K ..137 V £,(p) on the Coexistence Curve 138 B-2 Ethylene Coexistence Curve Data 139 Prism Data 139 Fringe Data 141 B-3 Hydrogen Coexistence Curve Data 150 Order Parameter Data 150 Coexistence Curve Diameter Data 155 B-4 Hydrogen Compressibility Data 159 Appendix C Computer Programs 160 C-1 BIGFIT - Coexistence Curve Data Analysis Program 160 Listing of BIGFIT 162 C-2 L L N E W E R and COEX86 - Prism Cell Data Analysis Programs . . . 179 Listing of L L N E W E R 180 Listing of COEX86 185 C-3 Temperature Sweeping Program 189 Listing of the Temperature Sweeping Program 190 vi L i s t o f T a b l e s I - l : Critical Exponents 5 I- 2: Universal Amplitude Ratios 5 II— 1: Scaling Laws 18 II—2: Corrections-to-Scaling Series 27 IV-1: Refractometric Virial Coefficients and Electronic Polarizability for Ethylene. 68 IV-2: Fits of the ethylene order parameter data to eq. (1-6) with A = 0.5. . . . 76 IV-3: Fits of the ethylene order parameter data to eq. (1-6) with A a free parameter. 78 IV-4: Fits to the Coexistence Curve Diameter of Ethylene 81 IV- 5:. Fits to the Coexistence Curve Diameter of Ethane 82 V - l : Fits of the hydrogen order parameter data to eq. (1-6) with A = 0.5. . . . 96 V-2: Fits of the hydrogen order parameter data to eq. (1-6) with A a free param-eter 98 V-3: Fits to the Coexistence Curve Diameter of Hydrogen 101 V - 4: Fits to the Compressibility of Hydrogen 107 V I - 1: Summary of Results 112 v i i L i s t of F i g u r e s 1- 1: The pressure-temperature phase diagram of a typical fluid 2 2- 1: The phase diagram of a van der Waals fluid 13 2- 2: A schematic representation of the Kadanoff scale transformation 19 3- 1: Gravity-induced density gradients in a fluid near the critical point 34 3-2: The optical setup for the fringe experiments 36 3-3: Formation of the Fraunhofer diffraction pattern 39 3-4: A block diagram of the ethylene apparatus 41 3-5: The temperature control circuit used in the ethylene experiment 43 3-6: The computer interface electronics 45 3-7: The cryostat used in the hydrogen experiment 47 3-8: The hydrogen cell assembly 48 3-9: The low temperature valve 50 3-10: The temperature controller for the hydrogen experiment 52 3-11: The gas handling system used in the hydrogen experiment 55 3-12: The optical setup for the hydrogen experiment 57 3-13: The optical arrangement for the prism experiment 59 3-14: A phase diagram showing the path followed in taking a data point in the prism experiment 60 3-15: Gravitational rounding 62 3-16: The Lorenz-Lorentz coefficient for H2, adapted from ref. 63 65 VIU 4-1: The Lorenz-Lorentz coefficient as a function of density for ethylene 69 4-2: The coexistence curve of ethylene and its diameter as measured in the prism experiment 71 4-3: A log-log plot of the order parameter vs. reduced temperature for the ethylene prism data 72 4-4: A log-log plot of Ap/t@ vs. reduced temperature for the ethylene prism data. 74 4-5: Residuals of fits to the order parameter data of fig. (4-4) 75 4-6: A schematic diagram of the coexistence curve diameter 79 4-7: The coexistence curve diameter of ethylene 80 4-8: The coexistence curve diameter of ethane 83 4-9: A log-log plot of the order parameter vs. reduced temperature for the ethylene fringe data 85 4-10: A log-log plot of Ap/t@ vs. reduced temperature for the ethylene fringe data. 86 4-11: Residuals for the fit shown in fig. (4-10) 87 4-12: A log-log plot of the order parameter vs. reduced temperature for both sets of ethylene data 88 4- 13: A log-log plot of Ap*/t^ vs. reduced temperature for both sets of ethylene data 90 5- 1: A log-log plot of the order parameter vs. reduced temperature for hydrogen. 93 5-2: A log-log plot of Ap*/t^ vs. reduced temperature for hydrogen 94 5-3: A log-log plot of Ap*/t^ vs. reduced temperature for set 250287.4 of the hydrogen data 97 5-4: Residuals for the fit shown in fig. (5-3) 98 5-5: The coexistence curve diameter of hydrogen as a function of reduced temper-ature 100 ix 5 - 6 : The coexistence curve diameter of hydrogen from data set 2 5 0 2 8 7 . 2 . . . . 1 0 2 5 - 7 : The effective compressibility of hydrogen vs. reduced temperature 1 0 6 6 - 1: The coexistence curve amplitude, Bo, for a number of fluids as a function of the quantum parameter Ay/fo 1 1 4 6-2: The first correction amplitude, B\, for a number of fluids as a function of A T / C O 1 1 5 6-3: The leading compressibility amplitude, TQ , for a number of fluids vs. XT/CO-1 1 7 6 - 4 : The coexistence curve diameter slope, A2, for several fluids vs. apc 1 2 0 6-5: The coexistence curve diameter slope, A2, vs. the coexistence curve amplitude, BQ, for a number of fluids 1 2 1 X A c k n o w l e d g m e n t s It is a pleasure to thank my supervisor, David Balzarini, for his advice and assistance at all stages of this work. I am also grateful to J. Carolan, M . Crooks, H. Gush and W. Hardy, whose loans of equipment, supplies and expertise were very much appreciated, and to J . Berlinsky for stimulating my interest in the field of critical phenomena. I have enjoyed and benefitted from numerous discussions with P. Palffy-Muhoray, S. Morris, E . Wishnow, M . Halpern and many others. I thank U. Narger for doing some shifts of data taking as well as for helpful discussions. I also thank the Natural Sciences and Engineering Research Council for several years' worth of scholarship support. Finally, I am grateful to my wife Sally for many discussions, for assistance in the preparation of this thesis, and for her un-failing support and patience throughout my involvement in this work. This thesis is dedicated to her. 1 CHAPTER I Introduction I - l An I n t r o d u c t i o n t o C r i t i c a l P h e n o m e n a i n F l u i d s Under certain conditions of temperature and pressure, a pure substance can exist in coexisting liquid, £, and vapour, v, phases. This situation is illustrated with a phase diagram in fig. ( l - l ) . The liquid-vapour coexistence curve traces out a path in the pressure-temperature plane from the triple point, where the two fluid phases coexist with a solid phase, to the critical point at the critical pressure, temperature and density Pc, Tc and pc. At a point on the coexistence curve the two phases are distinguished by their different densities pt and pv; beyond the critical point this distinction vanishes and only a single fluid phase exists. Near the critical point, the density difference behaves like a power law1: Ap* ^ = BotP (1-1) where t = \TC — T\/Tc is the reduced temperature and /? is a critical exponent. Experimental 2 - 4 and theoretical 5 - 9 determinations give /? ~ 0.327; the amplitude BQ is system dependent. Ap* is called the order parameter for the liquid-vapour 2 temperature Figure 1-1 The pressure-temperature phase diagram of a typical fluid. The heavy line is the liquid-vapour coexistence curve. The triple point is labeled T; the critical point, C. system, since it characterizes the ordering that occurs at the liquid-vapour phase transition. The behaviour of many thermodynamic properties in the critical region is qualitatively different from their behaviour elsewhere. For example, the dimen-sionless isothermal compressibility *cr = (Pcj'p\)[dp/dn)T, where fi is the chemical potential, diverges at the critical point as 4+ = r o i'1 (l"2a) as the critical point is approached along the critical isochore p = pc from above Tc, and as 3 as the critical point is approached along the liquid and vapour branches of the coexistence curve. The critical exponent 7 i s 2 ' 4 - 1 0 approximately 1.24, and the amplitudes are again system dependent. Similarly, the specific heat at constant volume Cv diverges at the critical point as C ± = F±t~a (1-3) with a ~ 0.11 and system dependent. Near the critical point, the critical isotherm can be described in terms of the reduced chemical potential Ap,* — (pc/Pc)(^fi[p,T) — p,{pc,T)^ and the reduced density by \Afi*\ = D0\(p-pc)/Pc\6- (1-4) S is predicted 5 - 9 to be 4.82 while Do is another system dependent amplitude. Finally, consider a system on the critical isochore with T > Tc. Far from the critical point, it is energetically unfavourable for regions of the fluid to be at densities larger or smaller than pc; local fluctuations in the density will therefore be very small. Closer to the critical point, density fluctuations become less unfavourable and their size accordingly increases. Finally, below Tc, these regions of high and low density become stable and the system phase separates. If we denote the typical dimension of the fluctuations by a correlation length f, then close to Tc £ diverges as £ = for" where fo is the bare correlation length a n d 5 - 9 v ~ 0.65. This divergence of the correlation length gives some insight into the unusual behaviour of a fluid near its critical point. Far from critical the fluid is well described by an equation of state involving some microscopic length scale a which characterizes the intermolecular (1-5) 4 interactions — for example the van der Waals equation. Close to the critical point, however, a is no longer the important length scale. Regions of size of order £ become correlated, and when f > a it is the behaviour of these fluctuations that determines the behaviour of the bulk fluid; thus £ becomes the dominant length scale. The details of the short-range ( «C f) intermolecular interactions become unimportant. It follows from this picture that all fluids should in some sense display the same critical behaviour, since for all fluids it is the divergence of £ that determines that behaviour. This is the principle of universality. It turns out that the critical exponents and certain combinations of the critical amplitudes are common to all members of a given universality class1 1. A system's universality class is determined by its dimensionality, rf, the number of components in its order parameter, and the general nature of the microscopic interactions in the system. Pure fluids belong to the universality class characterized by d = 3, a scalar order parameter, and short range interactions. Other systems in this class include the 3 -D Ising model, isotropic ferromagnets, and binary fluids. Calculated values of the critical exponents and universal amplitude ratios for this universality class are given in Tables I-l and 1-2. The exponent values given in Table I-l are "best" values based on the results of a number of theoretical studies using renormalization group (RG) and high temperature series (HTS) techniques. Also given in this table are the classical exponent values applicable to the van der Waals equation, to be discussed below. 1-2 S u m m a r y o f P a s t W o r k The study of critical phenomena began in the second half of the nineteenth century when experiments on C O 2 showed that above a certain temperature, liquid-vapour coexistence could not be observed12. This phenomenon was explained qual-itatively by van der Waals' (VDW) equation of state 1 3 , 1 4 , which predicted a region 5 Table 1-1 Critical Exponents Exponent Associated Quantity Classical Value Calculated Values R G a H T S b a Cv 0 0.11 0.11 & 1/2 0.325 0.327 1 1 1.241 1.238 6 AM*(P)|TC 3 4.82 4.79 V e 1/2 0.630 0.630 A Corrections — 0.50 0.52 to scaling a) References 5,6 b) References 8,9 Table 1-2 Universal Amplitude Ratios Ratio Calculated Values a R G HTS FQDQB^1  Foro/Bo 4.80 1.6 0.066 0.55 5.07 1.75 0.059 0.51 Bi/rt 0.5 — a) References 1,44 of liquid vapour coexistence and a critical point. The quantitative predictions of V D W theory, however, did not agree with measurements in the critical region — specifically, the critical exponents were wrong. 6 A clue as to the reasons for this discrepancy came from Onsager's exact solu-tion of the 2-D Ising model 1 5. While VDW-type theories involved thermodynamic functions that were everywhere analytic, Onsager's solution was nonanalytic at the critical point. Widom's formulation of a nonanalytic equation of state for fluids16 in 1965 and the subsequent development of the ideas of scaling 1 7' 1 8, discussed in Chapter II, led in the early 1970s to Wilson's 1 9' 2 0 development of the renormal-ization group approach to the theory of critical phenomena 2 1' 2 2. This approach provides a systematic way of dealing mathematically with the long range correla-tions that arise in a system as its correlation length diverges, and has been very successful both in helping our understanding of critical phenomena and in producing numerical results. At the same time as Wilson's theory was being developed, experimental tech-niques began to probe more closely the critical region in fluids. Prior to this time, most measurements of the coexistence curve gave /3 ~ 0.35, somewhat higher than the calculated values of approximately 0.32514. In 1972, Balzarini and O h m 2 3 re-ported measurements on SF6 indicating that /? was temperature dependent, tend-ing towards 0.33 for t ~ 10 - 4 . Other work around the same time led to similar results 2 4' 2 5. In 1976, Hocken and Moldover4 performed measurements on several fluids at t ~ 1 x 10 - 5 ; they found limiting values of 0 and 7 in the ranges 0.321-0.329 and 1.23-1.28 respectively, in good agreement with calculations for the 3-D Ising model. In 1972 Wegner26 introduced the idea of corrections-to-scaling. The critical power laws in eqs. ( l- l)-(l-5) are valid only asymptotically close to the critical point. At nonzero t, correction terms must be added. The observed temperature dependence of the effective order parameter exponent mentioned above is an indi-cation that these corrections are significant in pure fluids even for t ~ 10 - 4 . Wegner's theory 2 6 was applied to fluids by Ley-Koo and Green 2 7 in 1981. They obtained series expressions for various thermodynamic quantities of interest. For example, eq. ( l - l ) for the order parameter becomes 7 Ap* = Bot? ( l + Bit* + B2t2A + • • •) ( 1 - 6 ) while the isothermal compressibility of eq. ( 1 - 2 ) becomes 4 ± = r±r-» ( I + r ± t A + r f t 2 A + • • • ) ( 1 - 7 ) and the critical isotherm, eq. ( 1 - 3 ) is \An*\ = D0\(p - Pc)/Pc\S ( l + A K p - Pc)/Pc\A/P + •••). (1-8) The diameter of the coexistence curve, which is the average of the liquid and vapour densities, is given by Pi = = 1 + + A2t + Aatl-*+* + • • - . (1-9) The correction amplitudes B\, r\, etc. are system dependent, and the universal exponent A has a calculated 5' 7' 2 8' 2 9 and measured2 value of approximately 0.5 ± . 0 2 . Recently, many fluids have been studied in order to measure both the asymp-totic critical exponents and corrections-to-scaling effects 2' 3' 1 0' 3 0 , 3 1. The former goal requires precise measurements close to Tc; the latter, data covering a wide range of reduced temperature to permit meaningful fits to the corrections-to-scaling ex-pressions given above. The best experimental determinations of the exponents /3 and 7 are in good agreement with the theoretical results summarized in Table I—1. Results for other exponents are less precise but generally agree with the calculated values. Many previous determinations of amplitude ratios3 0 have relied on combining the results of different experiments. Since the values of the individual amplitudes tend to be sensitive to the choice of Tc and to the exponent values used in analysing 8 the data, the reliability of ratios determined in this way is limited. The measure-ments of Pestak and Chan 2 do not have this problem; their results for three ratios are in agreement with the theoretical predictions. Two techniques have proven very useful for measuring thermodynamic quan-tities near the liquid-vapour critical point. Both are based on the fact that, close to the critical point, the compressibility of the fluid becomes large enough that the weight of the fluid on itself is enough to cause a measurable density gradient in the sample. A capacitive method used by Pittman et al.3 for studies of 3 He and by Pestak and Chan 2 on N2, Ne and HD uses a stack of small capacitors to measure the dielectric constant as a function of height in the sample. The dielectric constant £ is related to the density by the Clausius-Mossotti equation, 47TNAO: 1 e — 1 . — = 1-10 3 pe + 2 K ' where a is the molecular polarizability and NA is Avogadro's number. This method has the advantage that capacitance can be measured very precisely using bridge techniques, but the disadvantage that the finite thickness of the capacitors limits reliable measurements to a region far enough from Tc that the density within a single capacitor is uniform 3 2. The second method is the one used in this work. It is an optical technique de-veloped by Balzarini 2 4 ' 3 3 ' 3 4 and since used to study a number of fluids4'23,30'31'35,36. In one version of the method, a laser beam is passed through a fluid sample and focussed by a lens; the Fraunhofer diffraction pattern due to the sample is observed in the focal plane of the lens. In a second version, the beam passing through the sample is mixed with a reference beam and the resulting interferogram viewed in the image plane of a lens. In each case changes in the observed fringe pattern with temperature give information related to the liquid-vapour refractive index difference 9 below Tc. In the case of the image plane experiment, the spacing between interfer-ence fringes can be related to the compressibility. The Lorenz-Lorentz coefficient £ = - -5 1-11) p n2 + 2 K ' is used to convert refractive indices n to densities. This technique can be used at smaller reduced temperatures than the capacitive technique, since a smaller height in the fluid is sampled by a ray of the laser beam 3 2 , but it loses precision close to Tc because of the "digitization error" inherent in counting small numbers of interference fringes. 1-3 Summary of This Work The experiments described in this thesis use the above optical techniques to study the critical behaviour of two fluids. Ethylene, C 2 H 4 , has its liquid-vapour critical point at 282.4K, slightly below room temperature. Hydrogen, H2, has its critical point at about 33 K. Ethylene was also studied using another optical technique in which the refraction of a laser beam by a prism-shaped sample was measured to determine £{p) and the densities of the coexisting phases 3 7 - 3 9 . These experiments were undertaken for two major reasons. First, it was hoped that measurements of several properties on the same sample would provide better values for some of the universal amplitude ratios given in Table 1—2. Second, we wished to investigate the effects of quantum mechanics on the critical behaviour of fluids. Because of the low mass and critical temperature of H2, quantum corrections to its critical behaviour may be significant. A theory, discussed in Chapter II, predicts that the critical exponents should not be affected, but that the critical amplitudes should change4 0. Our results in general show very good agreement with both theory and previ-ous experimental work on fluids near the critical point. From measurements of the 10 coexistence curves of both ethylene and hydrogen, we find values of the exponents /3 and A which agree within our experimental error with the theoretical predictions. The coexistence curve amplitude Bo shows at least qualitative agreement with the predicted effects of quantum mechanics, but our results for the correction amplitude B\ are not conclusive in this regard. Our measurements of the coexistence curve diameters of the two fluids show the presence of the predicted tl~a critical anomaly, and the hydrogen diameter data show strong evidence for quantum mechanical effects in this fluid. Finally, our results for the compressibility of hydrogen are less trustworthy because of experimental problems to be discussed below. Nevertheless we find values of the exponent 7 and the amplitude ratio Tq / i n agreement with theoretical predictions. The remainder of this thesis is organized as follows: Chapter II contains a more detailed discussion of the theory of critical phenomena. The experiments and important related considerations are described in Chapter I I I and the results presented in Chapters I V and V . Chapter V I is a discussion of our findings, which are summarized along with some concluding remarks in Chapter V I I . 11 C H A P T E R I I T h e o r y In this chapter the ideas behind the modern theory of critical phenomena are discussed. The development of the theory from the van der Waals equation to the renormalization group is outlined, with emphasis on the important concepts of universality, homogeneity of thermodynamic functions, and scaling. A complete exposition of the methods of calculating numerical results is beyond the scope of this thesis, but results relevant to this work will be presented. The chapter ends with a discussion of the possible effects of quantum mechanics on critical phenomena. I I - l The van der Waals Equation in the Critical Region Van der Waals' equation1 3 is the most successful of what are now called classical equations of state. While the V D W description of the critical point is incorrect, it is discussed here as a means of introducing several important ideas. The equation relates the pressure P , temperature T, and molar volume v of a fluid where R is the universal gas constant. Here a is a measure of the attractive inter-action between molecules and b represents the finite molecular size by a repulsive hard-core interaction. With a and b chosen to fit experimental data, this equation by (2-1) 12 of state describes the behaviour of real gases away from the critical point reasonably well. Eq. (2-1) predicts a liquid-vapour phase transition. The equation is cubic in v, and so in general has three solutions for a given T and P. At high temperatures, only one of these solutions is real, but below some critical temperature Tc all three are. In this region the compressibility, which is proportional to dv/dP\T, is negative near the middle root, and the system is thermodynamically unstable; it separates into a high density (low v) liquid phase and a low density (high v) vapour phase. Maxwell's construction removes the region of instability and determines the coexistence curve by imposing the condition / Pdv = P{v2 - vt) (2-2) J Wj on the coexisting phases 1 and 2. This forces equality of pressure, temperature and chemical potential on the two phases. The phase diagram of a V D W fluid is shown in Fig. (2-1). The parameters a and b depend on the microscopic properties of a given system. They can be eliminated from eq. (2-1) in favour of quantities related to the critical point, at which the three roots of eq. (2-1) coalesce to one. At this point, dv d2P = 0 a n d ^ (2-3) where the subscript c refers to the critical point. Using eqs. (2-3) and (2-1), the temperature, pressure and molar volume at the critical point are found to be 8a CL VC — 36, RTC = - , Pc = 777 c ' 276' 2762 (2-4) 13 molar volume Figure 2-1 The phase diagram of a van der Waals fluid. The fine solid lines are isotherms. The dashed portion indicates a region of negative compressibility, which is eliminated by Maxwell's construction (dotted line). The coexistence curve is shown as a heavy line and the critical point is labelled C. Using these relations to eliminate a and b from eq. (2-1) gives which depends only on the ratios of the thermodynamic variables to their critical values. This is known as the law of corresponding states. While of limited validity experimentally, it embodies the important idea of universality — different systems exhibit similar behaviour independent of their microscopic details. In particular, aspects of the critical behaviour of VDW fluids, including the critical exponents, should be universal. (2-5) 14 The critical exponents of the V D W equation can be calculated from eq. ( 2 - 5 ) . For example, the dimensionless isothermal compressibility at the critical density is * _ Pc dv K t ~ ~VedP T,vc where t = \TC — T\/Tc, so the exponent 7 = 1. On the critical isotherm, eq. (2-5) can be expanded in powers of p = (P — Pc)/Pc and Ap — (p — pc)/pc with p = u _ 1 to give p ~ A p 3 near the critical point, so S = 3. These and other exponents for the V D W equation are listed in Table I—1. These values are common to generalizations of the V D W equation and to mean field theories of the liquid-vapour critical point; they are referred to as the classical critical exponents. To make contact with results to be discussed later, it is useful to write eq. (2-5) in a different form. We use the chemical potential fi rather than the pressure; from basic thermodynamics41, changes in p,{T,P) are related to changes in the intensive variables by dn = -^dT+^dP ( 2 - 7 ) where S is the entropy, V the total volume, and N the number of molecules. At constant T then, pdfi = dP, ( 2 - 8 ) where p = N/V is the number density. Near the critical point we can write A M = -p\jt[p.T) - fi(Pc,T)J = ( 2 - 9 ) and using eq. ( 2 - 5 ) we calculate the right hand side of ( 2 - 9 ) to find Afji* = pc (etAp + ^ A p 3 ^ = w ( l + g ) 15 (2-10) where a\ = 3p c/2 and ai — 4. The critical isotherm, £ = 0, is therefore given by A more general equation of state permitting non-classical critical exponents, which will be discussed below, is often written in a form similar to that of eq. (2-10). It was clear as early as the end of the nineteenth century that the V D W equa-tion was quantitatively incorrect near the critical point 1 4. The strongest evidence for this came from measurements of the coexistence curves of various fluids, which seemed to be well described by a critical exponent (3 — | rather than the classical exponent ^. The cause of this discrepancy was not understood for several decades. In fact it is related to the importance of fluctuations and the change of length scale near the critical point, as discussed in Chapter I, and to the nonanalytic nature of the equation of state. II-2 H o m o g e n e i t y a n d S c a l i n g In 1965 Widom proposed a form for the equation of state near the critical point which was nonanalytic at the critical point and consistent with known non-classical critical behaviour1 6. His suggestion has since been given a firm theoretical basis as a result of the ideas of scaling and the development of the renormalization group, to be discussed below. An* = aiAp3, (2-11) while on the coexistence curve, Afi* = 0 and so Ap = ± a 2 / 2 | t | 1 / 2 (2-12) 16 The distance from the critical point in temperature is just T—Tc. The distance in density is p — pc, which can be expressed in temperature units by writing Tc-T(p)~\p-pc\l/P (2-13) where T(p) is the temperature on the coexistence curve corresponding to the density p. Widom suggested16'18 that comparable values of T— Tc and Tc — T(p) should have comparable effects on the thermodynamic properties of the system. Mathematically, he proposed that certain thermodynamic functions become homogeneous functions of T — Tc and Tc — T(p) near the critical point. (A homogeneous function <f>{x, y) of degree n has the property that <f>(x,y) = xn<p ( l , | ) = yn<p = yntp (^j , (2-14) i.e., <j> can be written as a power of one of its arguments times a function of the ratio of its arguments.) Widom's homogeneous equation of state can be written as A" - = A ' i A ' i { " l k ( j ^ j w ) - < 2 - 1 5 > The V D W equation as written in eq. (2-10) has the form (2-15) with classical exponent values and hci(u) = ai(l + a2« ) , (2-16) where u = t/ | A p | ! ^ . Widom's form, however, allows non-classical exponents and does not restrict h(u) to its classical form. The above suggests a rigorous expression of the universality hypothesis14: the critical exponents for all systems in a given universality class should be the same, and h(u) should be a universal function, except for two non-universal scale factors. 17 These set the scales of u and h; in the classical case, they are a2 and a\ respectively. The critical behaviour of various quantities, as well as scaling laws relating the critical exponents, follow from eq. (2-15). For example, on the critical isotherm, t = 0, and so Afi* = AplAp^^hiO) ~ A / , (2-17) as expected. As another example, consider the isothermal compressibility at p = pc. This quantity is expected to behave like t-1. Using (2-15), * - l d/i K T ~dp ~ Ap6-1 h{u) + • • • . (2-18) T Now, if K*T 1 is to be finite for all nonzero t at Ap — 0, the powers of Ap in (2-18) must cancel. Therefore we must have, for u —• oo, lim h{u) ~ u^-V, (2-19) so that K*T-' ~ A A V * * " 1 ) ~ Aps Ap6'1 ~ i^ 5 " 1 ) (2-20) from which follows the scaling law 1=P{6-1). (2-21) This and other scaling laws are tabulated in Table I I — 1 . 18 Table II-l Scaling Laws 2 - a = 2/3 + 7 = du Homogeneity of thermodynamic functions near the critical point follows nat-urally from the scaling ideas of Kadanoff 1 7' 1 8. Here it is convenient to consider an Ising model and to use the language of magnetism rather than of fluids. Instead of Afi*, we use the applied field h; instead of Ap, the magnetization M. Consider a lattice in d dimensions, with lattice spacing a, and with an Ising spin on each lattice site. The free energy per spin can be written as the sum of a regular and a singular part, f(t,h) = fr{t,h) + fs(t,h), (2-22) where fs contains all of the nonanalytic critical behaviour. We now divide the lattice up into blocks of side La, each containing Ld spins. We restrict L to values such that a < La <C f, where f is the correlation length; the spins on a block should thus be well correlated. Each block will contribute Ldfs(t,h) to the singular part of the free energy. We now rescale the system by replacing the Ld spins in each block by an average "block spin," and by measuring lengths in units of the block size La rather than a. This scale transformation is illustrated schematically in fig. (2-2). £ as measured in the units of the rescaled system is smaller than in the original system by a factor of L. Since £ grows like t~v as the critical point is approached (eq. ( 1 -5)), the rescaled system appears to be further from its critical point than was the 19 original system, and, since t = h = 0 at the critical point, the rescaled system must also appear to be at larger values of t and h. We assume that t and h scale like t — LH, Lxh, with x and y positive, when the system is scaled by a factor of Z, as above. The singu-lar part of the free energy per block spin in the rescaled system is then fe{Lvt, Lzh). —a— —La— 1 1 1 1 1 (a) (b) (c) Figure 2-2 A schematic representation of the Kadanoff scale transformation, a) The original lattice showing a block of size La; here L — 1. b) The lattice of block spins, c) The lattice of (b) rescaled by a factor L. These two contributions to the free energy are the same quantity expressed in terms of the system before and after scaling. Thus Ldfa{t,h) = fe(Ln,Lxh). (2-23) 20 Since the length L is arbitrary, eq. (2-23) must hold for all L. For this to be the case, fs must be a homogeneous function of the form f,{t1h) = td>y<P ( r * / ^ ) , or, returning to the language of fluids, /,(*, A/z*) = ti,v4> (r a / » A / i * ) • (2-24) This formulation in terms of the free energy is equivalent to that discussed above based on the equation of state. One can again derive the asymptotic critical power laws and various scaling laws from eq. (2-24), and express x and y in terms of the conventional critical exponents. Consider first the specific heat on the coexistence curve, where An* = 0: so d 2 - a. (2-25) y Also, on the critical isotherm, we know that Ap~^~(An*)1/S (2-26) so fs{0, Ap*) must behave like (A//*)^ + 1. For a finite t = 0 limit to exist we need lim^(u) "ii*'* (2-27) 21 so that the powers of t cancel: d/x /,(0,A/**) ( r x / » A M * ) * ~ ( A / / ) d / l (2-28) From this we get x 6 Finally, on the coexistence curve, d = \ + 1. (2-29) Ap ~ ~ td'n-*ly4>'{Q) ~ ^ (2-30) Combining eqs. (2-25),(2-29), and (2-30), we get d-x 6(2-a) 2 - a = 2 - a -r1 = r = /?, y l + <5 1+5 giving the scaling law 2 - a = p{6 + 1). (2-31) This result and the second scaling law given in Table II—1 can be derived as inequalities from thermodynamic or statistical mechanical arguments. The equali-ties in these two cases, as well as the other scaling laws, result from the scaling or homogeneity hypotheses. The above results allow us to rewrite fs using the conventional critical expo-nents: fs(t,An*)=t2-a<l>(J^y (2-32) Relations involving the correlation length exponent v can be obtained by applying similar arguments to the correlation function G(t,r)is. 22 I I - 3 R e n o r m a l i z a t i o n G r o u p The ideas discussed in the preceding pages provide some understanding of the nature of the liquid-vapour critical point. They allow prediction of the behaviour of thermodynamic quantities in the critical region and give scaling laws that relate the critical exponents. They do not, however, provide a means of calculating the exponents or other quantities. In 1971 Wilson 1 9 ' 2 0 developed the technique of the renormalization group (RG), which does provide that means, as well as lending a sounder basis to the ideas of homogeneity and universality. RG embodies scaling ideas in an essential way, but, unlike Kadanoff scaling, is formulated in a way that makes explicit calculations possible, at least in principle 2 1' 2 2. The R G is a set of transformations applied to the Hamiltonian of a system which are conceptually very similar to Kadanoff's scaling transformation discussed above. The properties of these transformations under certain special conditions are related to the critical behaviour of the system. The discussion here will be brief and very general, treating the concepts behind the R G and relating them to those introduced above. For more details, the reader is referred to textbooks on the subject by M a 4 2 and by Pfeuty and Toulouse43. While the discussion below applies equally to continuous systems, for simplic-ity we consider again a spin system on a lattice with lattice constant a. We can write down a very general Hamiltonian, where K is a vector containing the coupling constants Ki and {trx} is the set of values of the spin variables. We label the Hamiltonian by the vector K; this vector represents a point in a many-dimensional parameter space. The RG transformation (2-33) 23 Rs transforms the Hamiltonian represented by K to one represented by K': K — RSH.. (2-34) Rs involves a number of steps. First, the system is coarse-grained by averaging out all spin fluctuations of wavelength shorter than sa. This is analogous to the formation of block spins carried out above, and effectively increases the length scale by a factor s. Next, the spin variables and the length scale are rescaled such that This shrinks the system by a factor s, so the lattice constant is again a. Finally, the transformed Hamiltonian is rewritten in the form (2-33) and the new set of parameters K' determined. The operator RsRsi = Rssi is well defined if cs — sb with b independent of s, but the inverse transformation R~* is not. In general, the mathematics of the RG transformation will be complicated and a large number of new parameters K{ will be generated with each application of Rs. It is, however, possible, but by no means guaranteed, that there exist fixed points in the parameter space such that that is, points that are invariant under the transformation. The critical surface of K * is defined as the set of points K for which X —> sx (2-35) r> IT * XT* rCsJ\. — JY , (2-36) l im RSK = K*; (2-37) points on the critical surface of a fixed point are eventually driven to that fixed point by repeated application of the RG transformation. 24 To the extent that the Hamiltonian under consideration represents that of a real system, the mathematics of the RG is connected to the physics of critical phenomena by the hypothesis that a point in the parameter space representing a system at its critical point is on the critical surface of a fixed point. The behaviour of Rs near the fixed point determines the critical behaviour of the system. The concept of universality is a simple consequence of this hypothesis: critical points represented by different points on the same critical surface will display the same critical phenomena, since for each of these points the behaviour of Rs is governed by the same fixed point. Let us assume that a fixed point K* does exist and study the behaviour of Rs near that point. We can formally write If terms of order higher than linear in are ignored, then Rg can be approximated by a linear operator R^, and K = K* + SK (2-38) so K ' = RSK = Rs ( K * + 6K) = K* + Rs SK + 0(<5K) 2 = K * + SK'. (2-39) SK' = R^SK. (2-40) This linear operator can be represented by a matrix (2-41) with eigenvalues Xj(s) and eigenvectors ey, for which (2-42) 25 Since RsRsiej = Rssiej = \j(s)Xj(s')ej = Xj(ss')ej, we must have Xj(s) = syK (2-43) Using the eigenvectors as basis vectors near the fixed point, we can write 6K = ^ 'i ej> 6K' = 'Jey (2-44) 3 3 where t'j = \j[s)tj - sHtj. (2-45) Clearly if y ; > 0, repeated application of Rs will cause tj to grow, while if t/y < 0, tj will decrease. We refer to the tj as scaling fields. On the critical surface, repeated application of Rs drives the system towards the fixed point at 6~K — 0, therefore the scaling fields associated with positive y.j must be zero on the critical surface. These scaling fields are called relevant, since only they are relevant to the asymptotic critical behaviour. The eigenvectors associated with negative yys span the critical surface near the fixed point; the corresponding scaling fields are called irrelevant. For Hamiltonians in the universality class of the 3-D Ising model fixed points do exist; the fixed point associated with the liquid-vapour critical point has two relevant scaling fields. Since for a fluid the critical point is defined by t = 0, A/x* = 0, it seems reasonable from the above discussion to associate t and Afi* with these two fields. Now consider the free energy as a function of t and An* before and after application of Ra. Using arguments similar to those used in the discussion of Kadanoff scaling above, and noting that t and An* will scale as in eq. (2-45), we get sdfs(t,An*) = fa{s*H,s"An*), (2-46) 26 which is of the same form as eq. (2-23); fs is again a homogeneous function of its variables. The critical exponents are related to the exponents t/i and t/2 and to the dimensionality d. The R G can be used to calculate numerical values of these expo-nents, although in general this is a difficult task. It turns out for this fixed point, however, that the calculations are easy for d > 4; the exponents then assume their classical values. For d = 4 — e, a perturbation expansion in c = 4 — d is possible. This 6-expansion technique has led to good results for the critical exponents when extrapolated to c = 1, although the power series obtained appear to be poorly convergent. Other methods of calculating exponents apply field theoretical techniques to the R G equations5'6 or use high temperature series expansions 7 - 9. The results of these different calculations have begun to agree reasonably well with each other9, and the exponent values obtained can be regarded as quite reliable. Table I—1 gives some of these calculated exponents. II-4 C o r r e c t i o n s t o S c a l i n g It was pointed out in Chapter I that the simple critical power laws resulting from, for example, eq. (2-32) do not adequately describe experimentally observed behaviour except extremely close to the critical point. Corrections to these power laws are required; these arise from inclusion of the irrelevant scaling fields as vari-ables in the free energy2 6. Physically, these fields correspond to operators involving high order products of the spin variables in the Hamiltonian of eq. (2-33). The theory of corrections-to-scaling was developed by Wegner2 6 and applied to fluids by Ley-Koo and Green 2 7 in 1981. Consider the thermodynamic potential of a fluid to be a function not only of the two relevant scaling fields related to t and A/z*, which we write as gE and gh, but also of the two most important irrelevant scaling fields, gA and gs. As discussed 27 above, gE and will be zero on the critical surface, but gA and gs will not. Ley-Koo and Green 2 7 write this potential as P = Pr +Ps{9h,9E,9A,9s) (2-47) and require that the singular part, p s , be a homogeneous function of all of its vari-ables. They then use homogeneity properties and symmetry arguments to determine the form of ps in each region of the phase diagram. Substituting physical variables for the scaling fields in these expressions, they get equations for various quantities along thermodynamic paths of experimental interest. Equations (l-6)-(l-9) are examples. These corrections-to-scaling series are listed in Table I I — 2 . Table I I - 2 Corrections-to-Scaling Series On the coexistence curve, T < Tc, Ap* = Bot13 (1 + BitA + B2t2A + B3t + • • •) P i = 1 + Axtl-a + A2t + > M 1 - Q + A + A4t2-a-ps+AA + 4- = r i t - i ( I + r~tA + r~t2A + r~t^-1 + •••) On the critical isochore, T > Tc, 4+ = r+t-i ( I + r+tA + r+t2A + r+t2^-1) + • • •) On the critical isotherm, T = Tc, 6 A/** = Do P-Pc Pc ( ±1 + Di P-Pc Pc T D 2 P-Pc Pc A/P + The universal exponent A in these expressions is associated with the scaling field gs. The exponent AA associated with gA enters to higher order in the tabulated series and in expressions for other quantities. Values of A have been calculated using the methods mentioned above 2 8' 2 9 and are given in Table I—1. 28 I I - 5 U n i v e r s a l A m p l i t u d e R a t i o s The amplitudes in the expressions given in Table I I - 2 are system depen-dent. However, just as scaling laws exist which relate the critical exponents, certain combinations of these amplitudes are universal. There are only two independent exponents; similarly, there are only two independent leading amplitudes44. In ad-dition, the ratio of any two of the first correction amplitudes is universal 4 5' 4 6. In the discussion following eq. (2-15) it was noted that the equation of state involves a function h(u) which is universal except for two factors which determine the scales of h and u. We can write 4 4 h(u) = h0~h(u0u) (2-48) to show these scale factors, ho and uo, explicitly. It is possible through manipulation of eq. (2-15) to write all of the leading amplitudes BQ, TQ, etc. in terms of ho, UQ, and universal quantities. Combinations of these amplitudes that are independent of the two non-universal scale factors will thus be universal. The four universal combinations of the six amplitudes FQ, BO, TQ and Do so determined are given in Table 1-2 along with their values as calculated by e-expansion and series techniques. Aharony and Ahlers 4 5 showed that, to zeroth order in e, the ratio of the first correction amplitudes X\/Y\ for any two thermodynamic quantities X and Y was given by where Xx IS the asymptotic critical exponent corresponding to quantity X and Aox is the associated classical exponent. Chang and Houghton 4 6 used renormalized perturbation theory to calculate the first correction amplitudes to second order in e and confirmed that the ratio of any two of them was universal. Results for the ratios of interest in this work are given in Table 1—2. 29 I I - 6 Q u a n t u m M e c h a n i c a l E f f e c t s o n C r i t i c a l B e h a v i o u r Early measurements of the coexistence curves and compressibilities of the low temperature fluids 3 He and 4 He appeared to give exponent values that differed from those for room temperature fluids14. It is now clear that this was due to neglect of corrections-to-scaling effects. At the time, however, the question was raised as to whether quantum mechanics should have any effect on the critical behaviour of these fluids47. It is easy to see why quantum mechanics could modify the critical behaviour to some degree48. As discussed in Chapter I, for a classical fluid far from its critical point, the length scale that determines the physics of the system is the intermolec-ular interaction distance, a. Closer to the critical point the correlation length £ = £ot~u increases and in the critical region, where f ~> a, it is f that dominates the physics; the details of the short-range interactions are unimportant. For a quantum system, there is another fundamental length scale: the thermal De Broglie wavelength = ^~j=j=f, where h is Planck's constant divided by 2n, kg is Boltzmann's constant and m is the molecular mass. For a system in a state for which Ay > a and Ay > £, quantum modifications to the thermodynamic properties might be expected. Of course, since f diverges, it will eventually become much greater than Ay no matter how quantum mechanical the system is, and so the universality class of a quantum fluid must be the same as that of a classical fluid. The critical exponents and universal amplitude ratios should therefore not change. It would be reasonable, however, to expect the individual amplitudes to change systematically as the fluid becomes more quantum mechanical. A complete theory of the effect of quantum corrections to the equation of state near the critical point does not exist. A semi-quantitative theory has been developed by Young 4 0 ; while not very rigorous, with some numerical adjustment it does predict reasonably accurately the location of the critical points of the helium isotopes, their second virial coefficients, and other properties. Young 4 0 starts with 30 a Lennard-Jones 6—12 potential that describes well the behaviour of a classical fluid away from the critical point: where e is the potential well depth and a determines the length scale. He modifies this potential by averaging it over a gaussian "wavepacket" given by {a/n)2 e ' r - r ' , where yfct' = (SmlcsT/h2)1/2 ~ A^ 1 to obtain an effective potential V'r) = (?Pj j d3r'V{r')e-a'^-r'\2. (2-51) He evaluates V(r) near its minimum and defines parameters e and o, which are analogous to e and a above but describe the quantum mechanically modified po-tential V(r). These parameters are functions of a! and so of T, as well as of e and a, and contain all of the quantum corrections to V(r). Young's procedure then amounts to the following. He writes down an equation satisfied by the classical system. He rescales the energies and distances of the critical point properties by factors of eje and a fa respectively so that Tc —> Tce/e and Vc —> Vc (a/o)3, and then replaces T c , Vc and Pc by their classical values in terms of £ and a. For example, for the leading term in the expression for the density difference on the coexistence curve, (2-52) 31 since classically Tc = 1.26?. By using expansions for i and a about their values at the critical point, he finds for the coexistence curve and compressibility respectively where y is a positive parameter depending on e, o and Tc which vanishes in the classical limit. Thus the theory predicts that the leading coexistence curve ampli-tude should decrease and the leading compressibility amplitude increase as quantum corrections become more important. It also predicts no change in the asymptotic exponents, as expected. Similarly, Young finds that quantum corrections should reduce the first correction amplitudes: Ap = B0tp = B0{1 - yftP (2-53) K*r — T(jt 7 = r 0 ( i - y)-in (2-54) B1 = B1(l-y)A f i = r ! ( l - y ) A . (2-55) (2-56) These predictions will be compared with the results of this and other work in Chap-ter V I . 32 C H A P T E R I I I T h e E x p e r i m e n t s In this chapter, the optical experiments undertaken to study the critical be-haviour of ethylene and of hydrogen are described. These experiments measure var-ious thermodynamic quantities via measurements of the refractive index n. Three experimental techniques were used, as described briefly in Chapter I. These will be referred to as the image plane, the focal plane and the prism experiments. We begin with a discussion of the general principles and the optics of the first two of these. Detailed descriptions of the experiments performed on ethylene and hydrogen using these techniques are given in sections III— 2 and I I I - 3 . The prism experiment is described in section I I I - 4 . Section I I I - 5 is a discussion of the major experimen-tal limitations with which we were faced, and section I I I - 6 contains an outline of the data reduction procedures and a discussion of the accuracy of the raw data obtained. I I I - l T h e F o c a l P l a n e a n d I m a g e P l a n e E x p e r i m e n t s The ethylene and hydrogen experiments differed in detail because of the very different critical temperatures of the two fluids. The basic principles of the two experiments were the same, however. A sample container is filled with the experimental fluid to the critical density pc. The sample temperature is precisely 33 controlled to allow measurements very close to Tc. A laser beam is used to probe the sample using one of the two optical arrangements described below, and data are recorded on film as the temperature is changed. The data take the form of inter-ference fringes; the recorded information is converted to thermodynamic quantities as outlined below. The experiments described here exploit the fact that in the critical region the compressibility of the fluid becomes large. This leads to a significant density gradient — and hence a refractive index gradient — in the sample container due to the weight of the sample on itself. This is illustrated in fig. (3-1) for a cell filled to an average density pc. Above the critical point, the density changes continuously over the height of the cell (fig. (3-la)). Closer to Tc, the increased compressibility near p — Pc leads to a larger density gradient near the middle of the cell (fig. (3-lb)), and below Tc a discontinuity in the density appears as the system phase separates. This discontinuity grows as the temperature is further lowered (fig. (3-lc)). Two slightly different optical arrangements — the image plane and focal plane experiments — provide complementary measurements of this behaviour. The focal plane technique was first applied to the study of critical phenomena by Balzarini and Wilcox 2 4 ' 3 3 and the image plane technique by Balzarini 3 4; both have since been used to study a number of pure fluids4,23'30'31'35,36 and binary fluid systems 4 9 , 5 0. In the image plane experiment, the sample is placed in one arm of a Mach-Zender interferometer51 as shown in fig. (3-2). A spatially filtered, expanded and collimated HeNe laser beam is split by a beam cube. One beam passes through the sample while the second is directed around it. The two beams are then recombined and the exit plane of the cell imaged with a magnification m on the film plane of a modified 35 mm camera using a large aperture (306 mm focal length, //2.5) lens. The fringe pattern observed in the image plane is an interferogram of the sample; a difference of N fringes over a distance x in the interferogram corresponds to a density Figure 3-1 Gravity-induced density gradients in a fluid near the critical point, a) T > Tc, b)T> Tc, c)T<Tc. 35 change in the refractive index equal to 6n = NX/L over a height z = x/m in the sample. Here L is the sample thickness and A = 6328 A is the laser wavelength. In the one-phase region, where n(z) is continuous, the recorded interferograms can be interpreted as p,(p) isotherms. The chemical potential fi is related to pressure by eq. (2-9), and the change in pressure with height in the cell is P{p,T) - P(pc,T) = Pcg{z - z0), (3-2) where ZQ is the height at which p — pc and g is the acceleration due to gravity. This gives * */ , Pc9{z - zp) , s M ^Z' = —Tc—' ^ ' SN(z — ZQ) obtained as above can be related to Ap(z — ZQ) using eq. (3-6) below. The slope of these isotherms gives the isothermal compressibility /c*r+. Below Tc, as the temperature is lowered, fringes "disappear" into the image of the meniscus due to the growing difference in index between liquid and vapour. The number of fringes missing from each phase, NijV is related to the corresponding refractive index by Nt = j(nt - nc) Nv = j{nc - nv) where nc is the refractive index at the critical point. Thus ni-n^iNi + Ny)^ (3-4) and nt + nv- 2nc = (Nt - Nv) (3-5) 36 fa V S 0) (4 F i g u r e 3-2 T h e optical setup for the fringe experiments. T h e focal plane is at F; the image plane at I. these quantities are related to Ap* and p& respectively by 37 2pc (dn/dp + pcp*d d2n/dp2) and * Pt + Pv ~ 2pc Pd = TT = Pd ~ 1 2pc (ni + nv — 2nc) — \ d2n/dp2 (dp/dn)2 (ni — n c ) 2 + (n„ — n c ) 2 2pcdn/dp (3-7) Eqs. (3-6) and (3-7) and expressions for dn/dp and d2n/dp2 are derived in Appendix be determined from the fringe spacing below Tc in the same way that /c*r+ is above Tc. The image plane interference pattern can thus in principle yield data on Ap(T), Pd{T), p.{pt,v)\r and below Tc, and p>(p)\r and /c r + above Tc. The optical layout for the focal plane experiment is similar to that for the image plane experiment (fig. (3-2)) except that the reference beam is blocked. The filtered and collimated laser beam passes through the sample and is focussed by the lens, and the Fraunhofer diffraction pattern due to the cell contents is recorded in the focal plane of the lens. The electric field of the laser in the focal plane is the Fourier transform of that at the cell exit: Ef(kz) = j ' h/2 Es{z)eik*z dz, (3-8) h/2 where Ej and Es are the fields at the focal plane and cell exit respectively, and kz is the spatial frequency. If absorption and scattering by the sample are negligible, only the phase <f> of the laser beam changes on passing through the sample, and so Es{z) = E b e ' W ' H ^ ) = Eae'tr^*") (3-9) 38 where Eoelwt is the spatially uniform field of the laser entering the cell and <p{z) = 27iLn(z)/X is the phase shift caused by the sample. The intensity recorded in the focal plane is then given by which has a series of maxima at spatial frequencies for which the exponent in the integrand is zero. This pattern can also be understood by considering individual rays traversing the sample as illustrated in fig. (3-3). A ray entering the cell horizontally will be deflected by the refractive index gradient and emerge from the cell at an angle 9 from the horizontal given by 8 = - p (3-H) na dz where na is the refractive index of air and 0 is assumed to be small. This ray is focussed to a point in the focal plane a distance y from the optical axis, where / being the focal length of the lens. A ray passing through a given gradient in the top half of the cell will therefore be focussed to the same point as a ray passing through the same gradient in the bottom half of the cell. Their phases at this point will, however, be different because each ray encounters a different value of n. A fringe pattern will result, with each fringe corresponding to an additional 2n phase difference between interfering rays. In the one-phase region, fringes corresponding to phase differences of A<f> = <t>top — <f>bottom — 0,27r,47T,. . . exist in the diffraction pattern. As the density discon-tinuity appears at Tc, fringes disappear from the pattern as no pairs of rays leave (3-10) y = / tan 6 w f0, (3-12) Figure 3-3 Formation of the Fraunhofer diffraction pattern. Rays passing through the same refractive index gradient are refracted through the same angle 6 and focussed to the same point in the focal plane F. A fringe pattern results because of their different phases at that point. 40 the cell with A<f> = 0,2ir,... and so on. The number of missing fringes N gives the minimum value of A<j> in the pattern, corresponding to rays which pass through the sample just above and just below the liquid-vapour interface. This therefore gives the index difference between the phases: NX ni - nv = —j-, which is related to the order parameter by eq. (3-6). III-2 T h e E t h y l e n e E x p e r i m e n t A diagram of the apparatus used in the ethylene experiment is given in fig. (3-4). Much of this apparatus has been used in previous studies of the critical properties of room temperature fluids done in this laboratory 3 0' 3 1' 3 6. Modifications to the existing equipment for this work included computer control of the temper-ature changes and an improved temperature controller on the outer stage of the thermostatic housing. These additions and the rest of the experiment are discussed in this section. The ethylene sample cell body was made of Kovar and had sapphire windows 0.64 cm thick by 1.91cm in diameter brazed into it. The thickness of the fluid sample was 0.503 cm. The cell was evacuated, then filled to a density p > pc with research grade (99.94 mole %) C 2 H 4 5 2 while attached to a gas handling system. The sample density was adjusted to pc by observing the meniscus as the cell was warmed through the critical temperature. When the cell was overfilled, the meniscus rose to the top of the cell as the temperature increased. Fluid was bled out in steps until this no longer happened, but rather the meniscus vanished in the middle of the cell. When the cell was critically filled, a valve mounted on its top was closed and the cell disconnected from the gas handling system. The cell was placed in an aluminum jacket which fit snugly in a 5.1 cm diameter by 17.5 cm tall copper cylinder which was the innermost stage of the thermostat. (3-13) 41 circulating bath £ bath controller | laser | \ thermostat \ temperature monitor chart recorder quartz thermometer / \ \ 0 LED @ motor-driven camera temperature controller computer interface j computer ^ F igu re 3-4 A block diagram of the ethylene apparatus. 42 The next stage was a passive metal shield, and the next an outer cylinder wrapped with a coil of copper tubing. Each stage was isolated from the next by a layer of styrofoam. A final layer of styrofoam and a wooden box completed the housing. Small openings to allow passage of the laser beam were made in each layer; some of these were covered with microscope slides to eliminate air currents within the thermostat. The temperature of the outer copper jacket was controlled by circulating a mixture of water and ethylene glycol antifreeze through the coil of copper tubing, using a commercial refrigerated circulating bath 5 3 . The original temperature con-troller on the bath was replaced with an electronic controller which monitored the jacket temperature with a thermistor whose resistance was compared to that of an adjustable set resistance in a bridge. The error signal from the bridge controlled the duty cycle of a monostable multivibrator which switched on a heater in the bath. This controller held the jacket temperature stable to ±0 .005 K. The innermost copper cylinder was further temperature controlled using the proportional-integral controller shown in fig. (3-5). A thermistor embedded in a copper bolt and screwed into the cylinder monitored its temperature. The ther-mistor was one arm of a Wheatstone bridge, another arm of which was a decade resistance box. The bridge error signal was amplified by a Hewlett-Packard 419A D C null voltmeter whose output was fed to a Kepco OPS 7-2 operational power supply with the feedback network shown in the figure. The output of the OPS drove a heater wound non-inductively around the cylinder. This system controlled the cell temperature to better than ± 0 . 1 mK over periods of several hours. Temperature was measured by means of calibrated thermistors embedded in the copper cylinder. The thermistor resistance was determined to 0.01 H by nulling the error signal of a Wheatstone bridge, one arm of which was a General Radio decade resistor. This corresponds to a precision of ±20/^K in temperature for this experiment. Thermistors at the top and bottom of the cylinder allowed detection 44 of temperature gradients; these were determined to be less than 0.001 K over the height of the fluid sample. The thermistors were calibrated in a separate apparatus against a HP 2804A quartz thermometer, which was itself calibrated at the triple point of water using a Jarrett type B triple point cell. The quartz thermometer was also used to monitor the cell temperature, but since it was known to exhibit long-term drifts, the thermistors were used for the most accurate thermometry. Temperature changes were controlled by a Commodore Pet microcomputer which turned on a motor to drive a ten-turn potentiometer in the control bridge arms (fig. (3-5)), thus unbalancing the bridge and causing the cell to warm or cool. The computer also recorded the quartz thermometer temperature output at the end of each step and turned on LEDs to mark the data film at the beginning of each temperature step. Fig. (3-4) shows the computer connected to the rest of the experiment, and fig. (3-6) is a diagram of the interface electronics. The Pet has a parallel output port, to which a four bit binary number was sent by the control program. This number was input to a one-of-ten decoder, from which several relays were controlled. These switched on and off the motor, changed its direction, and controlled the LEDs. The computer program used is listed in Appendix C . The interference fringe data were recorded on photographic film. An old 35 mm camera was modified so that a synchronous motor drove the film continuously at a rate of roughly lOcm/h. The camera's lens was removed and replaced with a vertical slit. The film thus recorded the fringe pattern as a function of time. The output of the temperature monitor bridge and the quartz thermometer were recorded continuously on a chart recorder. The procedure for a run was straightforward. The sample was cooled to a starting temperature below Tc. The camera was loaded with 24 h worth of film and its motor started. The computer was programmed to execute a sequence of timed temperature sweeps followed by waiting periods sufficiently long that the fluid 45 Figure 3 - 6 The computer interface electronics. The relays control a motor which changes the temperature and LEDs which mark the data film. The 7442 chip is a one-of-ten decoder; the 7473 is a J-K flip-flop. 46 reached hydrostatic equilibrium. After 24 h, the film was removed and replaced and the process repeated until the fluid had warmed through its critical point. Only the focal plane experiment was performed for ethylene; our results are presented in Chapter I V . III-3 T h e H y d r o g e n E x p e r i m e n t While the hydrogen experiment was similar in principle to the ethylene ex-periment, the low critical temperature of H2 made the details of the experiments very different. The experiment was done in a commercial optical access cryostat5 4. The sample cell assembly was mounted on a removeable insert that fit into the central column of the cryostat as shown in fig. (3-7). The cell assembly is shown in detail in fig. (3-8). It consists of the cell itself and an outer jacket, both mounted below a copper platform. A valve and catalyst chamber are mounted on a second, smaller platform above the main one. The cell body was machined from O F H C copper. Sapphire windows 0.63 cm thick by 2.54 cm in diameter were mounted on the cell with copper flanges and indium O-ring seals. The enclosed sample space is 1.91 cm in diameter and 0.525 cm thick. The cell is suspended from the main platform by three thermally insulating nylon posts 0.48 cm in diameter; crushed thin-walled stainless steel (TWSS) tubing and a short length of copper wire connect the bottom of the cell to the outer jacket. This arrangement provides similar thermal links to the jacket at the two ends of the cell to help minimize thermal gradients. The copper outer jacket is 9.1cm high and 5.4 cm in diameter. Together with the main platform, it forms a thermal shield surrounding the cell. Openings in the jacket to allow passage of the laser beam were covered with microscope slide windows to limit circulation of the exchange gas in the central column between the 47 valve actuator — — T l hydrogen fill line Or" electrical feedthroughs pumping line • support rod LN, LHe exchange gas column windows Figure 3-7 The cryostat used in the hydrogen experiment. 48 valve actuator-support rod -valve capillary-cell ' WLrr I ->AAA- -J\i\K 4 K wall fill line • catalyst chamber upper platform — nylon posts -main platform — nylon posts ' outer jacket -sapphire window •centering points Figure 3-8 The hydrogen cell assembly. 49 outside and inside of the jacket. Three small nylon triangles positioned around the bottom of the jacket centred the assembly in the central column of the cryostat. The upper platform is mounted above the main platform on four 0.64 cm nylon posts. These serve to thermally isolate the upper platform from the rest of the assembly. The valve and catalyst chamber on the upper platform are part of the fill line connecting the hydrogen supply cylinder at room temperature with the cell. A 0.32 cm diameter TWSS tube enters the top of the cryostat through a vacuum seal and passes down the central column to the catalyst chamber. For most of its length a jacket of 0.64 cm tubing surrounds the smaller tube to prevent hydrogen from freezing in it. When the apparatus is cooled, the air in the 0.64 cm tube freezes, and the resulting vacuum insulates the inner fill line. The catalyst chamber is a cylindrical copper container of volume about 6 cm 3 , loosely filled with a commercial hydrogen catalyst5 5. Molecular hydrogen has a rotational state with total angular momentum J — 1 lying 172 K above the J — 0 ground state5 6. Hydrogen in the J = 1 state is called orthohydrogen (0-H2); in the J — 0 state, parahydrogen (P-H2). At room temperature the states are populated with an o : p ratio of 3 : 1 due to the three-fold degeneracy of the ground state. At the critical point near 33 K the equilibrium P-H2 concentration is 95 % 5 7 . The critical temperature and density are functions of the o : p ratio; catalysis is needed to convert the hydrogen from its room temperature concentration to its equilibrium concentration at 33 K because the very slow uncatalyzed conversion would cause Tc to vary with time. A length of 1 mm stainless steel (SS) capillary runs from an outlet at the bottom of the catalyst chamber to the low temperature valve, shown in fig. (3-9). This valve is similar in design to one described by Oversluizen5 8, with the difference that it is more easily disassembled, making replacement of the valve tip a simple procedure. The SS valve stem is soft soldered to the top of a SS bellows, the bottom of which is soft soldered to a copper collar piece. The valve tip is attached to the 50 stem with a drop of 5-minute epoxy, and can be easily removed for replacement if necessary. A SS tip was used for this work. The complete bellows assembly fits over a copper seat and is sealed to it with an indium O-ring at the shoulder of the seat. A copper jacket fits over the bellows and screws onto the base of the seat to put pressure on the indium and ensure a leak-tight seal. Assembled, the valve is 5.0 cm tall and 1.6 cm in diameter. actuator stem collar piece seat Figure 3-9 The low temperature valve. 51 The valve is actuated from room temperature by means of a 0.32 cm diameter rod that runs from the top of the cryostat down to the valve. The threaded end of this rod screws into the top of the valve jacket and pushes the stem down, causing the tip to seal the central hole in the seat. A 12 cm length of 0.5 mm SS capillary, stuffed with SS wire to reduce its volume, connects the valve outlet to the cell. It is important that the volume between the valve seal and the cell entrance be small, since near the critical point the fluid in this volume will become compressible and flow into the cell, changing the sample's average density and the level of the meniscus. In our case, this unwanted volume was 0.32 % of the cell volume. The cell was cooled by means of a weak thermal link between the cell assembly and a liquid helium bath at 4.2 K, provided by a small amount (~ 1 mtorr) of helium exchange gas in the central column of the cryostat. Two independent temperature controllers were used to maintain the cell tem-perature stable to ±30 /xK for periods of a few hours. A DC controller, similar to that used in the ethylene experiment, was used to control the outer jacket to about ± 0 . 0 0 5 K. A carbon resistor with a room temperature resistance of 1 kO was used as the temperature sensor; it was mounted on the underside of the main platform (see fig. (3-7)). At 30K, it had a temperature coefficient of 100fi/K. This sensor was one arm of a Wheatstone bridge, another arm of which was a decade resistance box. The error signal of the bridge was amplified by a home-made D C nullmeter based on the ICL 7605 instrumentation amplifier. The output of the nullmeter was fed to a Kepco PAT 7-2 operational power supply with a feedback network similar to that on the power supply in fig. (3-6), which powered a 50Q heater wound around the outer jacket. The inner cell was temperature controlled with an A C controller similar to that described by Ihas and Pobell 5 9; it is shown in fig. (3-10). The sensor, a Scientific Instruments type 6 germanium resistor with a temperature coefficient of 52 Figure 3-10 The temperature controller for the hydrogen experiment. The ther-mometer is one arm of an AC bridge. The bridge error signal is detected by the lock-in amplifier and shaped by the proportional-integral-differential controller, then fed back to a heater on the sample cell. 53 20Q/K at 30 K, was mounted in a hole drilled in the side of the cell body. A 100 fl coil of Evanohm wire, with a negligible temperature coefficient, was wound around the main platform and used as a standard resistance. These two resistors formed two arms of an A C bridge. The other two arms were provided by a Dekatran DT-72 seven decade ratio transformer. The bridge was excited at 430 Hz by the internal oscillator of an Ithaco 393 lock-in amplifier. The bridge error signal was detected by the lock-in and shaped by the proportional-integral-differential controller shown in fig. (3-10). The output of this went to a Kepco OPS 21-1 power supply which drove 50 heaters wound around the top and bottom of the cell. This arrangement allowed for the elimination of thermal gradients across the cell; the relative amounts of heat input to the two ends of the cell could be varied by adjusting a resistor at room temperature. The temperature was changed in small steps by changing the ratio transformer setting and thus the bridge balance. A number of resistance thermometers were mounted on the cell assembly to allow monitoring of the temperature in various places. The main thermometer — that used to measure the sample temperature — was a second SI type 6 germanium resistor mounted in a hole in the side of the cell body. Its resistance could be measured in a 4-wire configuration using a home-made constant current supply and a Keithley 177 digital voltmeter, or in a 3-wire configuration using a D C bridge with a decade resistor in one arm. By monitoring the error signal from this bridge with a HP419A voltmeter and a chart recorder, the thermometer resistance could be determined to ±0 .0003 0, corresponding to a temperature resolution of ± 2 0 / z K . This germanium thermometer had been calibrated by the supplier. While the calibration was occasionally roughly checked in this apparatus against the vapour pressure of H2, absolute temperatures were not of primary importance in this study. Temperatures measured with this thermometer are estimated to be accurate on an absolute scale to ± 0 . 0 2 0 K . 54 Four carbon resitors on the cell assembly were monitored occasionally with a 2-wire ohmmeter to check the temperature at various locations. These were located at the top and bottom of the cell, on the bottom of the main platform, and on the valve platform (see fig. (3-9)). Manually controlled 50 0 heaters powered by 12 V batteries were wound on the catalyst chamber (see below) and on the fill line, in case of blockage by frozen hydrogen. All automatically controlled heaters could also be manually controlled if necessary. The hydrogen used for the experimental samples was commercial6 0 ultra-high purity (99.999%) H2, contained in a pressurized gas cylinder at room temperature. It was passed through a cold trap at 77 K to remove any condensible impurities before entering the cryogenic system. The gas handling system is shown in fig. (3-11). A typical run of the experiment proceeded as follows. The vacuum spaces in the cryostat were evacuated and the helium dewar flushed with helium gas. The liquid nitrogen space was filled and the system pre-cooled for several hours. An automatic liquid nitrogen filler similar to that described by Reid and Hyde 6 1 was used to top up the nitrogen periodically throughout the run. A few millitorr of He exchange gas in the central column of the cryostat facilitated pre-cooling of the cell assembly. This gas was pumped out prior to transferring liquid helium. After the transfer, a small, measured amount of exchange gas was let into the column and the system allowed to cool to below 30 K. With the system cold, the low temperature valve was closed and sufficient H2 slowly let into the catalyst chamber, where it was allowed to sit for at least an hour. Occasional warming of the chamber ensured thorough mixing of the H2 in the catalyst. The valve was then opened and the fluid let into the sample cell. The amount of fluid in the cell was adjusted by manually heating the catalyst chamber to force fluid into the cell, and by either cooling the chamber or releasing H2 from 00 c •1 re CO I H t r fO g t r & 5' (ra cn co c t r rt> 5* Cu * i O era 3 to X T3 to <-i 3 n 3 helium cylinder © ©- to exchange gas to helium dewar column ($> 1PSI H X r H hydrogen cylinder 9_J 300 PSI <8> 1 <8> 1 ( ? ) cold trap to hydrogen cell ® shut off valve <8> needle valve £3 pressure relief valve P ) pressure gauge © en 56 the system at room temperature to draw fluid out of the cell. When the cell was filled so that the liquid-vapour interface disappeared in the middle of the cell when it was warmed up through the critical temperature, the valve was closed and data collection began. The optical setup for the H2 experiment, shown in fig. (3-12), allowed si-multaneous recording of both the focal plane and image plane patterns, using two continuously motor-driven cameras. The cell was cooled below Tc, then warmed in steps by adjusting the ratio transformer setting. The time required for the system to equilibrate after a change in temperature varied from less than one minute far from Tc to of order an hour very close to the critical point. Since the liquid helium had to be refilled roughly every 10 hours, and since helium transfers tended to disrupt the equilibrium of the sample, this limited the number of data points that could be obtained very close to the critical point. The more frequent nitrogen transfers had no effect on the sample. Relaxation times above the critical point were considerably longer than those below, being of order an hour even quite far from Tc. This rather seriously limited the amount of supercritical data obtained from a given run. The results of the hydrogen measurements are presented in Chapter V. III-4 The P r i s m Cel l Experiment The refractive index of a fluid is related to its density by eq. (1-10). While the Lorenz-Lorentz coefficient L is approximately constant for dilute gases, at higher densities £ becomes a function of density due to the density dependence of the molecular polarizability6 2. Z can then be written as a power series in the density, L(p) = AR(T) + BR(T)p + CR{T)p2 + • • • (3-14) where AR, BR, etc. are called the refractometric virial coefficients and are in gen-eral temperature dependent. In this section an experiment to measure £(/>) and the 5 7 F i g u r e 3-12 T h e optical setup for the hydrogen experiment. D a t a can be recorded i n the focal plane (F) and image plane (I) simultaneously. 58 coexisting fluid densities in the two-phase region is described; the results of measure-ments on ethylene are presented in section IV—2. The experiment was developed in this laboratory and has been used for similar studies of other fluids37-39. The sample cell, made of aluminum, consists of a wedge shaped head and a cylindrical body. Two sapphire windows in the head allow the cell to be used as a prism. The wedge angle of the prism was measured to be 20.733 ± .017° for these experiments. The sample density was determined from the mass of the cell, the empty mass and volume of which were accurately known. Fig. (3-13) shows the experimental setup. A spatially filtered, collimated laser beam was passed through the cell while a beam splitter and two pentagonal prisms directed a reference beam around the thermostatic housing. Both beams were directed by a micrometer-driven adjustable mirror (Lansing model 10-253) into a Davidson model D 275 autocollimating telescope. The cell was filled with ethylene, weighed on a Sauter monopan balance accurate to ±0 .0002 g and placed inside a temperature controlled housing similar to that described in section III—2. The refraction angle of the sample beam was determined by adjusting the micrometer-driven mirror to direct the beam into the telescope, where it was focussed to a spot. The micrometer was calibrated by measuring the positions of the diffraction spots from a 50 lines/inch Ronchi grating placed in the thermostat in place of the sample cell. £(p) was measured on the supercritical isotherm at 298.2 K and on the coex-istence curve. The measurements at 298.2 K were straightforward. The cell mass and beam refraction angle were measured, ethylene was bled out of the cell, and the process repeated until the cell was empty. The temperature was held constant using a controller similar to that described in section III—2. The coexistence curve measurements were only slightly more involved. The cell was filled and cooled until both phases were present in it. It was then warmed in small steps and the sample beam angle monitored. Since the fluid density and thermostat beam expander micrometer-driven mirror HeNe laser iris telescope 60 F igu re 3-14 A phase diagram showing the path followed in taking a data point in the prism experiment. The sample, with density pj , is cooled to a temperature T\ in the two-phase region (solid circles), then warmed until it re-enters the one-phase region at a temperature T2 (open circle). The refractive index is measured when the system has just moved off the coexistence curve. thus the refractive index changed as the cell was warmed, the refraction angle also changed. At some temperature the sample left the coexistence curve and entered the one-phase region of the phase diagram, as illustrated in fig. (3-14). The density of the single phase remained constant and so the refractive index changed very little as the cell was warmed further. At this point the micrometer reading was recorded and the cell weighed. Fluid was bled out and the process repeated. When the sample density was close to pc, refracted beams from both the liquid and vapour phases were observed. One of these eventually vanished as the system left the coexistence curve, which one depending on whether the average density was above or below pc. At p = pc the two refraction angles became equal at the critical temperature. At 61 this density, the coexistence curve was measured by recording the refraction angles of each of the two beams as a function of temperature below Tc. In both cases, when the cell was finally emptied of fluid it was evacuated and its empty mass determined. It was then filled with air at atmospheric pressure, placed in the thermostat, and the refraction angle measured; this provided a reference angle for use in the data reduction. The cell volume was measured by filling it with distilled water and reweighing it. I I I - 5 Experimental Limitations There are two major experimental problems that can affect the results of the above experiments close to the critical point. One is so-called gravitational round-ing, which is unavoidable; the second is thermal gradients, which can in principle be eliminated. Gravitational Rounding As discussed above, the contribution of the earth's gravitational field to the chemical potential causes the fluid density to vary with height in the cell. In the region close to the critical point where /c^  diverges, this variation becomes large. Any experimental measurement of the fluid density involves some sort of av-eraging over a finite height in the sample. In the case of the image and focal plane experiments, the averaging comes about because the laser beam is deflected by the refractive index gradient and so traverses a small vertical distance on passing hor-izontally through the sample, as shown in fig. (3-15). Close enough to the critical point, the gravitationally induced change in density with height becomes so large that the density varies significantly over the distance sampled by the beam. Under these conditions, the fringe patterns observed no longer provide information about the local density. This is known as gravitational rounding, or, for these optical experiments, the thick cell effect. 62 9 F i g u r e 3 -15 Gravitational rounding. A light ray is bent by the refractive index gradient in the fluid and so traverses a height h as it passes through the cell. If the sample density changes significantly over this height, the observed fringe pattern is affected. This problem has been analyzed in detail by Moldover et a / 3 2 . Using a para-metric representation of the equation of state in the critical region, they calculate how close to Tc data can be gathered that is accurate to a given precision. Following their procedure for the image and focal plane experiments and specifying a precision of 1 %, we find for ethylene that gravitational rounding should limit our measure-ments to t > tmin = 2.3 x 1 0 - 5 above T c , with a lower f m t r i below Te where the compressibility is smaller. In fact our C 2 H 4 data show no noticeable gravitational rounding down to the lowest reduced temperature of t = 1.5 x 10~6. For hydrogen, a similar calculation gives t m t n = 1.2 x 10~4 in the one-phase region, smaller in the two-phase region. 63 In the prism experiment the effect is much more serious, since the laser samples the entire sample height of approximately 1 cm. A calculation for ethylene gives = 1.9 x 10 - 4 above TC and 5.2 x 10~5 below. Experimentally, a smearing of the laser spot observed in the telescope was present for t < 10 - 4 and meaningful data could not be obtained for t < 3 x 10 - 5. T e m p e r a t u r e G r a d i e n t s A temperature gradient across the sample will cause a variation in chemical potential with height in addition to that due to gravity, given by Such a gradient would make determination of Kj difficult since the cell would not be isothermal; there would be contributions to dp/dp from both gravitationally and thermally induced density gradients. In H2 a thermal gradient of on the order of 0. 1 mK/cm will contribute to the chemical potential an amount similar to that due to gravity. The effects of thermal gradients on the coexistence curve measurements are not as serious, since these measurements are done at a single height in the cell, 1. e., the height of the liquid-vapour interface. The hydrogen cell was designed with the minimization of temperature gra-dients in mind. However, while the gradient could be varied by adjusting the the heat inputs to the two ends of the cell, we had no way of actually measuring it; in particular we had no way of knowing when it was zero. Our compressibility measurements, to be discussed below, were strongly affected by the small gradients present, and this made it impossible to determine the amplitudes TQ , TQ , etc. As shown in Chapter V, however, we were still able to obtain estimates of the exponent 7 and the amplitude ratio r$ /TQ . ( dp grav df~dz ~ p~c~&T~dz' (3-15) 64 I I I - 6 D a t a R e d u c t i o n a n d E x p e r i m e n t a l E r r o r Interference fringe data for the image plane and focal plane experiments were recorded on photographic film. Coexistence curve data were extracted by counting these fringes to determine the number of missing fringes after each temperature step. This was done manually with the aid of a travelling microscope or a magnifying eyepiece. A typical run might involve on the order of 500 fringes. The uncertainty in the count at low fringe numbers is about ± 0 . 5 fringes in the focal plane and ± 0 . 2 fringes in the image plane; at large fringe numbers it is about 0.5 % due to the difficulty in determining exactly when a fringe disappears. This translates into an uncertainty in pt — pv of from 0.5 % far from critical to 2 % at the lowest values of t. The fringe counts were converted to order parameter and coexistence curve diameter data using eqs. (3-4)-(3-7). For ethylene, C{p) and pc were measured using the prism cell apparatus de-scribed in section III-4. For hydrogen, was obtained from the measurements of Diller 6 3 . His results for £(p) on the coexistence curve are not accurate close to the critical point or on the vapour branch of the curve, therefore only his high density data were used in this analysis. Since his measurements were done at a wavelength of A = 5462 A while our work was at 6328 A, we scaled his L data down by a factor of 0.994. This factor was calculated from a linear interpolation of available refractive index data 5 7 at 6939 A and 5462 A. The scaled data was fitted to a quadratic in the density that was constrained to peak at the critical density, for which we used 5 7 pc = 0.03143g/cm3. Diller's scaled data and our fit to it are shown in fig. (3-16). The fit gives I (p) = 1.02565 + 0.05373p - 0.85469p2. (3-16) Index data from the prism experiment were obtained in the form of micrometer readings, which were converted to angles using the Ronchi grating calibration and 65 00 o u o o 00 o 1^ O 1 ^ CM O m co o r i 00 CD o o o O CM O O o o o • o in T2 Figure 3-16 The Lorenz-Lorentz coefficient for H2, adapted from ref. 63. The line is a fit to the high density data. Points used in the fit are marked x, other points are marked +. The large variations in t near the critical density are due to gravitational rounding. 66 then to refractive indices using the equation n = na I 1 + cos 0 — cos 0a + sin 6 — sin 8a (3-17) tan a obtained from a straightforward analysis of the prism cell optics. Here a is the prism angle and 8a the refraction angle measured with the cell filled with air. In the experiment to measure £(/>), densities were obtained from the full and empty masses of the cell, measured to ±0 .0002 g, and its volume, measured to ± 0 . 0 6 % . £{p) was calculated using eq. (1-10); the results have an estimated uncertainty of 0.1 % at higher densities and 0.3 % at low densities. For the coexistence curve measurements, £ was fitted to a quadratic in n and the density determined from the equation These densities are accurate to ± 0 . 2 % at high t, but become less accurate at very low t due to gravitational rounding. The precision of the compressibility data was limited by how accurately the spacing between fringes on the data films could be measured. Far from Tc, when the fringes were widely spaced, it was impossible to obtain any reliable data. Closer to the critical point, when several fringes appeared across the film, the compressibility could be determined to roughly 5 % at best. Temperature data were obtained from a continuous chart record of the tem-perature monitor bridge error signal. For both the room temperature and low tem-perature measurements, temperatures could be determined to a (relative) accuracy 1 n2 - 1 (3-18) P = £{n) n 2 + 2" of ± 2 0 /xK. 67 CHAPTER IV Results, Part 1: Ethylene Two experiments were performed on C 2 H 4 : the prism experiment of section III— 4 and the focal plane fringe experiment described in sections I I I - l and III— 2. The results of these experiments are presented in this chapter. The data are tabulated in Appendix B . I V - 1 T h e P r i s m E x p e r i m e n t The prism experiment produced two distinct sets of results. These are the Lorenz-Lorentz function £(p) and the liquid and vapour densities on the coexistence curve. The computer programs used to extract these results from the raw data are given in Appendix C . T h e L o r e n z - L o r e n t z F u n c t i o n £(p) was measured on the coexistence curve and on the supercritical isotherm at 298.2 K. The primary purpose of these measurements was to make possible the conversion of measured refractive indices to densities for analysis of the critical behaviour. The measurements of £ are, however, of some interest in themselves, since a knowledge of the refractometric virial coefficients of eq. (3-4) can lead to information on intermolecular forces 6 2 ' 6 4 - 6 6 . 68 Our results are plotted in fig. (4-1) along with fits to the expansion (3-14) carried to second order in the density, £{p) = AR + BRp + CRp2. (4-1) The fit parameters are given in Table TV—1. Since the measurements on the coexis-tence curve are limited to a small region around the critical density, the uncertainties in the fit parameters for these data are large. We find AR = 10.55 ± .10cm3/mole, BR = 32.8 ± 30.3 cm 6/mole 2 and CR = -2300 ± 2100cm9/mole3. Simply averaging the coexistence curve data gives LCoex — 10.665 ± .021 cm 3/mole. Table I V - 1 Refractometric Virial Coefficients and Electronic Polarizability for Ethylene Data Set i4 /?(cm3/mole) (cm6/mole2) Cj? (cm 9/mole3) «e(A 3) Coex. curve 10.55 ± .10 32.8 ± 30.3 -2300 ± 2100 4.14 ± .04 10.665 ± .021 (0) (0) 4.23 ± .01 298.2 K 10.50 ± .02 40.2 ± 8.7 (o) 4.162 ± .008 Both sets 10.48 ± .02 64.3 ± 9.1 -5100 ± 830 4.154 ± .008 It is, of course, not quite legitimate to use eq. (3-14) to describe L(p) on the coexistence curve. Since the temperature is not constant, the refractometric virial coefficients will in principle vary along the curve. It is, however, useful to parametrize the behaviour of £(p) in this way so that these results can be applied to the analysis of our critical phenomena data. The 298.2 K data agree in magnitude with the coexistence curve data although the density ranges covered by the two data sets do not overlap much. The results of fits to this data and to both sets of data combined are given in Table I V - 1 . 5 03 i — • B 3 n 01 95 n> O rt> X rt> 3 o •1 m lU I M H t r o rt) N i o •1 rt) 3 N S 8 < 35 rt) PJ. 3 O I 8 3 3 r> <n-5' 3 a. 3 5 ST 3 rt) 10.8 10.7 H OJ O 10.6 H 6 CO 10.5 H 10.4 H Ethylene X Coexistence curve • 298 K 10.3 -| 1 r 0.000 0.002 0.004 0.006 0.008 Density (mole/cm3) 0.010 70 St. Arnaud and Bose 6 5 find AR = 10.610 ± .009 cm 3/mole and BR = 40.8 ± 2.0cm 6/mole 2 for ethylene at 303 K; they see no evidence for a quadratic term in £(/>) up to densities of about 0.008 mole/cm 3. This density is higher than the critical density of ethylene, 0.0076 mole/cm 3. The 1 % difference in AR between our results and those of ref. (65) may be due to sample impurities or to a systematic error in density determination. The value of pc obtained with our apparatus and sample (see below) is 0.3 % higher than previously published values; this is not enough to account for the above difference. The zero density limit of £(p) is related to the molecular electronic polariz-ability ae by „/• \ . 47ra eNA , x £{0)=AR = —f-±. (4-2) From our values of AR we find ae = 4.16 ± .04 A on the coexistence curve and ae = 4.162 ± .008 A 3 at 298.2 K. T h e Coexis tence Cu rve The refractive index measurements on the coexistence curve are converted to density data as described in section I I I - 6 using the £,{p) data presented above. The resulting coexistence curve and its diameter are plotted in fig. (4-2). A log-log plot of the reduced density difference Ap* against the reduced tem-perature t is given in fig. (4-3). The critical density and temperature used for this plot were determined from fits to the data as described below. In fig. (4-4) the same data is plotted as Ap*/t^ vs. t on a log-log plot. Here the leading temper-ature dependence in eq. (1-6) has been divided out; the value of the exponent j3 used in this and other similar figures is that from the fit plotted in the figure. This type of plot 2 3 is very sensitive to deviations of the data from pure power law be-haviour. A straight horizontal line would imply Ap* oc t@'. Curvature of the plot as t increases indicates deviations from a power law due to corrections-to-scaling co W 2. 3 a H a -o o CD a r> m c> C m to a^ ST a f» 3 Cu CO Cw p' 3 to rt) i-l e 3 g 6 a* CD 284 282 H 280 H 278 H 5 276 H O Q- 274H 272 H 270 H 268 x* X X X X vapour liquid diameter X X ++ + + + + + + o o + + 0.002 0.004 0.006 0.008 0.010 0.012 0.014 density (mole/cm3) 72 F igu re 4-3 A log-log plot of the order parameter vs. reduced temperature for the ethylene prism data. 73 effects. An upturn in the data at low t could signal the presence of the gravitational rounding effects discussed in section I I I — 5 , although an incorrect choice of Tc would also cause an upturn or a downturn in the low t data. As the figure shows, this set of data deviates from a simple power law at high t. It was therefore analysed using the corrections-to-scaling series for the order parameter, eq. (1-6). It is well known that, with an incorrect choice of Tc, fits to eq. (1-6) or to its asymptotic power law form can be misleading. For example, a fit of the data in fig. (4-4) to a simple power law, Ap* = BtP, with B, Tc and /? all free parameters gives P = 0.355 and Tc = 282.3823 K. Fitting only the data with t < 7.2 x 10~4 gives P = 0.328 and Tc = 282.3750 K. The difference is due to neglect of corrections-to-scaling terms in the first fit, which includes data well out of the asymptotic critical region. Fig. (4-5a) is a plot of the residuals for the first of these fits; systematic deviations from the fit are visible. The critical temperatures used in the analysis of this and other data sets were determined from simple power law fits like the second one, using only data close to the critical point. These determinations of Tc were checked by fitting the data over the full range of t with the corrections-to-scaling expression. For the prism data under consideration here, a fit to eq. (1-6) with two correction terms and P fixed gives Tc = 282.3754 ± .0004 K for P = 0.327 ± .002, consistent with the result obtained above. The value Tc = 282.3750 ± .0010 K was therefore adopted for this data set. Table IV—2 gives the results of several fits to this data which cover the range 8.7 x 10~5 < t < 4.5 x 10 - 2 . The fits in this table were performed with the correction exponent fixed at the value 0.5, roughly its calculated value 7 , 9 and the value conventionally used in analyses of this type 2 , 3 . Fits were done using one or two correction terms and with P either fixed at values near its predicted value 5 - 9 or free. Parameter values enclosed in parentheses were fixed for that fit. 74 I o U/A) 6o| Figure 4-4 A log-log plot of Ap/t& vs. reduced temperature for the ethylene prism data. The line is fit 1 from Table IV-2. 75 o 73 'tn -2--4 10 - 6 X x X x X * * X x x rfx X* X i . . — a — : x X I I I I I I l l | I I I I I I l l | I I I I I I l l | I I I I I I II v-5 1 Q - 4 1 Q - 3 1 0 "2 10" 10"1 reduced temperature D 3 0 0) -2--4 icr 6 ,V • * X X icr 5 • 111 J 111 i 10 -4 10 r3 mi i 10-2 10-1 reduced temperature Figure 4-5 Residuals of fits to the order parameter data of fig. (4-4). a) Residuals for a fit to a simple power law (fit 6 of Table IV-2). Note the systematic behaviour of the residuals, b) Residuals for fit 1 of table rV-2, which includes two corrections-to-scaling terms. 76 T a b l e I V - 2 Fits of the ethylene order parameter data to eq. (1-6) with A = 0.5 data set fit# B0 Bi prism data 1 0.3286 1.586 0.89 -1.58 2 (0.325) 1.538 1.09 -2.01 tmax = 4.5 X 10 3 (0.327) 1.564 0.98 -1.77 tmin = 8.7 X 10~5 4 0.3399 1.749 0.25 (0) 5 (0.327) 1.598 0.535 !° 6 0.3549 1.924 (0) (o) fringe data set #2 7 (0.327) 1.490 1.65 -19.7 8 0.3278 1.511 0.85 (0) tmax = 7.5 X 10 4 9 (0.325) 1.467 1.22 (0) tmin = 2.5 x l O - 6 10 (0.327) 1.498 0.95 (0) set #3 11 (0.327) 1.493 1.16 -1.30 12 0.3272 1.498 1.05 (0) tmax = 4.4 X l O " 3 13 (0.325) 1.468 1.19 (0) tmin = 2.1 X 10"5 14 (0.327) 1.495 1.06 (0) set #4 15 (0.327) 1.513 -0.10 !°1 16 0.3268 1.509 (0) (o) tmax = 1-4 X l O - 4 17 (0.325) 1.482 (0) (0) tmin = 2.8 X l O " 6 18 (0.327) 1.512 (0) (0) set #5 19 (0.327) 1.496 1.09 -1.34 20 0.3269 1.499 0.97 (0) tmax = 7.1 X l O " 3 21 (0.325) 1.473 1.09 (0) tmin = 1.5 X 10-6 22 (0.327) 1.500 0.97 (0) all sets 23 (0.327) 1.496 1.12 -1.49 24 0.3269 1.497 0.99 (0) tmax = 7.1 X l O " 3 25 (0.325) 1.471 1.12 (0) tmin = 1.5 X 10"6 26 (0.327) 1.499 0.99 (0) all data 27 0.3288 1.585 0.96 -1.89 28 (0.325) 1.530 1.23 -2.64 tmax = 4.5 X 10~2 29 (0.327) 1.559 1.09 -2.24 tmin = 1.5 X l O " 6 30 0.3359 1.699 0.35 (0) 31 (0.327) 1.585 0.61 (0) 77 The computer program used for curve fitting is listed and described in Ap-pendix C . It is based on a standard non-linear least squares fitting routine called N L 2 S N 0 6 7 . Fits to this data set give 0 = 0.328 ± .004, B0 = 1.59 ± .05, B\ = 0.90 ± .15 and B2 = —1.6± .3. Fits with B2 = 0 had significantly larger x 2 values than did fits with Bi free, indicating the need for at least two correction terms in the analysis. The quoted uncertainties are several times larger than the statistical uncertainties from the fits and are intended to include uncertainties due to correlations between the parameters and to the uncertainty in Tc. Correlations between parameters are strong, as can be seen from inspection of Table I V - 2 . Fit 1 from the table is shown in fig. (4-4); residuals for this fit are plotted in fig. (4-5b). Fits were also performed with A a free parameter. The data were not precise enough to permit meaningful fits with both /? and A free, but with /? fixed at values close to its predicted value, A could be determined. The results of these fits are given in Table I V - 3 . We find for this data that A = 0.44 ± .02 for /? = 0.327 ± .002. The amplitudes obtained from these fits are significantly lower than when A = 0.5; B\ decreases by about 20 % and B2 by about 50 %. The leading amplitude BQ decreases by about 1%. The Coexistence Curve Diameter The corrections-to-scaling expression for the coexistence curve diameter was given in eq. (1-9): pd = ei±£l = 1 + A l t i - + A2t + A3t>~a+A + •••. (1-9) Classically, the diameter is rectilinear, i.e., only the amplitude A2 is non-zero. In the case of a V D W fluid, A2 — 2/5. The term ~ t1~a+A is a corrections-to-scaling term, and the f 1 - a term arises due to thermodynamic field-mixing. Although predicted by various theoretical treatments 6 8 - 7 0 , this t*~a anomaly has proven very difficult to 7 8 Table IV-3 Fits of the ethylene order parameter data to eq. (1-6) with A a free parameter data set fit# & Bo Bi Bi A prism data 32 (0.325) 1.520 0.84 -1.02 0.42 33 (0.327) 1.553 0.81 -1.07 0.44 fringe data 34 (0.325) 1.461 0.80 (0) 0.42 35 (0.327) 1.496 0.88 (0) 0.47 all data 36 0.299 1.034 1.52 -0.76 0.23 37 (0.325) 1.512 0.85 -1.02 0.40 38 (0.327) 1.548 0.83 -1.14 0.43 observe experimentally , mainly because it is quite weak; its exponent, 1 — a, is only slightly smaller than unity. Separation of the anomaly from the analytic background thus requires precise data in the vicinity of the critical point. The predicted shape of the coexistence curve diameter is illustrated in fig. (4-6). In Section III-3 the physical variables t and A/x* were associated with the two relevant scaling fields of the liquid-vapour critical point. For real fluids the scaling fields are not simply equal to t and A/x*, but are analytic functions of them that vanish at the critical point 7 1. In the simplest expression of this "revised" scaling7 6, the scaling fields are t + cA/x* and A/z*, where c is a universal constant. This field-mixing is a manifestation of the lack of particle-hole symmetry in real fluids. It has been argued that its microscopic origin is many-body interactions in the fluid; a theory based on three-body, primarily repulsive Axilrod-Teller interactions has been found to agree well with experimental results 7 3' 7 4. The diameter data from fig. (4-2) for ethylene are replotted in fig. (4-7) on an expanded scale. While there is some scatter in the data, a small (0.1 %) systematic deviation from the classical rectilinear behaviour is seen for ( < 3 x 1 0 - 3 . 79 I 01 density Figure 4-6 A schematic diagram of the coexistence curve diameter. The dashed line is the classical rectilinear diameter; the dotted line shows the expected 1 — a anomaly. The critical density for ethylene was determined by fitting these data to eq. (4-3) below and adjusting pc to give p& = 1 at t = 0. This procedure gives pc = 0.007659 ± .000008 mole/cm 3. The diameter data were analyzed by fitting to the form pd = A0 + Ait1'* + Ait (4-3) 80 F i g u r e 4-7 T h e coexistence curve diameter of ethylene. T h e line is a fit of the d a t a in the range 0.02 > t > 0.008 to a straight line. T h e inset shows the deviations f r o m this fit at low t. These are systematically negative for t < 0.002. 81 with Ao, Ai and A2 parameters. The exponent a was fixed at its theoretical value of 0.11. The corrections-to-scaling term was not included in the analysis because the precision of the data did not justify the introduction of additional free parameters. Fits were performed over an "inner" temperature range, 0.0003 < t < 0.002, an "outer" range, 0.008 < t < 0.02, and an "entire" range, 0.0003 < t < 0.02. The data shown in fig. (4-7) include points outside these ranges; the fits were done as described for consistency with the analysis of data for other fluids in refs. (73) and (74). Results of these fits are given in Table rV-4. Data in the outer region were fit to the form pd = A0 + Ait (4-4) to determine the slope of the diameter away from the critical point; we find A2 — 0.878 ± .037. Fits over the inner and entire ranges show that the observed deviation from rectilinear is consistent with the predicted tx~a anomaly with A\ positive, although a determination of the exponent was not possible because of the scatter in the data and the weakness of the critical anomaly. Table rV-4 Fits to the Coexistence Curve Diameter of Ethylene range of t Ao A2 x 2 outer 1.00058(52) (0) 0.878(37) 1.20 inner 1.00017(26) 0.517(121) (0) 1.02 inner 1.00027(23) (0) 0.988(226) 0.99 entire 1.00008(11) 0.576(7) (0) 1.14 entire 1.00057(11) (0) 0.885(11) 1.27 entire 1.00058(19) 0.401(182) 0.269(280) 1.14 82 In fig. (4-8) the coexistence curve diameter of ethane, C 2 H 6 is shown. This set of data was obtained by Burton in 1973 3 8' 7 5 in the same way as the ethylene data, but the coexistence curve diameter was not carefully analysed until now. These data also show an anomaly for t < 3 x 10 - 3 . Results of fits of eq. (4-3) to this data over the same temperature ranges as for ethylene are given in Table IV—5. Here the slope of the diameter away from Tc is A2 = 0.795 ± .038; again the anomaly is consistent with the predicted form. Table IV-5 Fits to the Coexistence Curve Diameter of Ethane range of t A0 Ai x2 outer 1.00159(55) (0) 0.795(38) 1.58 inner 1.00013(20) 0.567(82) (0) 1.00 inner 1.00027(19) (0) 1.053(155) 1.03 entire 1.00017(10) 0.564(8) (0) 1.14 entire 1.00062(11) (0) 0.865(14) 1.44 entire 1.00010(17) 0.661(180) -0.150(276) 1.08 I V - 2 The Fringe Experiment The focal plane fringe experiment yields the difference in refractive index between the coexisting phases, ni — nv. This quantity is converted to the reduced density difference Ap* using the results for £(/>) presented in the previous section, and eq. (3-6); the algebraic details of the conversion are given in Appendix A and the relevant computer programs listed in Appendix C . Four sets of fringe data were obtained for ethylene. For each set the critical temperature was determined as above to ± .0001 K; the four values obtained were in the range 282.4882 ± .0024 K. These data are shown on a log-log plot in fig. (4-9) 83 Figure 4-8 The coexistence curve diameter of ethane. The line is a fit of the data in the range 0.02 > t > 0.008 to a straight line. The inset shows the deviations from this fit at low t. These are systematically negative for t < 0.004. 84 and on a sensitive plot in fig. (4-10). The different symbols in the figures identify data from the different runs. Fits to these data sets individually, as well as inspection of fig. (4-10), indicate that the four sets are consistent with each other. All four sets were therefore analysed together. We found that one correction term in eq. (1-6) was sufficient to describe the data, which extended to a maximum reduced temperature of tmax = 8 x 10 - 3 . If a second correction term was included, its value was not well determined and varied greatly with the temperature range of the fit. The results of fits to these data are given in Table I V - 2 . Since the data extend very close to Tc, (5 could be determined quite precisely. The fits give f3 = 0.3269 ± .0010, B0 = 1.50 ± .03, and Bi = 1.0 ± .1 with A = 0.5. Fit 24 from the table is shown in fig. (4-10), and a plot of the residuals for this fit is given in fig. (4-11). Gravitational rounding effects can lead to and be masked by an incorrect value of Tc, as discussed above. For this reason, fits were also performed excluding those points at the lowest values of t, where rounding was expected to be most important. The results of these fits were consistent with fits to the entire data set, implying that such effects are small in this case. These data were also fitted with A a free parameter; we find A = 0.47 ± .05 for f3 = 0.327 ± .002. As before, the correction amplitudes decrease significantly from their values when A = 0.5. Results of these fits are given in Table IV—3. I V - 3 C o m b i n e d R e s u l t s The order parameter data from the two ethylene experiments differ by 0.11 K in critical temperature. In addition, the fringe data are systematically 4 % lower than the prism data in the region where the two sets overlap, as shown in fig. (4-12). The cause of these differences is not known. One possibility is a difference in impurity content between the two samples. The sample used in the fringe experiment was from a bottle purchased in 1968 which 85 F i g u r e 4-9 A log-log plot of the order parameter vs. reduced temperature for the ethylene fringe data. F i g u r e 4-10 A log-log plot of Ap/t^ vs. reduced temperature for the ethylene fringe data. T h e line is fit 24 from Table IV-2. 87 • o »o • J> • . - - m*. • • • • » * — i i i 1 1 1 1 1 H I i 1 1 1 1 1 H I i 1 0 - 6 1 0 - 5 1 0 - 4 - m i i i i 11M! 10 - 3 lO"2 reduced temperature 10-1 F igu re 4-11 Residuals for the fit shown in fig. (4-10). had been unused until this work began. The sample used in the prism experiment was purchased in 1985. A possible impurity in ethylene is ethane. Following the procedure of Hastings et a / . 7 6 , the amount of ethane impurity required to cause the observed difference in Tc is estimated to be 0.48 %, which is large. Even this amount of impurity, however, would only cause a shift in pc of 0.06% and in nc of about 0.02 %, nowhere near the 4 % required to reconcile the two sets of data. Mass spectra of ethylene samples from both the old and new bottles were virtualy identical, apparently ruling out impurities as the source of the discrepancy. Possible systematic errors stemming from the calibration of the prism appa-ratus are expected to be at worst 0.2%. Other possible errors, for example in the determination of pc, would affect both sets of data in the same way and so cannot explain the differences between them. Figure 4-12 A log-log plot of the order parameter vs. reduced temperature for both sets of ethylene data. 89 Since the sample used in the prism experiment was nominally more pure and also much newer, the order parameter data from the fringe experiment were scaled up by 4.0%. When this was done, the fringe and prism data agreed within the scatter. A sensitive plot of the combined data is given in fig. (4-13). Results of fits to these combined data are given in Table rV-2. From these fits we get 0 = 0.3288 ± 0.0012, B0 = 1.58 ± .03, Bx = 0.96 ± .15 and B2 = -1.9 ± .3 with A = 0.5. Fit 27 from the table is shown in fig. (4-13) as a dashed line; the solid line is fit 24 to the fringe data only. With A free we find A = 0.43 ± .03 for 0 = 0.327 ± .002, and again the correction amplitudes decrease. It can be seen from fig. (4-13) that there are some difficulties with the fits obtained using eq. (1-6). Specifically, the unphysical "turnover" of the fitted curve at t ~ 7 x 1 0 ~ 2 seems to indicate the need for an additional, positive correction term at reduced temperatures in this vicinity. Small systematic deviations of the fit from the "best fit" as determined by eye are also evident. We believe these problems to be indications of crossover effects, caused by the fact that the corrections to scaling series does not converge to any classical limit away from the critical region. This point will be discussed further in Chapter V I . 91 CHAPTER V Results, Part 2: Hydrogen The coexistence curve of hydrogen and its diameter were measured using the focal plane and image plane techniques, as described in sections III— 1 and III— 3 . These results are presented below in sections V—1 and V - 2 ; the data are tabulated in Appendix B . Our attempts to measure the isothermal compressibility from the image plane data were not completely successful due to the presence of temperature gradients. We found the compressibility to vary substantially for runs with differ-ent temperature gradients. Despite this, the compressibility measurements provide useful information; they are presented in section V - 3 . V—1 The Coexistence Curve The focal plane and image plane fringe counts were converted to densities as described in Chapter III. The values of Ap* obtained from the two techniques for a given run should be identical, within fringe counting errors, since the data were recorded simultaneously. Although the image plane fringes could be counted with slightly better precision, adding the liquid and vapour fringe counts to obtain n^ — nv made the overall counting uncertainties about the same for the two techniques; they are as discussed in section I I I - 6 . The Ap* data analysed in this chapter are from the focal plane counts; the image plane data were used as a check. Several experimental runs were done for hydrogen. The critical temperature, determined as for the ethylene data, covered a range of 32.85 K < Tc < 33.30 K as 92 measured with the calibrated Ge resistance thermometer. Five of the seven runs analysed had Tc in the range 32.9735 K <TC < 33.0030 K with a mean and standard deviation of 32.984 ± .013 K. Tc for each run could be determined to°a precision of from ± 0 . 0 0 0 2 K to ±0 .0005 K, depending on how many near-critical data points there were in each data set. The observed variations in Tc are probably due to changes in the thermometer calibration on thermal cycling, and possibly also to slightly different ortho-para concentrations in the samples used for different runs. The accepted value 5 7 of Tc for pure P - H 2 is 32.976K, and for normal (i.e., room temperature composition) H 2 it is 33.19K. Assuming that Tc varies linearly with P - H 2 concentration, the critical point of equilibrium (95% para) H 2 should be at 32.990 K. The observed critical temperatures are consistent with this value. Ap* obtained from the focal plane data is plotted against reduced tempera-ture on a log-log plot in fig. (5-1). Fig. (5-2) is a sensitive plot of the same data. Data from seven runs are shown; the different symbols distinguish data from the different runs. There are slight (< 0.5 %) systematic differences between the various runs which may be due to variations in o-p concentration (although no systematic variation of BQ with Tc was observed) or, more likely, to systematic errors in fringe counting. Basically, however, the different runs overlap within the expected exper-imental error. The systematic differences do not have a very significant effect on the analysis of the data, as discussed below. No systematic variation of Bo with the temperature gradient was observed, indicating as expected that the coexistence curve measurements are relatively insensitive to that problem. The hydrogen data plotted in figs. (5-1) and (5-2) have a larger tmax (7.0 x 10 - 2 ) and tmin (3.2 x 10 - 5 ) than do the corresponding ethylene data, for two reasons. First, since the critical temperature of hydrogen is roughly a factor of 10 lower than Tc for ethylene, the reduced temperature scale for hydrogen is stretched by a factor of 10 compared to ethylene's. Second, the lower refractive index of hydrogen5 7 93 Figure 5-1 A log-log plot of the order parameter vs. reduced temperature for hydrogen. 94 7 Figure 5-2 A log-log plot of Ap* /t& vs. reduced temperature for hydrogen. The different symbols distinguish data from different runs. The line is fit 27 from Table V - l . 95 (n c = 1.049) means that fewer interference fringes per unit reduced temperature are observed; this limits how close to Tc accurate data can be obtained. These data were fitted to eq. (1-6) in the same way as were the ethylene data. Table V - l shows the results of several fits to the individual data sets and to all seven sets combined. For two sets with tmax = 1.4 x 1 0 - 3 and 1.3 X 1 0 - 3 respectively, a simple power law described the data adequately. The other sets required two correction terms; setting B% = 0 resulted in a significant increase in the x 2 value for the fit. Since the individual data sets have relatively few data points, especially at low t, the uncertainties in the fit parameters due to the uncertainty in Tc are large. Typically, 0 varied by ±0 .008 and Bo by ±0 .07 when Tc was varied by ± 0 . 0 0 0 2 K. Simply averaging the values of the parameters obtained from these fits with the best values of Tc gives 0 = 0.324 ± .002, B0 = 1.17 ± .03, Bi = 1.4 ± .3 and Bi = —2.4 ± 1.3. One data set gives Bi = —4.5, which is quite different from the other results; excluding this value from the average gives Bi = —1.9 ± .6. The uncertainties here are standard deviations. The most extensive data set (250287.4) and a fit to it (no. 16 from the table) are shown in fig. (5-3); residuals for this fit are plotted in fig. (5-4). The results of fits to the seven data sets combined are also given in Table V - l . We find from these fits that 0 = 0.327 ± .004, B0 = 1.20 ± .04, By = 0.98 ± 0.18 and Bi — —1.1 ± .4. Here the errors are twice the statistical uncertainties from the fits. These values all agree within error with the results given above for the individual data sets. Fit 27 from the table is shown in fig. (5-2). In an effort to determine the effect of the small systematic differences between the various runs, the individual data sets were scaled slightly so that they overlapped more exactly. A fit to this scaled data is shown in the table; the parameters are slightly but not significantly different from those for the unsealed data. The combined data were also fitted to the corrections-to-scaling series with A a free parameter. Results of these fits are given in Table V—2. The unsealed Table V - l Fits of the hydrogen order parameter data to eq. (1-6) with A = 0.5 data set fit# 0 Bo B2 211186.1 1 0.3255 1.229 (0) (0) tmax = 1.3 X 10 - 3 2 (0.325) 1.225 (0) (0) tmin = 4.9 x IO"5 3 (0.327) 1.243 (0) (0) 171286.1 4 0.3265 1.168 1.67 -4.51 5 (0.325) 1.153 1.78 -4.85 tmax = 1.6 X 10 - 2 6 (0.327) 1.173 1.64 -4.40 tmin = 2.6.x 10"4 7 (0.327) 1.204 0.88 (0) 250287.1 8 0.3216 1.129 1.48 -2.73 9 (0.325) 1.161 1.30 -2.28 tmax = 3.2 x 10~2 10 (0.327) 1.181 1.19 -2.02 tmin = 9.0 X 10~5 11 (0.327) 1.203 0.74 (0) 250287.2 12 0.3262 1.175 1.09 -1.21 13 (0.325) 1.164 1.14 -1.29 tmax = 7.0 X IO"2 14 (0.327) 1.182 1.06 -1.16 tmin = 7.9 x IO"5 15 (0.327) 1.210 0.68 (0) 250287.4 16 0.3248 1.177 1.13 -1.68 17 (0.325) 1.178 1.12 -1.67 tmax = 6.1 X IO"2 18 (0.327) 1.198 1.03 -1.49 tmin = 3.2 X IO"5 19 (0.327) 1.225 0.60 (0) 070487.1 20 0.3208 1.135 1.35 -2.06 21 (0.325) 1.177 1.12 -1.57 tmax = 4.2 X 10~2 22 (0.327) 1.197 1.01 -1.34 tmin = 9.2 x IO"5 23 (0.327) 1.218 0.66 (0) 070487.2 24 0.3210 1.183 (0) (0) tmax = 1-4 X IO"3 25 (0.325) 1.218 (0) (0) tmin = 4.0 X IO"5 26 (0.327) 1.237 (0) (0) all sets 27 0.3270 1.197 0.98 -1.11 28 (0.325) 1.177 1.07 -1.27 tmax = 7.0 X IO"2 29 (0.327) 1.196 0.98 -1.11 tmin = 3.2 X IO"5 30 (0.327) 1.218 0.65 (0) all sets, scaled 31 0.3283 1.214 0.91 -0.89 97 I Figure 5-3 A log-log plot of Ap*/t^ vs. reduced temperature for set 250287.4 of the hydrogen data. The line is fit 16 from Table V - l . 98 4 X 2-X *xx x" "Aw / f i ; x —2 I i i i i 11111 i i t 111111 i i i i 11111 i i i i i n n ' 1 M i n i to-6 icr5 icr4 icr3 icr2 ioH reduced temperature Figure 5-4 Residuals for the fit shown in fig. (5-3). data give A = 0.46 ± .02 for (3 = 0.327 ± .002 and, as with ethylene, the critical amplitudes decrease significantly from their values with A = 0.5. The scaled data give A = 0.44 ± .02, not significantly different from the unsealed result. T a b l e V - 2 Fits of the hydrogen order parameter data to eq. (1-6) with A a free parameter data set fit# Bo B2 A all data 32 0.248 0.370 3.63 -0.16 0.20 33 (0.325) 1.166 0.88 -0.70 0.44 34 (0.327) 1.190 0.86 -0.76 0.46 all data, scaled 35 (0.325) 1.169 0.81 -0.48 0.42 36 (0.327) 1.193 0.80 -0.53 0.44 99 V-2 The Coexistence Curve Diameter Fig. (5-5) is a plot of the coexistence curve diameter of hydrogen obtained from the image plane data. Data from six runs are shown. There are some system-atic differences between the various data sets but despite this, the data sets agree within acceptable scatter. Shifts in the recorded interference pattern due to, for example, changes in room temperature, have a large effect on the diameter data and probably cause the observed systematic differences. These data are strikingly different from the diameter data for ethylene and ethane presented in the previous chapter, and for other fluids 7 3' 7 4. A critical anomaly is again present for t < 2 x 10 - 3 , but in this case it is larger (~ 0.5% compared to 0.1-0.2% for other fluids) and in the opposite direction, i.e., the di-ameter bends towards higher densities as the critical point is approached. It is shown in ref. (74) that, for fluids having a diameter anomaly in this direction, the liquid and vapour compressibilities should obey the inequality Our compressibility data (see below) clearly show this to be the case for hydrogen. These results are consistent with the presence of attractive many-body interactions in fluid hydrogen, possibly quantum mechanical exchange interactions. This idea is discussed further in Chapter V I . The hydrogen diameter data were analysed in the same way as the data for the other fluids. For consistency, fits to the form •T,t *— — K, * — < 0. (5-1) Pt + Pv - 2pc 2pc = AQ + Ait 1-a + A2t (5-2) were done over the inner, outer, and entire t ranges as defined in section IV—1, but because of the distribution of the data points and the nature of the anomaly in this case, fits were also done over broader ranges to include the several data points 100 0.015 0.010-0.005-I •o 0.000-hydrogen • 250287.1 * 250287.2 o 250287.6 * 260387.1 • 260387.2 ' 070487.1 X X fi * « — -0.005--0.010 3 0.5- • ?> 0 -0.5 0.000 0.004 0.008 0.00 0.01 0.02 0.03 0.04 reduced temperature 0.05 F i g u r e 5-5 T h e coexistence curve diameter of hydrogen as a function of reduced temperature. T h e different symbols distinguish data from different runs. T h e inset shows the deviations of the near-critical d a t a f r o m a straight line fit to the dat a away from c r i t i c a l . 101 close to T c , and to get a better value of the diameter slope away from the critical point. The results of fits to the data set 250287.2, which covers the largest range of t, are given in Table V -3 . Fits to the other data sets and to all of the data together were qualitatively the same but were slightly affected by the systematic effects mentioned above. Table V—3 Fits to the Coexistence Curve Diameter of Hydrogen range of t Ao A2 x 2 outer -0.00492(35) (0) 0.315(25) 0.73 inner -0.00377(34) -0.136(149) (0) 1.81 inner -0.00381(31) (0) -0.255(284) 1.82 entire -0.00466(10) 0.183(8) (0) 1.68 entire -0.00451(9) (0) 0.283(12) 1.36 entire -0.00406(16) -0.519(157) 1.080(242) 1.02 0.008-0.04 -0.00537(17) (0) 0.344(7) 0.88 0-0.002 -0.00203(40) -0.859(215) (0) 9.58 0-0.002 -0.00217(39) (0) -1.642(443) 10.30 0-0.002 -0.00009(40) -17.1(2.8) 32.2(17.4) 2.97 0-0.04 -0.00428(22) 0.200(10) (0) 14.22 0-0.04 -0.00410(20) (0) 0.292(13) 11.67 0-0.04 -0.00310(23) -0.914(157) 1.612(718) 6.81 From a fit to the data in the range 0.008 < t < 0.04 we find the slope away from critical to t>e 0.344 ± .007. The intercept of this straight line fit is Ao = —0.0054 ± .0002, indicating that the critical density obtained by extrapolation of the rectilinear diameter is 0.54 % too low; the critical density of hydrogen implied by this result is pc = 0.03160 g/cm 3 . Data in the temperature range 0 < t < 0.002 are reasonably well described by the functional forms AQ + A2i and Ao + Ait1-01, with the latter slightly preferred on the basis of the x 2 value, but much better described by an expression including 102 0.015-1 0.010-0.005-I 0.000--0.005-hydrogen * 250287.2 -0.010-1 0.00 0.01 0.000 -I -0.005-r r 0.000 0.004 —I— 0.02 0.03 0.04 reduced temperature 0.008 0.05 F i g u r e 5-6 The coexistence curve diameter of hydrogen from data set 250287.2. The solid line is a straight line fit to the data away from the critical point, the dashed line is a three-parameter fit to the data close to critical and the dotted line is a three-parameter fit to all of the data. The inset shows the near-critical data in more detail. 103 both t and t terms. A fit to this expression does not extrapolate well to higher t, though, and fits to the entire data set show strong systematic deviations from the data in the near-critical region. Inclusion of a correct ions-to-scaling term propor-tional to £ 1 - a+A d o e s n o t improve the situation. Fig. (5-6) shows the data from set 2 5 0 2 8 7 . 2 along with three fits — the straight line fit away from the critical point mentioned above, and fits with both t and f 1 - a terms over the ranges 0 < t < 0 . 0 0 2 and 0 < t < 0.04. It is clear that the fit over the large temperature range does not describe the observed anomaly adequately. This may be another example of crossover effects. V - 3 The Compressibility As mentioned above, our compressibility measurements were strongly affected by the presence of thermal gradients in the hydrogen sample. For example, values of the amplitude TQ determined from two different runs differed by a factor of three and were 3-10 times larger than values of TQ for other fluids. This implies that the gravitationally induced density gradient was enhanced by a thermally induced gradient, and so that the cell bottom was colder than the top. Despite this, a fair amount of information can be extracted from our data. The total density gradient in the sample is the sum of density gradients due to gravity and to thermal gradients. Thus ( 5 - 3 ) where we have used eq. (3-15). Rearranging, we get 104 Pc dp p\g d z ' ^ f f - ^ y 1 * pcgdT 1 dP (5-4) P~Pc which shows that analysis of the image plane fringe spacing yields an effective com-pressibility K*ejp which depends on dT/dz. Assuming that the quantity in paren-theses in eq. (5-4) is constant, or at least slowly varying, in the critical region, /c* .^ will have the same t dependence K *± ± r 7 + (5-5) This implies that the amplitude ratio r+ o,eff 1 0 r (5-6) o,eff which is universal. This shows that our data can provide the exponent 7 and the amplitude ratio TQ /TQ despite the presence of temperature gradients. In addition, if the true compressibility is known, eq. (5-4) can be used to obtain an estimate of the size of the gradient. We begin by calculating dT/dz, assuming it to be uniform over the sample. Our best compressibility data set (see below) was obtained below Tc; it gives PQRJJ — 0.10. Tables of the thermophysical properties of H 2 7 T give dP/dT\p~Pc — 1.89 x 1 0 5 J / m 3 K at the critical point. Assuming that TQ for H 2 is equal to 0.014, the value obtained by Pestak2 for HD, we find from eq. (5-4) that 105 o,eff - 0.010 K / m P=Pc (5-7) which corresponds to a temperature difference of 0.0002 K between the top and bottom of the cell. This is by most standards a small gradient. We turn now to the compressibility data itself. Unfortunately not much super-critical data was obtained due to the long times required (typically several hours) for the fluid to achieve hydrostatic equilibrium following a change in temperature. Good data was, however, obtained in the two-phase region. The best data was recorded with the interferometer reference beam (see fig. (3-12)) slightly misaligned. This caused a number of fringes to be present in the interferogram even with a uniform sample, and so the fringe spacing could be accurately measured over a reasonable range of t. Without this misalignment, the fringes became too widely spaced to measure for t > 1 x 10 - 4 . Fig. (5-7) shows compressibility data from several runs for which the temper-ature gradients, characterized by the outer jacket temperature and the heat input to the cell, were approximately the same. The critical temperatures used in analysing these data were obtained from the coexistence curve analyses presented in section V—1 above. It is clear from the figure that /c* .^ is larger in the vapour phase than in the liquid phase, as mentioned above in connection with the diameter results. A critical divergence of the difference K^V — T, depending on temperature like t^~l is predicted as a consequence of thermodynamic field-mixing71; our data are not precise enough to show this clearly. The two-phase compressibility data were fitted to the critical power law (5-5) both with 7 free and fixed at its theoretical value of 1.24. The results of these fits are given in Table V-4. From fits to the liquid and vapour data together, 106 Figure 5-7 The effective compressibility of hydrogen vs. reduced temperature. The solid and dashed lines are fits 6 and 8 from Table V-4 to the sub- and super-critical compressibilities respectively. 107 Table V-4 Fits of the hydrogen compressibility data to eq. (5-5) data set fit# 1 Fo,eff o,eff vapour 1 1.231 0.133 — 2 (1.24) 0.122 — liquid 3 1.135 0.223 — 4 (1.24) 0.082 — both phases 5 1.192 0.161 — 6 (1.24) 0.102 — 260387.1 7 1.192 — 0.894 >TC 8 (1.24) — 0.532 250287.6 9 1.125 — 0.456 10 (1.24) — 0.113 we find with 7 free that r~ „ = 0.16 ± .02 and 7 = 1.19 ± .10; with 7 = 1.24, ' o,ejj ' i i i r~ ejj  = 0.102 ± .006. The uncertainties here are approximate and take no account of uncertainties in TC. Fit 6 from the table is shown in fig. (5-7). Results of fits to the liquid and vapour data separately are also given in the Table V—4. Fig. (5-7) also shows some data for the supercritical compressibility, taken during one of the runs from which the plotted subcritical compressibility was ob-tained. Fits to this data are also given in the table and fit 8 is shown in the figure. We find r~ „ = 0.89 ± .02 and 7 = 1.19 ± .10; with 7 = 1.24, T~ „= 0.532 ± .023. Taking the values of obtained with 7 fixed at its theoretical value, we find the amplitude ratio r+ = 5.2 ± .4. (5-8) Fo,eff Also given in the table are the results of fits to a set of supercritical data from a different run. In this case rt „ = 0.113 ± .002 when 7 = 1.24. The amplitude 108 here is much lower than that for the first data set, demonstrating the effects of changing the temperature gradient. 109 CHAPTER VI Discussion In this chapter the results presented above are discussed both in relation to previous results for the particular fluids and in terms of the general picture of critical phenomena in fluids. Evidence for quantum mechanical effects from our hydrogen data is considered, and, finally, some limitations of the corrections-to-scaling approach to the analysis of critical behaviour are discussed. VI-1 C o m p a r i s o n w i t h P r e v i o u s W o r k The thermophysical properties of both ethylene and hydrogen have been extensively studied because of the important industrial applications of these fluids. Most previous work, however, has been done relatively far from the critical point and so is not directly comparable with the present results. Some P V T studies of ethylene have extended into the critical reg ion 7 6 ' 7 8 ' 7 9 . The measurements of Hastings et al.76 reach a minimum reduced temperature of 7 x 1 0 - 4 ; their coexistence curve results over the range 7 x 1 0 - 4 < t < 1 0 - 2 give an effective critical exponent 0eg = 0.338 when fitted to a simple power law. This is higher than the expected asymptotic value of 0 due to neglect of corrections-to-scaling effects. When their data are fitted to a scaled equation of state that includes one corrections-to-scaling term 1 , they find BQ = 1.56 ± .03 and B\ = 1.06 ± .02, assuming 0 = 0.325 and A = 0.5. This value of BQ agrees well with the prism results in Table IV-2, and the value of B\ is in agreement with both sets of ethylene results. 110 Douslin and Harrison 7 8 find /? = 0.35 from a simple power law fit to coexistence curve data in the range 1.4 x 10~3 < t < 1.4 x 10 _ 1 . Levelt Sengers et a/. 8 0 have analysed a number of experimental results for ethylene. Fitting these to a scaled equation of state as above they find Bo — 1.54 and B\ — 1.22 for the same exponent values; again these results agree within error with the present results. Both of the above analyses rely on fitting the data to a given form for the equation of state and assuming specific values for the critical exponents. In the present work no assumptions were made concerning the exponent values; they were obtained as parameters from our fits. From the data tabulated in ref. (80), the slope of the coexistence curve diam-eter away from the critical point can be calculated to be 0.79. This is somewhat lower than our value of 0.88 ± .04. The difference may be due to the different temperature ranges and analysis procedures used in the two cases. The critical temperature of ethylene determined in ref. (76) is 282.3452 ± .0017 K, obtained as a parameter in a fit to the above equation of state. Moldover8 1 obtained a value of 282.344 ± .004 K by visual observation of the disappearance of the meniscus. Our value of 282.3750 ± .0010 K obtained from the prism experiment is slightly higher than these results, but agrees with that of Thomas and Zander 7 9, who find Tc = 282.37 ± .02 K. The difference may be due to differences in sample purity, in data analysis procedures, or, possibly, to an unknown systematic difference in temperature scales. Measurements of the thermophysical properties of hydrogen, while extensive away from the critical region 5 7 ' 7 7 ' 8 2 , have apparently not been thouroughly analysed using a scaled equation of state that includes corrections-to-scaling terms. Estimates of critical amplitudes tabulated in the literature 3 2' 8 4 give BQ = 1.28 and 1.35 for pure P-H2. Fig. (6) of ref. (69) gives Bo = 1.65; this value comes from an unpublished pure power law analysis of the data of Goodwin et al 8 3 . This estimate is higher than the present result, BQ = 1.19 ± .03, no doubt due to the different analysis Ill procedures and to neglect of corrections-to-scaling. An estimate of r+ = 0.093 is given in ref. (84). The diameter of the coexistence curve away from critical can be calculated from the tabulated data of Roder et a/. 7 7, giving A2 = 0.347. Fig. (6) of ref. (69) gives a value of 0.37. Our value of 0.344 ± .007 is in good agreement with both of these results. As discussed in Chapter V , our values of TC for hydrogen are in agreement within error with the expected value for equilibrium H2. V I - 2 Universal Cri t ica l Parameters Table V I - 1 summarizes our results for the various critical exponents, am-plitudes and the amplitude ratio TQ /TQ , along with the appropriate theroetical values, if any. We find /? = 0.327 ± .002 for ethylene and 0.326 ± .003 for hydro-gen. These agree within experimental error with each other, with results for other fluids2-4, and with the theoretical value o f 5 - 9 0.327. This supports the widely-held belief in critical point universality in fluids, and the belief that the liquid-vapour critical point belongs to the universality class of the 3-D Ising model. The hydrogen result demonstrates that a fluid for which quantum effects may be significant in fact belongs to the same universality class as do other fluids. For the correction exponent A we find A — 0 .46±.04 for ethylene and 0 .46± .02 for hydrogen. These two results agree within error. They are slightly but proba-bly not significantly lower than previously measured values2 and the theoretical value 5' 7' 2 8' 2 9 of close to 0.5. These results confirm the universality of this exponent within our experimental error. Our value of 7 from the hydrogen compressibility data is less precise and also less reliable due to the problems with thermal gradients. We find 7 = 1.19 ± .05, which agrees within error both with previous results 2' 4' 1 0 and theory 5 - 9 , again confirming the predictions of critical point universality within our experimental 112 Table V I-1 Summary of Results C 2 H 4 H 2 Theory 282.3750 ± .0010 32.984 ± .013 0.327 ± .002 0.326 ± .002 0.327 1 — 1.19 ± .05 1.238 A 0.46 ± .04 0.46 ± .02 0.50 — 5.2 ± .4 4.80, 5.07 Bo 1.56 ± .03 1.19 ± .03 Bj 0.98 ± .10 0.98 ± .09 B 2 -1.9 ± .3 -1.1 ± .2 A 2 0.88 ± .04 0.344 ± .007 error. The universal ratio TQ /TQ was found to be 5.2 ± .4. This result is also somewhat unreliable due to the effects of thermal gradients, but nonetheless agrees within error with theoretical predictions44 and previous determinations2. The fact that these two quantities are in agreement with their predicted values suggests that the assumptions made in using eq. (5-4) are justified. VI-3 N o n - U n i v e r s a l P r o p e r t i e s a n d Q u a n t u m E f f e c t s T h e C o e x i s t e n c e C u r v e a n d t h e C o m p r e s s i b i l i t y The critical coexistence curve amplitudes determined from our data are sum-marized in Table V I-1. Young's theory of quantum effects near the critical point 4 0 predicts systematic variations of these amplitudes as quantum mechanics becomes more important. To investigate this, the amplitudes J50 and B\ from this work and for other fluids are plotted as a function of the dimensionless quantum parameter 113 Ar/£o> where Ar is the thermal De Broglie wavelength and £o is the bare correlation length, in figs. (6-1) and (6-2). Values of B 0 for the fluids 3 He, Ar, K r , H 2 0 , D 2 0 , C O 2 , SF6 and iso-C 4Hio were taken from Table 1 of ref. (1), as were values of Co-Values of BQ for Ne, N 2 and HD were taken from ref. 2, and for Xe, GeH4 and C 2 H6 from refs. (30), (31) and (85) respectively. The B\ values plotted in fig. (6-2) for the fluids H 2 0 , D 2 0 , i so -C 4 Hi 0 , C 2 H 4 , HD, Ne, N 2 , 3 He, Xe, GeH 4 and C 2 H 6 are from refs. (1,2,3,30,31 and 85). Some of these data come from corrections-to-scaling fits, others from fits to an equation of state, but all come from analyses involving at least one correction-to-scaling term in some form. There is a considerable amount of scatter in the Bo data for the various "clas-sical" fluids, clustered at low values of Ay/Co in fig- (6-1). This may be due in part to the different analysis procedures used in the different cases, but may also reflect the predicted non-universality of Bo and its dependence on factors affecting the in-termolecular interactions. It is interesting that the points for water and heavy water are significantly higher than those for the other classical fluids, which are clustered in the range Bo = 1.4-1.65. These other fluids have roughly spherical molecular symmetry, whereas intermolecular interactions in the waters will presumably be influenced by hydrogen bonding; this may account for their higher Bos. The low temperature fluids Ne, HD, H 2 and 3 He show a clear decrease in Bo with an increase in the quantum parameter, in at least qualitative agreement with the predictions of ref. (40). The heavy line plotted in fig. (6-1) shows the predicted trend, Bo = Bg(l-y)fi, (6-1) where the "classical" amplitude BQ has (arbitrarily) been set equal to 1.437 so that the theoretical curve passes through the data point for neon. The fine line is a straight line, Bo = 1.514(1 - 0.134^), (6-2) Co 114 o F i g u r e 6-1 T h e coexistence curve amplitude, BQ, for a number of fluids as a function of the quantum parameter \T/ZO- Results from the present work are shown as open circles. T h e heavy line shows the behaviour predicted in ref. (40); the fine line is a straight line fit to the four d a t a points with \T/£O > 0.75. Figure 6-2 The first correction amplitude, J5j, for a number of fluids as a function of \T/£O- Results from the present work are shown as open circles. The line shows the behaviour predicted in ref. (40). 116 where the numerical factors were obtained from a least-squares fit to the data for the four low-temperature fluids. It is amusing that the straight line describes the data better than the predicted form does, but the physical content of this fact is not clear. The B\ data plotted in fig. (6-2) are less precise than the corresponding data for the leading amplitude. The predicted trend, is shown by the solid line. B\ has been set equal to 1.033, again to fit the neon data. While the predicted trend seems to describe the data within the error bars, the agreement is not very impressive, and the large error bars make it difficult to make any quantitative statements. Indeed, it is almost fair to say that B\ = 1 describes the data within experimental error. Fig. (6-3) is a plot of values of vs. XT/(O for the thirteen fluids tabulated in ref. (l). The data are clustered around r£ = 0.055 with the exception of the point for 3 He, which is a factor of three higher. The trend predicted by Young's theory, is also shown; TQ has been set equal to 0.053. The data appear to follow this trend at least qualitatively, but this appearance is based entirely on the high value of T Q for 3 He; the other data are essentially constant. A good value for H 2 would obviously have been helpful in assessing the effects of quantum mechanics on the compressibility. The values of ^ t g obtained from this work are not shown in the figure; they lie well above the theoretical curve. Bx = B\l{\-y)*, (6-3) rtcl(i-y)-\ (6-4) 117 o Figure 6-3 The leading compressibility amplitude, r£, for a number Ar/Co- The line shows the behaviour predicted in ref. (40). 118 T h e Coexistence Curve Diameter The behaviour of the coexistence curve diameters of several fluids (including ethane and ethylene) has been analysed recently by Goldstein, Pestak et al73'74. They explain the origin of the thermodynamic field-mixing that leads to the presence of the tl~a anomaly in terms of many-body interactions in the fluid. By modifying the free energy of a V D W gas to include a term due to three-body interactions, they predict an anomaly of the correct form. This process introduces a new energy scale into the system, characterized by a dimensionless parameter x — q/ab where q is the integrated strength of the three-body potential and a and b are the usual V D W parameters, x is thus the ratio of three-body to two-body interaction strengths. For the special case of the primarily repulsive Axilrod-Teller triple-dipole interaction86, x is proportional to the product, apc, of the polarizability and the critical density. Pestak et al.74 derive expressions for various thermodynamic quantities from this model. In particular, they find that the diameter slope is 2 22 * = S + i 5 * + - ( ^ ) and that the order parameter amplitude and the slope of the diameter are related by Fig. (6-4) is a plot of the diameter slope as a function of apc for the six fluids analysed in ref. (74) (SF6, C 2 H 4 , C 2 H 6 , N 2 , Ne and HD) as well as for hydrogen. The first six fluids show the predicted linear relationship. A least-squares fit to these data gives Ai = 0.41 + 25.2apc. (6-7) This fit is plotted in the figure; the intercept is in surprisingly good agreement with the V D W value of 0.4. The point for H 2 is clearly significantly below the line defined 119 by the other fluids. A fit to all seven points still lies well above the hydrogen point and describes the rest of the data less well. In fig. (6-5), Bo is plotted against Ai for the same fluids. Here the predicted linear relationship is followed by all of the data. A least-squares fit gives Bo = (0.892 ± .011) + (0.826 ± .005) A2. (6-8) As mentioned in Chapter V , the critical anomaly in the hydrogen diameter is in the opposite direction to those for other fluids, and the compressibility in the vapour phase is larger than that in the liquid. The fact that the hydrogen data fit the linear relation between BQ and A2 suggests that the three-body interaction model remains valid for hydrogen, but fig. (6-4) implies that the relevant interactions in this case are not of the Axilrod-Teller type. In addition, the diameter slope for H 2 is slightly lower than the V D W value of 0.4. Eq. (6-5) thus says that the parameter x is negative for hydrogen; i.e., that the three-body interactions in hydrogen are attractive. While no critical anomalies in the diameters of the helium isotopes have been observed, their diameter slopes are substantially lower than the V D W value, indicating even stronger attractive interactions in these fluids69'74. It is known that three-body exchange interactions, which are primarily attractive, are important in helium. Our data provide evidence that such an attractive interaction is also significant in hydrogen. V I - 4 C r o s s o v e r E f f e c t s There are some minor problems with the corrections-to-scaling fits to the coexistence curve of ethylene and to the coexistence curve diameter of hydrogen. For example, the fitted functions in figs. (4-4) and (4-13) turn over and start to decrease above a certain temperature because the highest order term in the fit is negative. The beginnings of similar behaviour are seen in the plots for hydrogen. This behaviour is clearly unphysical. In addition, there are small but systematic 120 Figure 6-4 The coexistence curve diameter slope, A2, for several fluids vs. apc. The line is a fit to the data excluding the point for H 2 . Figure 6-5 The coexistence curve diameter slope, A2, vs. the coexistence curve amplitude, Bo, for a number of fluids. The line is a fit to the data. 122 differences between the fitted function and the best fit as determined by eye in the ethylene fits. Inclusion of additional terms, of order tZA and higher, to the series could improve the quality of the fits over a given temperature range. The physical meaning of these terms soon becomes lost, however, since the full corrections-to-scaling series contains several terms of order £ 1 - 5 (t3A, f 1 + A , etc.), r 2 , and so on. In fact, even the meaning of our t2A term is obscure. A similar, but more obvious, problem arose in fitting the hydrogen diameter data. While the theoretical form described the data well over a restricted temper-ature range, it could not simultaneously describe the anomaly close to the critical point and the diameter away from critical. The problem in both cases is that the corrections-to-scaling series do not converge to any classical limit away from the critical region. In fact, it is not clear that the series converge at all. Since a classical description of fluid behaviour is correct far from the critical point, there is clearly a missing ingredient in the theoretical description. Recently, much effort has been directed towards the developement of physi-cally reasonable "crossover" theories — descriptions of fluid behaviour that cross over from scaling behaviour in the critical region to classical behaviour away from critical 1. Early efforts in this direction predicted unphysical behaviour in the crossover region 8 7. More recently, however, a formalism based on RG ideas has been developed by Albright and co-workers88. It includes a crossover function that depends on two parameters — one related to the wavelength cut-off of the RG transformation, i.e., the size of the smallest important fluctuations, and one related to the rate of convergence of the appropriate corrections-to-scaling series. This for-malism has been used successfully to describe measurements of the specific heat of carbon dioxide8 9. 123 It is probable that our data, which extend from very close to the critical point to several degrees away from it, cover a large enough range that the corrections-to-scaling approach is being stretched close to the limits of its applicability in our analysis. It would be of interest to reanalyse our data in terms of such a crossover theory. 124 CHAPTER VII Conclusions In this work we have presented measurements of the coexistence curve and its diameter near the critical points of hydrogen and ethylene, and measurements of the compressibility of hydrogen in the critical region. Our results are in good quantitative agreement with the predictions of the theory of critical phenomena and support the idea of critical point universality in fluids. Our values of the critical exponents /? = 0.327 ± .002 and 7 = 1.19 ± .05 agree with the theoretical values. Our result for the corrections-to-scaling exponent A = 0.46 ± .02 is slightly lower than the predicted value of 0.5, but was the same within error for both fluids. The hydrogen results show evidence for the presence of quantum mechanical effects. The coexistence curve amplitudes BQ for H 2 and other fluids decrease as quantum effects become more important, as predicted. On the other hand, no systematic trend in the correction amplitudes B\ was observed within experimental error. The data for the coexistence curve diameters of ethylene, hydrogen and ethane show the importance of many-body effects near the critical point. The hydrogen results, in particular, are consistent with the presence of attractive many-body interactions, possibly quantum mechanical exchange. It would be useful to obtain better compressibility data for hydrogen using a sample cell which allowed more complete elimination of thermal gradients. This 125 would permit a better assessment of the effects of quantum mechanics on the com-pressibility amplitudes. It would also be interesting to extend this work to lower critical temperatures and presumably larger quantum effects by studying 4 He. The data quality for such a study using these techniques would be limited, however, by the low refractive index of helium. 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B 36, 877 (1987). 132 APPENDIX A Conversion of Refractive Indices to Densities In this Appendix, eqs. (3-6) and (3-7), used to convert refractive index-related quantities to density-related quantities in the analysis of the fringe data, are derived. We deal first with the calculation for the order parameter Ap*. We start by expanding n — nc in a Taylor's series about p — pc for both liquid and vapour phases: where the derivatives are evaluated at the critical point. Taking the difference between these two equations gives an expression for n.£ — nv, which is the quantity determined from the experimental fringe counts: nv — nc ni — nc (A-1) (A-2) n t - nv = — (Pi - pv) + (A-3) 133 where . * Pt~ Pv , . Pt + Pv - 2pc Ap = — — — and pd = -2Pc 2pc Solving eq. (A-3) for Ap* gives Ap* = ni — nv 2pc (dn/dr + pcpd<iPn/dp2) (3-6) The evaluation of the derivatives of n will be treated below. In the case of the coexistence curve diameter p*d, we add eqs. (A-l) and (A-2) to get . . . . dn , . 1 d2n r. .2 / >2l / * (nt - n c) + (n„ - n c) = — (pt + pv - ^P^ + Ydp2 VPt ~ Pc' + *Pv ~ Pc> \' ^ ' But to first order in p — pc, n t ~ n c = ~d~p ^Pt ~ Pc^ n" ~~ n° ~ ~d~p ^Pv ~ Pc^' ^A_6^ Substituting the expressions (A-6) into the second term of eq. (A-5) gives / x dn . n , \d2n(dn\ 2 \ . . 2 / \2 (nt + n v - 2nc) = — (pt + pv - 2pc) + I ^ ) [ ( n* ~ n c ) + ( n » -(A-7) so (nt + nv - 2nc) - \ d2n/dp2 (dp/dn)2 J^ (n/ - n c ) 2 + (n„ - n c ) J p* = 2pc dn/dp (3-7) The quantities — n c and nv — nc are determined from the image plane fringe counts. 134 To determine dn/dp and d2n/dp2, we write n as a function of p and £ using eq. (1-11): 2p£Vf2 pt) ' (A-9) Then dn dn dn A l dp = ~dp + d£ (A-10) and d?n dp2 d % n ~„l ^ n /iff &n I /if\2 (A- l l ) where £ f = d£/dp\c and £ " = d2£jdp\ are obtained from a fit to £{p) on the coexistence curve. The various partial derivatives required are calculated in a straightforward manner. In the experessions below, the relation Pc ( « 2 + 2) has been used to write these expressions in terms of n c and p c . The results are dn dp dn dl d2n dp2 d2n dpd£ (n 2 - 1) (n 2 + 2) 6n cp c Pc (n 2 + 2) 2 6ne (n 2 - I ) 2 (n 2 + 2) (3n2 - 2) 36n 3p 2 (n 2 + 2) 2 (3n* + n2 + 2) 36M? (A-12) (A-13) (A-14) (A-15) and d2n 6\C2 p 2 ( n 2 c + 2 f ( S n 2 c - 2 ) 36nf (A-16). 136 APPENDIX B Data Listings 137 B - l Lorenz-Lorentz Coefficient for Ethylene £(p) at 298.2 K L i s t i n g o f - 0 a t 1 5 : 2 4 : 3 0 o n S E P 1 2 . 1987 f o r CC1d>DBRU 1 C E L L MASS • 1 6 8 . 7 7 0 0 C E L L V O L . • 1 2 . 0 0 5 0 2 MOLECULAR WT • 2 8 . 0 S 4 0 4 A L P H A ' 2 0 . 4 1 7 0 5 MICROMETER F I T P A R A M E T E R S : - 0 . 3 7 7 3 3 1 6 1 0 O E - O 4 - 0 . 6 5 4 3 9 9 B O O O E - 0 2 0 . 0 6 N O . OF P T S . • 21 7 B C a l l • a s s M 1 c . RdO A n g l e O a n s 1 t y n L L c o e f t B <B> ( d a g ) ( n o 1 a / c m 3 ) ( c a 3 / a o 1 e ) 10 11 1 9 0 . 4664 S . . 4235 0 . .030022 0 . .005037 1 .O81801 10. 6 6 4 2 1 5 12 1 9 0 . 4031 5. . 6023 0 . 028851 0 . O 0 4 8 4 9 1 .078692 10. 6 6 3 1 1 5 13 1 9 0 . 3 4 1 0 S , . 7743 0 . 0 2 7 7 2 5 0 . 004665 1 . 0 7 5 6 9 9 10. 6 6 9 4 3 5 14 1 9 0 . 2 8 8 0 5. . 9220 0 . 0 2 6 7 5 9 0 O045O7 1 .073128 10. .672388 15 1 9 0 . 2258 6. .0968 0 . .025614 0 .004323 1 .070084 10. .671582 16 1 9 0 . 1640 6. 2725 0 . .024464 0 . . 004139 1 . 0 6 7 0 2 3 10 .664354 17 1 9 0 . 1009 6, . 4495 0 . .023306 0 .003952 1 . 0 6 3 9 3 8 10 .662263 18 1 9 0 . 0331 6. 6 3 8 0 0 . 0 2 2 0 7 2 0 . 0 0 3 7 5 0 1 .060651 10 .663856 19 1 8 9 . 9 6 6 8 6 . . 8213 0 . .020872 0 , .003554 1 . 0 5 7 4 5 3 10 .667821 26 1 8 9 . 8 9 9 2 7 . . 0113 0 . .019628 0 . 0 0 3 3 5 3 1 .054137 10 .660647 21 1 8 9 . 8 2 1 3 7 . .2278 0 . .018211 0 .003122 1 . 0 5 0 3 5 7 10 .658643 2 2 189 . 7591 7 . 3 9 8 0 0 . 017O97 0 . 002937 1 .047384 10 .665971 23 1 8 9 . 7 0 2 3 7 , . 5613 0 . .016028 0 . 002768 1 . 0 4 4 5 3 0 10 .639908 24 1 8 9 . 6 4 1 3 7 . . 7285 0 . .014934 0 . 002587 1 .041606 10 . 643195 25 1 8 9 . 5844 7 . . 8823 0 .013928 0 . 002418 1 .038916 10 .655876 26 189 . 5214 8 . 0 6 5 0 0 .012732 0 .002231 1 . 0 3 5 7 1 9 10 .606764 27 1 8 9 . 4 6 1 8 8 . 2245 0 . 011688 0 .002054 1 .032927 10 .625418 28 1 8 9 . 4 0 4 7 8 , . 3838 0 . . 010646 0 0O1B85 1 . 0 3 0 1 3 7 10 .605461 29 1 8 9 . 3408 8 . 5 5 4 0 0 . .009464 0 . 0 0 1 6 9 5 1 . 0 2 6 9 7 5 10 .561324 3 0 1 8 9 . 2144 8 .8968 0 . 007193 0 . 0 0 1 3 2 0 1 .020892 10 .517488 31 1 8 9 . 1307 9 , . 1308 0 . . 005662 0 .001071 1 . 0 1 6 7 8 7 10 . 419365 138 o n t h e C o e x i s t e n c e C u r v e I s t l n g o f - 0 U T 2 ( 6 7 ) e t 1 4 : 1 5 : 5 0 o n J U L 1 2 . 1987 f o r C C I d ' D B R U 6 7 68 6 9 7 0 71 72 73 74 75 76 77 78 7 9 8 0 61 82 83 84 85 8 6 87 88 89 9 0 91 92 9 3 94 95 96 97 98 99 too 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 L o r e n z - L o r e n t z c o e f f i c i e n t d a t a f o r e t h y l e n e o n t h e c o e x i s t e n c e c u r v e . C E L L MASS • MOLECULAR WT ALPHA" 1 8 8 . 8 1 9 6 2 8 . 0 5 4 0 C E L L V O L . « 1 2 . 0 0 5 0 N O . OF > 2 0 . 7 3 3 0 IETER ! F I T P A R A M E T E R S : - 0 . 1 6 5 4 1 9 2 3 0 0 E - 0 4 - O . 6 5 2 8 7 7 0 2 0 O E - 0 2 0 . 0 ; PTS 59 C a l 1 mass M 1 c . Rdg. A n g l e D e n s t t y n L L c o e f f (0) ( d e g ) ( n o l e / c m 3 ) ( c m 3 / m o ! e ) 191 . 8524 - 8 . 6 0 5 0 . 0 5 6 1 9 4 0 . 0 0 9 0 0 5 1 . 148126 1 0 . 6 4 8 8 3 5 1 9 1 . 8273 - 8 . 555 0 . 0 5 5 8 6 8 0 . 0 0 8 9 3 1 1 . 147285 1 0 . 6 7 8 7 7 4 1 9 1 . 8 0 3 0 - 8 . 4 7 5 0 . 0 5 5 3 4 0 0 . 0 0 8 8 5 8 1 . 145922 1 0 . 6 6 9 4 1 3 1 9 1 . 7912 - 8 . 432 0 . 0 5 S 0 6 1 0 . 0 0 8 8 2 3 1. 145199 1 0 . 6 6 0 4 4 7 1 9 1 . 7 6 0 0 - 8 . 342 0 . 0 5 4 4 7 2 0 . 0 0 8 7 3 1 1 . 143678 1 0 . 6 6 4 3 9 3 1 9 1 . 7 4 1 9 - 8 . 289 0 . 0 5 4 1 2 8 0 . 0 0 8 6 7 7 1 . 142790 1 0 . 6 6 6 1 9 6 1 9 1 . 7 1 7 0 - 8 . 2 2 3 0 . O S 3 6 9 4 0 . 0 0 8 6 0 3 1. 141667 1 0 . 6 7 6 0 0 3 191 . 6 9 4 5 - 8 . . 160 0 . 0 5 3 2 8 2 0 . 0 0 8 5 3 6 1. 140602 1 0 . 6 8 1 2 2 7 1 9 1 . 6 6 0 0 - 8 . 0 4 6 0 . 0 5 2 5 3 8 0 . 0 0 8 4 3 4 1. 138678 1 0 . 6 6 7 6 1 3 191 . 6 4 5 8 - 7 . ,877 0 . 0 5 2 0 8 9 0 . 0 0 8 3 9 2 1. . 137517 1 0 . 6 3 4 1 5 0 191 . 6051 - 7 . .851 0 . 0 5 1 2 6 5 0 . 0 0 8 2 7 1 1. .135384 1 0 . 6 2 7 2 1 8 191 . 5 5 0 0 - 7 . . 703 0 . 0 5 0 2 9 4 0 . 0 0 8 1 0 7 1. . 132871 1 0 . 6 4 6 3 8 0 191 . 5 2 5 0 - 7 . . 6 3 0 0 . O 4 9 B 1 9 0 . O 0 8 0 3 3 1. 131642 1 0 . 6 4 8 2 8 4 191 . 4 9 5 3 - 7 . . 560 O . 0 4 9 3 6 0 0 . 0 0 7 9 4 5 1. .130452 1 0 . 6 7 1 9 4 7 191 . 4 6 3 9 - 7 . .471 0 . 0 4 8 7 7 9 0 . 0 0 7 8 5 2 1. .128947 1 0 . 6 7 7 6 0 5 191 . 4098 - 7 .314 0 . O 4 7 7 5 6 0 . O 0 7 6 9 1 1, . 126296 1 0 . 6 8 2 7 5 2 191 . 3773 - 7 , . 215 0 . O 4 7 1 0 7 O . O 0 7 5 9 4 1. . 124615 1 0 . 6 7 8 4 4 6 191 . .3414 - 7 . . 101 0 . 0 4 6 3 6 2 0 . 0 0 7 4 8 8 1. .122682 1 0 . 6 6 6 8 9 7 191 . 3002 - 6 .967 0 . 0 4 5 4 8 5 O . 0 0 7 3 6 5 1 . 120408 1 0 . 6 4 8 3 7 1 191 . .2552 - 6 .845 0 . 0 4 4 6 8 4 0 . 0 0 7 2 3 2 1 .118331 1 0 . 6 6 2 7 9 3 191 . .2306 - 6 . 776 0 . 0 4 4 2 3 5 0 . 0 0 7 1 5 9 1 .117164 1 0 . 6 6 8 0 4 2 191 . . 1865 - 6 .651 0 . 0 4 3 4 2 1 0 . 0 0 7 0 2 8 1 . 115O50 1 0 . 6 7 5 5 9 2 191 . . 1640 - 6 . 596 0 . O 4 3 0 6 3 0 . 0 0 6 9 6 1 1 . 114122 1 0 . 6 9 3 2 1 7 191 . . 1334 - 6 . 506 0 . 0 4 2 4 7 1 0 . 0 0 6 8 7 0 1. .112584 1 0 . 6 9 2 1 5 9 191 . . 1025 - 6 . 4 0 5 0 . 0 4 1 8 1 4 0 . O O 6 7 7 8 1 . 110876 1 0 . 6 7 6 3 7 7 191. 0 6 7 7 - 6 . 3 1 6 0 . 0 4 1 2 3 1 0 . 0 0 6 6 7 5 1 .109361 1 0 . 6 9 6 9 7 9 191. .0261 - 6 . 188 0 . 0 4 0 3 9 6 0 . O O 6 S 5 2 1 . 1 0 7 1 9 0 1 0 . 6 8 7 2 3 8 190 . 9873 - 6 . 0 7 3 0 . 0 3 9 6 4 4 0 . O 0 6 4 3 6 1 . 1 0 5 2 3 6 1 0 . 6 8 4 6 3 8 190. . 9 6 2 0 - 6 . 0 0 0 0 . 0 3 9 1 6 6 0 . 0 0 6 3 6 1 1 . 103992 1 0 . 6 8 5 8 1 7 190. .9164 - 5 .851 0 . 0 3 8 1 9 5 0 . 0 0 6 2 2 6 1 .101464 1 0 . 6 5 8 4 6 1 190. .8721 - 5 . 6 9 3 0 . 0 3 7 1 5 9 O . O O 6 0 9 4 1 . 0 9 8 7 6 9 1 0 . 6 0 5 2 1 6 190 .7972 - 5 . 5 0 5 0 . 0 3 5 9 3 2 0 . 0 0 5 8 7 2 1 . 095574 1 0 . 6 5 7 8 8 4 190. 7125 - 5 . 2 5 5 0 . 0 3 4 2 9 9 0 . O 0 S 6 2 0 1 . 0 9 1 3 1 8 1 0 . 6 4 8 3 1 6 190. . 6 1 7 0 - 4 . 9 6 9 0 . 0 3 2 4 3 3 . 0 . 0 0 5 3 3 7 1 .086451 1 0 . 6 2 7 0 1 3 190 .5062 -4 . 6 5 6 0 . 0 3 0 3 8 9 O . O 0 5 0 O 8 1 . 0 8 1 1 1 8 1 0 . 6 3 8 0 8 7 190. . 4 3 1 0 - 4 . 4 5 2 0 . O 2 9 O 5 6 0 . 0 0 4 7 8 5 1 . 0 7 7 6 3 5 1 0 . 6 6 3 8 7 0 190 .3312 -4 . 176 0 . 0 2 7 2 5 1 0 . 0 0 4 4 8 8 1 . 0 7 2 9 1 9 1 0 . 6 8 7 4 0 3 139 B - 2 E t h y l e n e C o e x i s t e n c e C u r v e D a t a P r i s m D a t a Listing of -M at 14:41:27 on AUG 16. 1987 for CCId-DBRU 1 4 Ethylene prism cell data. at 3 Number of data points 77 4 e C r i t i c a l Temperature 1 • 282 .3750 9 6 Temp (K) t rho V rho 1 rho d 7 o (mol/cmS) (mo1/cm3) (mo1/cm3) o 9 269. 83740 0. 4646E -01 0.0030741 0.0128468 0. 0079604 10 270. 25635 0. 4484E -01 0.0031176 0.0127714 0. 0079445 11 271. 49194 0. 4009E -01 0.0032741 0.0125655 0. 0079198 12 272. 80786 0. 3507E -01 0.OO34454 0.0123207 0. 0078831 13 272. 81299 0. 3505E -01 0.0034519 0.0123167 0. 0078843 14 273. 74341 O. 3153E -01 0.0035819 O.0121371 0. 0078595 15 273. 74878 0. 3151E -01 0.0035884 0.0121310 0. 0078597 16 274. 72900 0. 2783E -OI 0.0037527 O.0119309 0. O078418 17 275. 26099 0. 2584E -01 0.0038325 0.0118227 0. 0078276 18 275. 50342 0. 2494E -01 0.0038843 0.0117573 0. 0078208 19 275. 76074 0. 2399E -01 0.0039382 0.0117042 0. 0078212 20 276. ,01587 0. 2304E -01 0.0039813 0.0116552 0. 0078183 21 276. .53809 0. 2111E -01 0.0040804 0.0115244 0. 0078024 22 276. .85352 0. 1994E -01 0.0041536 0.0114426 0. 0077981 23 277. .07397 0. 1913E -01 0.0042009 0.0113915 0. 0077962 24 277. .44165 0. 1778E -01 0.0042718 0.0112728 0. 0077723 25 277. .62842 0. 1710E -01 0.0043190 0.0112278 0, .0077734 26 277. .97632 0. 1582E -01 0.0044027 0.0111317 0. .0077672 27 278. .19702 0. 1502E -01 0.0044585 0.0110682 0. .0077633 28 278, .36646 O. 1440E -01 0.0045121 0.0110047 0.0077584 29 278. .73633 0. 1305E -01 0.0046085 0.0108941 0. .0077513 30 278. .78296 0. 1288E -01 0.0046192 0.0108778 0. .0077485 31 279. .10742 0. 1171E -01 0.0047112 0.0107651 0 .0077381 32 279. .23047 0. 1126E -01 0.0047693 0.0107413 0. .0077553 33 279, .33545 0. 1088E -01 0.0047881 0.0106851 0. .0077366 34 279 .48584 0. 1034E -01 0.0048546 0.0106473 0. .0077510 35 279 .49048 0. 1032E -01 0.0048394 0.0106236 0. .0077315 36 279 .74365 0. 9406E -02 0.0049367 0.0105442 0. .0077404 37 279 .87671 o. 8926E -02 O.0049845 0.O104657 O. .0077251 38 279 .90332 0. 8830E -02 0.0049781 0.0104677 0 .0077229 39 280 .00562 0. 8462E -02 O.O050336 0.0104390 0 .0077363 40 280 .27002 0. 7511E -02 0.0051295 0.0103035 0, .0077165 41 280, .27026 0. 7510E -02 O.O0514O0 0.0103255 0.0077327 42 280. .43384 0. 6922E -02 0.0051955 0.0102317 O. .0077136 43 280. .53809 0. 6548E -02 0.0052494 0.0101907 0 .O0772O0 44 280. .54443 0. 6525E -02 0.0052465 0.0101721 0.0077093 45 280. .81152 0. 5568E -02 0.0053905 0.0100588 0 .0077246 46 280. .81692 o. 5552E -02 0.0053805 0.0100262 0 .0077033 47 281. .08740 0. 4581E -02 0.0055293 0.0098951 0 .0077122 48 281.09204 0. 4564E -02 0.0055291 0.0098637 0 .0076964 49 281 .25342 0. 3988E -02 0.0056330 0.0097873 0 .0077102 50 281 .37085 0. 3569E -02 O.O056881 0.0096908 0 .0076895 51 281 .42188 0. 3387E -02 0.0057398 0.0096682 0 .0077040 52 281 .48145 0. 3174E -02 0.0057665 0.0096043 0 .0076854 53 281 .59204 0. 2780E -02 0.0058666 0.0095367 0 .0077016 54 281 .64893 0. 2578E -02 0.0059019 0.0094621 0 .0076820 55 281 .65210 0. 2567E -02 0.0059040 0.0094662 0 .0076851 56 281 .70605 0. 2375E -02 0.0059625 0.0094315 O .0076970 57 281 .76636 0. 2160E -02 0.0060012 0.0093549 0 .0076781 58 281 .82080 o. 1966E -02 0.0060648 0.0093192 0 .0076920 140 59 281 .82642 0 . 1947E -02 0.0060646 0.0092909 0 .0076778 60 281 , .93604 0. 1557E -02 0.0061890 0.0091873 0. .0076881 61 281 .95898 O. 1475E -02 0.0062587 0.0090886 0. O076736 62 281 .99878 0 . 1334E -02 0.0062482 0.0090844 0 .0076663 63 282. .05176 0 . 1146E -02 0.0063361 0:0090429 0. .0076895 64 282. .05688 0 . 112BE -02 O.0063304 0.0090039 0. .0076671 65 282. .10986 0 . 9398E -03 0.0064254 0.0089507 0. .0076880 66 282. .11548 0 . 9199E -03 0.0064251 0.0089129 0. .0076690 67 282. . 16821 0 . 7329E -03 0.0065313 0.0088502 0. .0076908 68 282. .17358 0 . 7138E -03 0.0065156 0.0088095 o. .0076625 69 282. .20337 0 . 6082E -03 0.0065850 0.0087494 0. .0076672 70 282. .22632 0 . 5268E -03 0.0066466 0.0087270 o, .0076868 71 282. .23096 0 . 5104E -03 0.0066628 0.0086749 0. .0076688 72 282. .23315 0 . 5026E -03 0.0066460 0.0086832 0. .0076646 73 282. .25879 0 . 4117E -03 0.0067111 0.0086169 0 .0076640 74 282. .27051 0. 3702E -03 0.0067586 0.0086078 0. .0076832 75 282. .27661 0 . 3486E -03 0.0067636 0.0085610 0. .0076623 76 282. .29785 0 . 2733E -03 0.0068307 0.0084988 0. 007664B 77 282. .31494 0 . 2127E -03 0.0069154 0.0084462 0. .0076808 78 282. .31592 0 . 2093E -03 0.0069020 0.0084221 0. .0076621 79 282. .32544 0 . 1755E -03 0.O069251 0.0083806 0. .0076528 80 282. .33301 0. 1487E -03 0.0069754 0.0083474 o. .0076614 81 282. .33618 0. 1375E -03 0.0070089 0.0083266 0. .O076678 82 282. .33765 0. 1323E -03 0.0070219 0.0083390 0. O076805 83 282. .34253 0. 1150E -03 0.0070299 0.0082893 0. .0076596 84 282. .34448 0. 1081E -03 0.0070689 0.0083040 0. .0076864 85 282. .35059 0. 8647E -04 0.0070927 0.0082249 0. .0076588 86 282. .36963 o. 1902E -04 0.0072789 O.O08O462 0. .0076626 141 F r i n g e D a t a L i s t i n g o f -0UT3 a t 15:08:27 on JUL 12. 1987 f o r CC1d=DBRU 1 E t h y l e n e f r i n g e d a t a , s e t ul. 2 3 Number o f d a t a p o i n t s • 42 4 C r i t i c a l T e m p e r a t u r e * 282 .48940 K 5 6 Temp (K) t F r i n g e # D e l t a r h o 7 8 282 .48870 0 . 2478E - 0 5 46 .0 0 .022198 9 282 .48660 0 . 9912E - 0 5 72 .0 0 .034883 10 282 .48440 0 . .1770E -04 87 .0 0. .042201 11 282 .48260 0 . 2407E -04 96 .0 0 .046592 12 282 .48040 0 . 3186E -04 105 .0 0. .050983 13 282 .47830 0 . 3929E -04 113 .O 0 .054886 14 282 .47610 0. 4708E -04 119 .0 0 .057813 15 282, .47390 0 . 5487E -04 125 .0 0. .060741 16 282 .47160 0. 6301E -04 131 .0 0 .063668 17 282 .46940 0 . 7080E -04 136 .0 0. .066107 18 282 .46720 0. 7859E -04 141 .0 0. .068546 19 282 .46510 0. 8602E -04 145 .0 0. .070498 20 282 .46210 0. 9664E -04 151 .0 0. .073425 21 282 .46060 0. 1020E -03 154 .0 0 .074889 22 282 .45720 O. 1140E -03 160 .0 0. .077816 23 282 .45410 0. ,1250E - 0 3 165 .0 0 .080255 24 282 .45140 0 . 1345E - 0 3 169 .0 0. .082207 25 282 .448O0 0. 1466E -03 173 .0 o. .084158 26 282 .44470 0. . 1582E - 0 3 178 .0 0 .086598 27 282 .44120 0 . ,1706E -03 182 .0 0. .088549 28 282 .43840 0. 1805E -03 186 .0 0 09O5O0 29 282 .43510 0 . , 1922E -03 190 .0 0. .092452 30 282 .43190 0 . 2035E -03 194 .0 0. .094403 31 282 .42850 0. 2156E - 0 3 197 .0 0. .095867 32 282 .42530 0 . 2269E -03 201 , .0 0. 097818 33 282 422O0 0. 2386E -03 204 .0 0. 099282 34 282 .41840 0 . 2513E -03 207 .0 0 . ,100746 35 282 .41450 0 . 2651E -03 212 .0 0. 103185 36 282, .41020 0 . 2804 E -03 216. .0 0. 105136 37 282 .40580 0 . 2959E -03 220. .0 0 . 107088 38 282 .40130 0 . 3119E -03 224. .0 0. 109039 39 282 .39710 0 . 3267E -03 227. 0 0 . 110503 40 282 .39300 0 . 3413E -03 230 0 0 . .111966 41 282. .38010 0 . 3869E -03 240. 0 0 . 116845 42 282. .36720 0 . 4326E -03 250. .0 0 . 121723 43 282. 35400 0 . 4793E -03 259. .0 0 . 126114 44 282. 34180 0 . 5225E -03 266. 0 0 . 129529 45 282. 32860 0 . 5692E -03 274. .0 0 . 133431 46 282. 31580 0 . 6145E -03 281 . 0 0 . 136846 47 282. 30270 0 . 6609E -03 287. .0 0 . 139773 48 282. 28940 o. 7080E -03 293. .0 o. 142700 49 282. ,27620 o. 7547E -03 300. 0 0 . 146115 142 L i s t i n g o f - 0UT3(50 ) a t 15 :11 :15 on JUL 12. 1987 f o r CC1d=DBRU 50 E t h y l e n e f r i n g e d a t a , s e t #3. 51 52 Number o f d a t a p o i n t s «= 91 53 C r i t i c a l T e m p e r a t u r e • 282 .48580 K 54 55 Temp (K) t F r i n g e 0 D e l t a r h o 57 282. .48000 0 . 2053E-04 88 .0 0 . 044153 58 282. .47885 0 . 2460E-04 93 .0 0 . 046592 59 282. .47800 0 . 276 IE-04 97, .0 0 . 048544 60 282. .47679 0 . 3190E-04 102. .0 0 . 050983 61 2B2. .47640 0 . 3328E-04 104 .0 0 . 051959 62 282. .47520 0 . 3752E-04 108 .0 0 . .053910 63 282. .47430 0 . 407 IE-04 111 .0 0 . 055374 64 282. .47350 0 . 4354E-04 114 .0 0 . 056838 65 282. .47250 0 . 4708E-04 117 .0 0 . 058301 66 282 .47130 0 . 5133E-04 120 .0 0 . 059765 67 282 .47010 0 . 5558E-04 123 .0 0 . 061228 68 282 .46900 O. 5947E-04 126 .0 0 . 062692 69 282 .46800 0 . 6301E-04 128 .0 0 . 063668 70 282 . 46690 0 . 6 6 9 I E - 0 4 131 .0 0 . .065131 71 282 .46590 0 . 7045E-04 133 .0 0 . 066107 72 282 .46486 0 . 7413E-04 135 .0 0 . 067083 73 282 .46390 0 . 7753E-04 138 .0 0 . 068546 74 282 .46290 0 . 8107E-04 140 .0 0. 069522 75 282 .45790 0 . 9877E-04 149 .0 0 . .073913 76 282 .45360 0 . 1140E-03 157 .0 0 . ,077816 77 282 .44920 0 . 1296E-03 164 O 0 . 081231 78 282 .44460 0 . 1458E-03 171 .0 0 . 084646 79 282 .44180 0 . 1558E-03 174 .0 0. ,086110 80 282 .43980 0 . 1628E-03 176 .0 0 . 087085 81 282 .43780 0 . 1699E-03 179 .0 0 . .088549 82 282 .43630 0 . 1752E-03 181 .0 0. .089525 83 282 .43430 0 . 1823E-03 183 .0 0 . .090500 84 282 .42990 0 . 1979E-03 189 .0 0 . 093428 85 282 .42570 0 . 2128E-03 194. .0 o. 095867 86 282 .42130 o. 2283E-03 198 .0 0 . 09781B 87 282 .41820 0 . 2393E-03 201 .0 0 . 099282 88 282 .40780 0 . 2761E-03 212 .0 0 . 104648 89 282 .40100 0 . 3002E-03 217 .0 0 . 107088 90 282. .38800 0 . 3462E-03 228. 0 0 . 112454 91 282. .38460 0 . 3582E-03 231 . .0 0 . 113918 92 282. .37710 0 . 3848E-03 237 . .0 0 . 116845 93 282. .36560 0 . 4255E-03 245. .0 0 . 120747 94 282. .36120 0 . 4411E-03 248. .0 o. 122211 95 282. .34843 0 . 4863E-03 257 .0 0 . 126602 96 282. .33850 0 . 5214E-03 263. .0 0 . 129529 97 282. .33460 0 . 5352E-03 264. .0 0 . 130016 98 282. . 36520 0 . 4269E-03 245. .0 0 . 120747 99 282. .36080 0 . 4425E-03 247 .0 o. 121723 100 282. .35260 0 . 4715E-03 253. .0 0 . 124650 101 282. .34830 o. 4868E-03 256 .0 o. 126114 102 282. .33630 0 . 5292E-03 263 .0 0 . 129529 103 282. .33460 o. 5352E-03 264 .0 0 . 130016 104 282. .32900 0 . 555 IE -03 268. .0 0 . 131968 105 282. .31860 0 . 5919E-03 274. .0 0 . 134895 106 282. .30510 0 . 6397E-03 281. .0 0 . 138310 107 282. .29160 0 . 6875E-03 288. .0 0 . 141724 108 282 .27860 0. 7335E -03 295 .0 0. 145139 109 282 .27010 0. 7636E -03 298 .0 0. 146603 110 282 .26830 0. 77O0E -03 300 .0 0. 147578 111 282 .25240 0. 8262E -03 308 .0 0. 151481 112 282 .25520 0. B163E -03 306 .0 0. 150505 113 282 .23930 0. 8726E -03 314 .0 0. 154408 114 282 .22910 0. 90B7E -03 318 .0 0. 156359 115 282 .21590 0. 9554E -03 324 .0 0. 159286 116 282 .20290 0. 1001E -02 329 .0 0. 161725 117 282. .18710 0. 1057E -02 . 335 .0 0. 164652 118 282 .17960 0. 1084E -02 338 .0 0. 166115 119 282. .14020 0. 1223E -02 353 .0 0. 173432 120 282. .10080 0. 1363E -02 367 .0 0. 180261 121 282 .06230 0. 1499E -02 379 .0 0. 186115 122 282 .03570 0. 1593E -02 387 .0 0. 190017 123 281 .98801 0. 1762E -02 400 .0 0. 196358 124 281 .94900 0. 19O0E -02 411 .0 0. 201723 125 281 .91080 0. 2036E -02 421 .0 0. 206601 126 281 .86820 0. 2186E -02 432 .0 0. 211966 127 281 .82990 0. 2322E -02 441 .0 0. 216356 128 281 .81370 0. 2379E -02 445 .0 0. 218306 129 281 .79400 0. 2449E -02 449 .0 0. 220257 130 281 .75560 0. 2585E -02 458 .0 0. 224647 131 281 .72460 0. 2695E -02 465 .0 0. 228061 132 281 .68610 0. 283 IE -02 473 .0 0. 231962 133 281 .64730 0. 2968E -02 481 .o 0. 235864 134 281 .62120 0. 3061E -02 487 .0 o. 238790 135 281 .59010 0. 3171E -02 493 .0 0. 241716 136 281 .54750 0. 3322E -02 500 .0 0. 245130 137 281 .50950 o. 3456E -02 508 .o 0. 249032 138 281 .46979 0. 3597E -02 514 .0 0. 251957 139 281 .44180 0. 3696E -02 521 .o 0. 255371 140 281 .42410 0. 3758E -02 524 .0 0. 256834 141 281 .40530 0. 3825E -02 527 .0 o. 258297 142 281 .38900 0. 3883E -02 528 .0 o. 258784 143 281 .37000 0. 3950E -02 533 .0 0. 261223 144 281 .34410 0. 4042E -02 537 .0 0. 263173 145 281 . 30620 0. 4176E -02 544 .0 0. 266587 146 281 .28330 0. 4257E -02 547 .0 0. 268050 147 281 . 22880 0. 4450E -02 555 .0 0. 271950 144 >t 1 ng o f - 0 U T 3 O 4 1 ) e t 15 : 14: 09 on JUL 12. 1987 f o r CC1d=DBf 141 E t h y l e n e f r i n g e d a t a , s e t 04 142 143 Number o f d a t a p o i n t s » 66 144 C r i t i c a l T e m p e r a t u r e « 282 . 48710 K 145 146 Temp (K) t F r i n g e 0 D e l t a r h o 147 148 282. 48631 0 . 2797E-05 48 .0 0 . .023174 149 282. 48595 0 . 407 IE -05 53 .5 0. ,025857 150 282. 48572 0 . 4885E-05 57 . 0 0. .027565 151 282. .48535 0 . 6195E-05 62 .0 0. .030004 152 282. .48504 0 . 7292E-05 65 .0 0, .031468 153 282. .48471 0 . 846 IE -05 68 .0 0 .032932 154 282. .48440 0 . 955BE-05 71 .O 0. .034395 155 282. .48403 0 . 1087E-04 74 .0 0 .035859 156 282. .48382 O. 1161E-04 76 .0 0. .036835 157 282. .48350 O. 1274E-04 78 . 0 0. .037810 158 282 .48316 0 . 1395E-04 80 .5 0. .039030 159 2B2. .48291 0 . 1483E-04 82 .0 O. .039762 160 282. .48259 0 . 1597E-04 84 .0 0. .040738 161 282. .48233 0 . 1689E-04 86 .0 0 .041713 162 282. .48219 0 . 1738E-04 87 . 0 0. .042201 163 282. .48201 0 . 1802E-04 88 .6 0 .042689 164 282. .48173 0 . 1901E-04 69 .5 0. .043421 165 282 .48134 0 . 2039E-04 91 .O 0 .044153 166 282 .48109 0 . 2128E-04 92 .5 0 .044885 167 282. .48085 0 . 2212E-04 94 .0 0. .045616 168 282 .48044 0 . 2358E-04 95 .5 0 .046348 169 282. .48019 0 . 2446E-04 96 .5 0 .046836 170 282 .47984 0 . 2570E-04 98 .5 0 .047812 171 282. .47961 0 . 2651E-04 99 .5 O .048300 172 282 .47933 0 . 2751E-04 101 .0 0 .049032 173 282. .47905 0 . 2850E-04 102 .5 0. .049763 174 282. .47885 0 . 2920E-04 103 .0 O. 050O07 175 282. .47854 0 . 3030E-04 104 .0 0 .050495 176 282. .47804 o. 3207E-04 105 .0 O. .050983 177 282 .47764 0 . 3349E-04 107 .0 0 .051959 178 282. .47696 0 . 3590E-04 109 . 0 0 .052935 179 282. .47648 o. 3759E-04 111 .0 o. .053910 180 282. .47598 0 . 3936E-04 113 .0 0. .054886 181 282. .47549 0 . 4110E-04 114 .0 0 . .055374 182 282. .47512 o. 4241E-04 116 .o 0. 056350 183 282. 47468 o. 4397E-04 117 .o o. .056838 184 282. .47418 0 . 4574E-04 119 .0 0. .057813 185 282. 47368 o. 4751E-04 120 .o 0. .058301 186 282. 47312 0 . 4949E-04 122 .o o. .059277 187 282. 47274 0 . 5083E-04 123 .0 0. .059765 188 282. 47221 o. 5271E-04 124 .0 0 . O60253 189 282. .47176 0 . 5430E-04 125 .0 0. .060741 190 282. .47091 0 . 5731E-04 127 .0 0. .061716 191 282. .47055 0 . 5859E-04 128 .o o. ,062204 192 282. .47004 0 . 6039E-04 130 . 0 0. .063180 193 282. .46928 0 . 6308E-04 132 .0 0. .064156 194 282. .46831 0 . 6 6 5 2 E - 0 4 134 .0 0 .065131 195 282 .46722 0 . 7037E-04 136 .0 0. .066107 196 282 . 46656 0 . 727 IE -04 138 .0 0 .067083 197 282, . 46560 0 . 7611E-04 140 .0 0 .068059 198 282 .46466 0 . 7944E-04 142 .0 0 .069034 199 282 .46416 O. 8121E -04 143 .0 0. .069522 200 282 .46300 0. 8531E -04 146 .0 0. .070986 201 282 .46203 0 . 8875E -04 147 .0 0 .071474 202 282 . 46088 O. 9282E -04 149 .O 0 .072449 203 282 .46061 0 . 9377E -04 150 .O 0. .072937 204 282 .45946 0 . 9785E -04 152 .0 0 .073913 205 282 .45824 0 . .1022E - 0 3 154 .0 0 .074889 206 282 .45738 0 . 1052E -03 155 .0 0 .075377 207 282 .45635 0 . 1089E - 0 3 157 .0 0. .076352 208 282 .45511 0 . 1132E - 0 3 159 .0 0. .077328 209 282 .45413 0 . 1167E -03 161 .0 0 .078304 210 282 .45210 0 . 1239E - 0 3 164 .0 0 .079767 211 282 .45077 0 . 1286E -03 166 .0 0. .080743 212 282 .44872 0 . 1359E - 0 3 169 .0 0 . .082207 213 282 .44789 0 . 1388E - 0 3 170.0 0. .082695 146 L i s t i n g o f -0UT3(418) a t 15 :16:58 on JUL 12, 1987 f o r CC1d=DBRU 418 E t h y l e n e f r i n g e d a t a , s e t 05. 419 420 Number o f d a t a p o i n t s • 199 421 C r i t i c a l T e m p e r a t u r e - 282 .49060 K 422 423 Temp (K) t F r i n g e 0 D e l t a r h o 424 425 282. .49017 0 . 1522E -05 39 .0 0. .018783 426 282 .48986 0 . 2620E - 0 5 47 .0 0, .022686 427 282 .48957 0 . 3646E -05 52 .0 0 . .025126 428 282 .48922 0 . 4885E -05 57 .0 0. .027565 429 282 .48898 0 . 5735E -05 60 .0 0. .029029 430 282 .48657 0 . 7186E -05 64 .0 0 . .030980 431 282 .48824 0 . 8354E -05 68 .0 0. .032932 432 282 .48793 0 . 9452E - 0 5 70 .0 0 .033907 433 282 .48776 0. .1005E -04 72 .0 0 .034883 434 282 .48748 0 . , 1104E -04 75 .0 0, .036347 435 282 .48717 0 . 1214E -04 77 .0 0 .037323 436 282 .48686 0 . 1324E -04 79 .0 0. .038298 437 282 .48661 0 . 1412E -04 81 .0 o. 039274 438 282 .48636 0 . 1501E -04 83 .0 0. .040250 439 282 .48602 0. . 1621E -04 85 .0 0. .041226 440 282 .48571 0 . 1731E -04 87 .6 0. .042201 441 282 .48539 0 . . 1844E -04 89 .0 0. .043177 442 282 .48497 0. . 1993E -04 90 .0 0. .043665 443 282 .48466 0. 2103E -04 92 .0 0. .044641 444 282 .48430 0 . 2230E -04 94 .0 0. .045616 445 282 .48403 0. 2326E -04 95 .0 0. .046104 446 282 .48371 0. 2439E -04 96 .0 0 .046592 447 282 .48346 0. 2528E -04 97 .0 0. .047080 448 282 .48314 o. 2641E -04 99 .0 o .048056 449 282 .48291 0. 2722E -04 100 .0 0 .048544 450 282 .48258 0. 2B39E -04 102 .0 0, .049519 451 282 .48234 o. 2924E -04 102 .o o. .049519 452 282 .48217 0. 2984E -04 103 .0 0 .050007 453 282 .48175 0. .3133E -04 104 .0 0 .050495 454 282 .48140 o. ,3257E -04 106 .o o .051471 455 282 .48110 0. . 3363E -04 107 .0 0 .051959 456 282 .48083 o. 3459E -04 108 .0 0. .052447 457 282 .48064 o. 3526E -04 109 o o. .052935 458 282 .48038 0 . 3618E -04 110 .0 0 .053422 459 282. .48015 0 . 3699E -04 111 .0 0. .053910 460 282. .47983 0 . 3813E -04 112 .o o, .054398 461 282. .47953 0 . 3919E -04 113. .0 0 . .054886 462 282. .47928 0 . 4007E -04 114. .0 0 . ,055374 463 282. 47895 o. 4124E -04 115. .0 0. ,055862 464 282. .47839 o. 4322E -04 116. .0 0. 056350 465 282. .47809 0 . 4428E -04 117, .0 0 . .056838 466 282 .47781 o. 4528E -04 118 .0 0. .057325 467 282 .47754 0 . 4623E -04 119 .0 0. .057813 468 282 .47695 o. 4832E -04 121 .0 0. .058789 469 282 .47670 0 . 4921E -04 122 .0 0. 059277 470 282 .47641 0 . 5023E -04 123 .0 0 .059765 471 282 .47613 o. 5122E -04 123 .o o. ,059765 472 282 .47575 0. 5257E -04 124 .0 0 .060253 473 282 .47529 0. .5420E -04 125 .0 0 .060741 474 282 .47501 0 5519E -04 126 .0 0 .061228 475 282 .47473 0. 5618E -04 127 .0 0 .061716 476 282 .47412 0. 5834E -04 128 .0 0 .062204 477 282 .47385 0. 5929E -04 129 .0 0. .062692 478 282 .47352 0. 6046E -04 130 .0 0 .063160 479 282 .47299 0. .6234E -04 131 .0 0 .063668 480 282 .47271 0 6333E -04 132 .0 0 .064156 481 282 .47217 0. 6524E -04 133 .0 0 .064644 482 282 .47171 O. 6687E -04 134 .0 o .065131 483 282 .47141 0. 6793E -04 135 .0 o, .065619 484 282 .47101 0. 6935E -04 136 .0 0 .066107 485 282 .47053 0. 7105E -04 137 .0 0 .066595 486 282 .46993 0. 7317E -04 138 .0 0 .067083 487 282 .46930 0. 7540E -04 140 .0 0 .068059 488 282 .46857 0. 7798E -04 141 .0 0 .068546 489 282 .46810 0. 7965E -04 142 .0 0 .069034 490 282 .46746 0. 8191E -04 143 .0 0 .069522 491 282 .46679 0. 8429E -04 145 .0 0 .070498 492 282 .46614 0. B659E -04 146 .0 0 .070986 493 282 .46547 0. 8896E -04 147 .0 0 .071474 494 282 .46449 0. 9243E -04 149 .0 0 .072449 495 282 .46385 0. 9469E -04 150 .0 0 .072937 496 282 .46261 0. 9908E -04 153 .0 0 .074401 497 282 .46150 0. 1030E -03 155 .0 0 .075377 498 282 .46082 0. 1054E -03 156 .6 0. .075865 499 282 .45971 0. 1093E -03 158 .0 0. .076840 500 282 .45886 0. 1124E -03 159 .o 0 .077328 501 282 .45852 0. 1136E -03 159 .0 0. .077328 502 282 .45813 o. 1149E -03 160 .0 0. .077816 503 282 .45755 o. 1 170E -03 162 .0 0. .078792 504 282 .45677 o. 1198E -03 163. .0 0. 079280 505 282 .45570 0. 1235E -03 164. 0 0. .079767 S06 282 .45491 0. 1263E -03 166 .0 0. 080743 507 282 .45382 0. 1302E -03 167 .0 0. 081231 508 282 .45283 o. 1337E -03 169. .0 o. .082207 509 282. .45194 0. 1369E -03 170 .0 0. 082695 510 282 .45075 0. 1411E -03 172. .0 0. 083670 511 282 .44976 o. 1446E -03 173. o o. 084158 512 282. .44877 0. 1481E -03 175. .0 0. 085134 513 282 .44779 0. 1515E -03 176, .0 0. .085622 514 282. .44590 0. 1582E -03 179 .0 0. 087085 515 282. .44241 0. 1706E -03 183. .0 0. ,089037 516 282. ,44102 0. 1755E -03 185. 0 0. 090013 517 282. .43773 o. 1872E -03 189. .0 o. 091964 518 282. 43604 0. 1931E -03 190. .0 0. 092452 519 282. .43344 0. 2023E -03 194. .0 0. 094403 520 282. .43035 0. 2133E -03 197 . o 0. 095867 521 282. .42746 0. 2235E -03 200, 0 0. 097331 522 282. .42467 0. 2334E -03 203. .0 0. 098794 523 282. .42128 0. 2454E -03 206. .0 0. 100258 524 282. .42160 0. 2443E -03 207. 0 0. 100746 525 282. .41480 0. 2683E -03 213. 0 0. 103673 526 282 . 40920 0. 2882E -03 218. 0 0. 106112 527 282. .39980 0. 3214E -03 226. .0 0. 110015 528 282. .39520 0. 3377E -03 229. 0 0. 111478 529 282. .39040 0. 3547E -03 233. 0 0. 113430 530 282. .38390 0. 3777E -03 238. .0 0. 115869 531 282. .37800 0. 39B6E -03 242. .0 0. , 117820 532 282. 37060 0. 4248E -03 248. 0 0. 120747 533 282. . 36020 0. 4616E -03 255. 0 0. 124162 534 282 .35130 0 .4931E -03 261 .0 0. .1270B9 535 282 .34340 0 .5211E -03 265 .0 0. .129041 536 282 .33330 0 .5568E -03 271 .0 0 . 131968 537 282 .32330 0 .5922E -03 276 .0 0. .134407 538 282 .31580 0 .61BBE -03 281 .0 0. . 136846 539 282 .31460 0. 6230E -03 283 .0 0. .137822 540 282 .30530 0 6560E -03 287 .0 0. 139773 541 282 .29680 0 .6860E -03 291 .0 0. 141724 542 282 .28640 0. 7229E -03 296 .0 0. . 144164 543 282 .27830 0 7515E -03 300 .0 0. . 146115 544 282 .27280 0. 7710E -03 302 .0 0. 147091 545 282 .26270 O. 8O68E -03 307 .0 0. 149530 546 282 .25600 0 8305E -03 310 .0 0. 150993 547 282 .24570 0. 8669E -03 315 .0 0. 153432 548 282 .23870 0. 8917E -03 318 .0 0. .154896 549 282 .22820 0. 9289E -03 322 .0 0. 156847 550 282 .21820 0. .9643E -03 326 .0 0. 158798 551 282 .21560 0. 9735E -03 327 .0 0. 159286 552 282 .20850 0. 9986E -03 330 .0 0. 160749 553 282 .19980 0. .1029E -02 334 .0 0. 162701 554 282 .18990 0. .1064E -02 338 .0 0. . 164652 555 282 .18970 0. .1065E -02 339 .0 0. 165140 556 282 .17559 0. .1115E -02 344 .6 0. 167579 557 282 .16258 0. . 1161E -02 349 .0 0. 170018 558 282 .14957 0. . 1207E -02 353 .0 0. 171969 559 282 .14047 0. . 1239E -02 356 .0 0. 173432 560 282 .12506 0. . 1294E -02 361 .0 0. 175871 561 282 . 11745 o. . 1321E -02 364 .0 0. . 177335 562 282 .10434 o. .1367E -02 368 .o 0. 179286 563 282 .09123 0. 1414E -02 373 .0 o. 181725 564 282 .08102 0. . 1450E -02 376 .0 0. 183188 565 282 .06601 0. , 1503E -02 381 .0 0. 185627 566 282 .05250 0. 1551E -02 385 .0 0. 187578 567 282 .04210 0. 1588E -02 388 .0 o. 189041 568 282 .02779 0. . 1638E -02 392 .0 0. 190992 569 282 .02088 0. 1663E -02 394 .0 0. 191968 570 282 .00747 0. 1710E -02 398 .0 0. 193919 571 281 .99376 0. 1759E -02 402 .0 0. 195870 572 281 .98265 0. 1798E -02 406 .0 0. 197821 573 281 .97084 0. 1840E -02 409 .0 0. 199284 574 281 . .95753 0. 1887E -02 412 .0 0. 200748 575 281. .94433 0. 1934E -02 416 .0 0. 202699 576 281 . 93582 o. 1964E -02 418 .0 0. 203674 577 281. 92201 0. 2013E -02 422 .0 0. 205625 578 281. 90960 0. 2057E -02 425 .0 0. 207088 579 281 . 89729 0. 210OE -02 427 .o o. 208064 580 281 . 89750 0. 2100E -02 428 .0 0. 208552 581 281. 89328 0. 2114E -02 429 .0 0. 209039 582 281 . 87305 0. 2186E -02 434 .0 0. 211478 583 281 . 85293 0. 2257E -02 439 .0 0. 213917 584 281 . 83291 0. 2328E -02 444. .o 0. 216356 585 281 . 81289 0. 2399E -02 449. .0 0. 218794 586 281. .79356 0. 2467E -02 453 .0 0. 220745 587 281 . 77344 0. 2539E -02 458 .0 o. 223184 588 281 . 75370 o. 2609E -02 462 .o o. 225135 589 281. .73375 0. 2679E -02 467 .0 0. 227573 590 281. .71353 0. 2751E -02 471 .0 0. 229524 591 281 . 69301 0. 2823E -02 476 .0 0. 231963 592 281 .67298 0. 2894E -02 480 .0 0. 233913 593 281 .65256 0. 2967E -02 484 .0 0. 235864 594 281 .65256 0. 2967E -02 485 .0 0. 236352 595 281.63484 0. 3029E -02 489 .0 0. 238303 596 281 .62522 0. 3063E -02 490 .0 0. 238790 597 281 .59430 0. 3173E -02 496 .0 0. 241716 598 281 , .57958 0. 3225E -02 499 .0 0. 243179 599 281. .54896 0. 3333E -02 505 .0 0. 246106 600 281 .52605 0. 3414E -02 509 .0 0. 248056 601 281, .49533 0. 3523E -02 515 .0 0. 250982 602 281.46851 0. 3618E -02 519 .0 0. 252933 603 281 .43759 0. 3728E -02 524 .0 0. 255371 604 281. .40545 0. 3841E -02 529 .0 0. 257809 605 281. .37513 0. 3949E -02 535 .0 0. 260735 606 281 .31474 0. 4162E -02 545 .0 0. 265611 607 281. .28457 O. 4269E -02 551 .0 0. 268537 608 281 .25420 0. 4377E -02 555 .0 0. 270488 609 281 . 22395 0. 4484E -02 559 .0 0. 272438 610 281 . 19698 0. 4579E -02 563 .0 o. 274388 611 281 . 17871 0. 4644E -02 566 .0 0. 275851 612 281. .15974 0. 4711E -02 569 .0 0. 277314 613 281 . 12916 0. 4819E -02 574 .0 • 0. 279752 614 281 .09862 0. 4928E -02 578 :o 0. 281702 615 281 .03225 0. 5162E -02 588 .0 0. 286578 616 280. .96514 0. 54O0E -02 599 .0 0. 291942 617 280 .89772 0. 5639E -02 607 .0 0. 295842 618 280 .82990 0. 5879E -02 616 .0 0. 300230 619 280 .76229 o. 6118E -02 625 .0 0. 304618 620 280 .69477 0. 6357E -02 632 .0 0. 308030 621 280. .62695 0. 6597E -02 641 .o 0. 312418 622 280 .55963 0. 6836E -02 648 .0 0. 315830 623 280 .49142 0. 7077E -02 656 .0 0. 319730 B-3 Hydrogen Coexistence Curve Data Order Parameter Data L i s t i n g o f -0UT1 a t 13 :37 :26 on AUG 14. 1987 f o r CC1d-DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 211186.1 2 3 Number o f d a t a p o i n t s • 12 4 C r i t i c a l T e m p e r a t u r e • 33 .3004 K 5 6 7 Temp (K) • t F r i n g e ** O e l t a r h o 9 8 33. .29876 0 . 4925E-04 41 . .0 0 .049536 9 33. .29656 0 . 1153E-03 53 .5 0.O64824 10 33. .29346 0 . 2084E-03 63. .5 0 .077055 11 33. .29052 0 . 2967E-03 71 . .5 0 .086840 12 33. .28682 0 . 4078E-03 79, .5 0 .096625 13 33. .28369 O. 5018E-03 85. .0 0 .103352 14 33 .27992 0 . 6150E-03 91 .0 0 .110690 15 33. .27648 0 . 71BSE-03 96. .0 0 .116806 16 33. .27266 0 . 8330E-03 100. .5 0 .122310 17 33 .26794 0 . 9748E-03 106. .0 0 .129036 18 33. .26317 O. .1118E-02 1 lO . .5 0 .134540 19 33 .25846 0 . 1259E-02 115 .0 0 .140044 L i s t i n g o f -0UT1 a t 13 :39 :10 on AUG 14. 19B7 f o r CC1d=DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 171286.1 2 3 Number o f d a t a p o i n t s » 22 4 C r i t i c a l T e m p e r a t u r e • 32 .9918 K 5 e Temp (K) t F r i n g e * D e l t a r h o f 8 32. .98314 O . 2625E-03 66 .5 0 . 080725 9 32. .97495 0 . 5107E-03 84 .0 0 . 102129 l O 32. .96713 O . 7478E-03 96 .0 0 . 116806 11 32. .95459 0 . 1128E-02 110 .0 0 . 133929 12 32. .94177 0 . 1516E-02 121 .5 0 . 147994 13 32. .92977 0 . 1880E-02 131 .5 0 . 160224 14 32, .91730 0 . 2258E-02 140 .0 0 . 170620 15 32 .90115 0 . 2748E-02 150 .0 0 . 182851 16 32 .88499 0 . 3237E-02 159 .0 0 . 193858 17 32 .86896 0 . 3723E-02 167 .5 0 . 204253 18 32. .85259 0 . 4220E-02 175 . 0 0 . 213425 19 32 .82884 0 . 4939E-02 185 .5 0 . 226267 20 32 .80528 0 . 5654E-02 194 .5 O. 237273 21 32 .76533 0 . 6864E-02 208 .5 0 . 254394 22 32 .72507 0 . B085E-02 221 .5 0 . 270291 23 32 .68462 0 . 9311E-02 233 .0 0 . 284354 24 32 .64546 0 . 1050E-01 243 .0 0 . 296582 25 32 .60632 0 . 1168E-01 252 .0 0 . 307587 26 32 .56789 0 . 128SE-01 261 .0 0 . 318591 27 32 .52912 0 . 1402E-01 269 .5 0 . 328984 28 32 .49065 0 . 1519E-01 277 .5 0 . 338766 29 32 .45204 0 . 1636E-01 285 .0 0 . 347935 151 L i s t i n g o f -0UT1 a t 13 :40 :57 on AUG 14, 1987 f o r CC1d=DBRU 1 H y d r o g e n f o c a ) p l a n e f r i n g e d a t a , s e t 250287.1 •£ 3 Number o f d a t a p o i n t s • 45 4 5 C r i t i c a l T e m p e r a t u r e • 32 .9769 K 6 7 Temp (K) t F r i n g e # D e l t a r h o 8 32 .97372 0 . 9643E-04 49 .0 0 . 059320 9 32 .97113 0 . 1750E-03 58 .5 0 . 070940 10 32 ,96719 0 . 2944E-03 70 .0 0 . 085005 11 32 .96325 0 . 4139E-03 78 .5 0 . 095402 12 32 .95902 0 . 5422E-03 85 .5 0 . 103963 13 32 .95036 0 . 8 0 4 8 E - 0 3 97 .0 0 . 118029 14 32 .94186 0 . 1063E-02 107 .0 0 . 130260 15 32 .93366 0 . 1311E-02 115 .0 0 . 140044 16 32 .92550 0 . 1559E-02 121 .5 0 . 147994 17 32 .91713 0 . 1812E-02 128 .5 0 . 156555 IB 32 .91279 0 . 1944E-02 132 .0 0 . 160836 19 32 .90847 0 . 2075E-02 135 .0 0 . 164505 20 32 .90431 0 . 2201E-02 138 .0 0 . 168174 21 32 .90183 0 . 2276E-02 139 . 0 0 . 169397 22 32 .89935 0 . 2352E-02 141 • P 0 . 171843 23 32 .89681 0 . 2429E-02 142 .0 0 . 173066 24 32 .89450 0 . 2499E-02 144 . 0 0 . 175512 25 32 .89199 0 . 2575E-02 145 .0 0 . 176735 26 32 .68870 O. 2675E-02 147 .o 0. 179181 27 32 .88543 0 . 2774E-02 149 . 0 0 . 181627 28 32 .88128 0 . 2900E-02 152 .0 0 . 185296 29 32 .87704 O. 3028E-02 154 .o 0. 187742 30 32 .86883 O. 3277E-02 158 .o o. 192634 31 32 .86074 0 . 3522E-02 162 .0 0 . 197526 32 32 .84476 0 . 4007E-02 170 .0 0 . 207310 33 32 .82676 0 . 4492E-02 177 .o 0 . 215871 34 32 .81286 0 . 4974E-02 184 .0 0 . 224432 35 32 .78082 0 . 5946E-02 195 .5 0 . 238496 36 32 .74888 0 . 6915E-02 206 .5 0 . 251948 37 32 .71696 0 . 7882E-02 216 .5 0 . 264176 38 32 .68638 0 . 8B10E-02 225 .5 0 . 275182 39 32 .64617 0 . 10O3E-O1 236 .5 0 . 288633 40 32 .60690 0 . 1122E-01 246 .5 0 . 300861 41 32. .56674 o. 1244E-01 256 .O o. 312477 42 32. .52769 0 . 1362E-01 264 .0 0 . 322259 43 32. .49637 0 . 1457E-01 270 .5 0 . 330206 44 32. .45045 0 . 1596E-01 280 .0 0 . 341821 45 32. ,40434 o. 1736E-01 287 .5 0 . 350990 46 32. 34283 0 . 1923E-01 298 .0 0 . 363827 47 32. 28167 0 . 2108E-01 309 .0 0 . 377275 48 32. 22252 o. 2288E-01 318 .5 o. 388888 49 32. 14699 0 . 2517E-01 329 .0 0 . 401723 50 32. .07264 0 . 2742E-01 340 .0 0 . 415169 51 32. ,00015 0 . 2962E-01 351 .0 0 . 428614 52 31 . .92745 0 . 3182E-01 360 .0 0 . 439614 152 L i s t i n g o f -0UT1 a t 13 :42 :08 on AUG 14, 1987 f o r CC1d«DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 250287.2 2 3 Number o f d a t a p o i n t s » 58 4 C r i t i c a l T e m p e r a t u r e • 32 .9766 K 5 6 Temp (K) t F r i n g e * D e l t a r h o 7 8 32 . .97399 0 . 7915E-04 45 .5 0 . 055040 9 32 . 97235 0 . 1289E-03 53 .0 0 . 064213 IO 32. .97058 0 . 1826E-03 59 .5 O. 072163 11 32 . ,96797 0 . 2617E-03 67 .0 0 . 081336 12 32 . 96523 0 . 3448E-03 73 .5 O. 089266 13 32 . 96261 0 . 4242E-03 78 .5 0 . 095402 14 32. .96014 0 . 4 9 9 I E - 0 3 83 .0 0 . 100906 15 32. .95776 0 . 5713E-03 87 .0 0 . 105798 16 32 .95404 0 . 684 IE -03 92 .0 0 . 111913 17 32. .95017 0 . 8015E-03 97 .0 0 . 118029 18 32. ,94578 0 . 9346E-03 101 .5 0 . 123533 19 32. .94182 0 . 1055E-02 106 .0 0 . 129036 20 32. .93683 0 . 1206E-02 111 .0 0 . 135152 21 32. .93175 0 . ,1360E-02 116 .0 0 . 141267 22 32. .92495 0 . 1566E-02 121 .5 0 . 147994 23 32. .91748 0 . 1793E-02 128 .0 0 . 155944 24 32. .90925 0 . 2042E-02 134 .0 0 . 163282 25 32 .90091 0 . 2295E-02 139 .0 0 . 169397 26 32. 89208 0 . 2563E-02 145 .0 0 . 176735 27 32 .88238 0 . 2857E-02 150 .5 0 . 183462 28 32. .87193 0 . 3174E-02 156 .0 0 . 190188 29 32. .86146 0 . 3492E-02 161 .5 0 . 196915 30 32 . 85096 O. 3810E-02 166 .0 0 . 202418 31 32 .83893 0 . 4175E-02 172 .0 0 . 209756 32 32. .81528 0 . 4892E-02 182 .0 0 . 221986 33 32 .79129 0 . 5619E-02 191 .5 0 . 233604 34 32. .76724 0 . 6349E-02 200 .O 0 . 243999 35 32 .73519 0 . 7321E-02 210 .5 0 . 256839 36 32 .69671 o. 8488E-02 222 .5 0 . 271514 37 32 .65702 0 . 9691E-02 233 .0 0 . 284353 38 32. .61784 o. ,1088E-01 243 .0 o. 296581 39 32. .56972 0 . 1234E-01 254 .5 0 . 310643 40 32. .51512 0 . 1399E-01 266 .0 0 . 324704 41 32. .45301 0 . 1588E-01 279 .0 0 . 340599 42 32. ,39117 o. 1775E-01 290 .0 0 . 354047 43 32. 33009 0 . 1961E-01 301 .0 0 . 367495 44 32. 26881 0 . 2146E-01 311 .0 0 . 379720 45 32 . 21045 0 . 2323E-01 320 .0 0 . 390722 46 32 . 14992 0 . 2507E-01 329 .5 0 . 402335 47 32 . 09132 0 . 2685E-01 338 .0 0 . 412724 48 32 . 03215 0 . 2864E-01 346 .5 0 . 423114 49 31 . 97410 0 . 3040E-01 354 .0 0 . 432281 50 31 . 91345 o. 3224E-01 362 .O 0 . 442058 51 31 . 85501 0 . 3401E-01 369 .0 0 . 450613 52 31 . 78175 o. 3623E-01 379 .0 o. 462834 53 31 . .71472 0 . 3827E-01 386 .5 0 471999 54 31 . .64357 0 . 4042E-01 395 .5 0 . 482997 55 31 . .56788 0 . 4272E-01 404 .5 0 . 493995 56 31. .50149 0 . 4473E-01 411 .5 0 . .502548 57 31 .42623 0 . 4701E-01 420 .0 0 . .512933 SB 31 .35672 0 . 4912E-01 426 .5 0 . ,520874 59 31 .28788 0 5 1 2 1 E - 0 1 433 .5 0 .529426 6 0 31 .18943 0 .5420E-01 443 .5 0 .541642 61 31 .08377 0 . 5 7 4 0 E - 0 1 451 .0 0 .550800 6 2 30 .95415 0 6 1 3 3 E - 0 1 462 . 0 0.564235 6 3 3 0 . 8 1 6 7 0 0 . 6 3 5 0 E - 0 1 4 7 3 . 0 0 .577668 64 30 .75567 0 .6735E-01 479 . 5 0 .585607 65 30 .68196 0 .69S8E-01 484 .0 0 .591100 153 U s t i n g o f -0UT1 a t 1 3 : 4 3 : 5 0 o n AUG 14, 1987 f o r CC1d=DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 250287.4 2 3 Number o f d a t a p o i n t s • 54 4 C r i t i c a l T e m p e r a t u r e * 32 .9735 K 5 6 Temp (K) t F r i n g e * D e l t a r h o 8 32. .97245 0 . 3184E-04 35 .0 0 . 042197 9 32. .97072 O . B431E-04 46 .0 0 . 055651 10 32 .96979 0 . 1125E-03 51 .5 0 . 062378 11 32. .96834 0 . 1565E-03 57 .5 0 . 069717 12 32. .96684 0 . 2020E-03 63 .0 0 . 076444 13 32 .96540 0 . 2457E-03 66 .5 0 . 080725 14 32. .96436 0 . 2772E-03 69 .0 0 . 083782 15 32 .96257 0 . 3315E-03 73 .0 0 . 088675 16 32. .96098 0 . 3797E-03 76 .5 0 . 092956 17 32 .95925 0 . 4322E-03 80 .0 0 . 097236 18 32 .95737 0 . 4B92E-03 83 .5 0 . 101517 19 32. .95586 0 . 5350E-03 85 .5 0 . 103963 20 32 .95429 0 . 5826E-03 88 .0 0 . 107021 21 32 .95187 0 . 6560E-03 91 .5 0 . 111302 22 32 .94951 0 . 7276E-03 95 .O 0 . 115583 23 32 .94701 0 . 8034E-03 98 .0 0 . 1 19252 24 32 .94237 0 . 944 IE -03 104 .0 0 . 126590 25 32 .93661 0 . 1119E-02 109 .5 0 . 133317 26 32 .93104 0 . 1288E-02 115 .5 0 . 140656 27 32 .92518 0 . 1465E-02 121 .0 0 . 147382 28 32 .91939 0 . 1641E-02 125 .5 0 . 152886 29 32 .91290 0 . 1838E-02 130 .5 0 . 159001 SO 32 .90638 O. 2036E-02 135 .O o. 164505 31 32 .89819 0 . 2284E-02 140 .5 0 . 171232 32 32 .89002 0 . 2532E-02 146 .0 0 . 177958 33 32 .88029 0 . 2827E-02 151 .5 0 . 184685 34 32 .86982 0 . 3144E-02 157 .0 0 . 191412 35 32 .84950 0 . 3761E-02 167 .0 0 . 203642 36 32 .82939 0 . 4370E-02 176 .5 0 . 215260 37 32 .80464 0 . 5121E-02 186 .O o. 226878 38 32 .78174 0 . 5816E-02 195 .0 0 . 237884 39 32 .75778 0 . 6542E-02 203 .5 0 . 248279 40 32 .72593 o. 7508E-02 214 .0 0 . 261120 41 32 .69441 0 . 8464E-02 223 .5 0 . 272737 42 32 .65518 0 . 9654E-02 234 .5 0 . 286188 43 32 .61574 0 . 1085E-01 244 .5 0 . 298416 44 32 .57613 0 . 1205E-01 253 .5 0 . 309420 45 32. .52913 0 . 1348E-01 264 .0 0 . 322259 46 32. .48234 0 . 1490E-01 273 .5 0 . 333874 47 32. .43605 0 . 1630E-01 283 .0 0 . 345490 48 32. .37431 0 . 1817E-01 294 .0 0 . 358938 49 32. .30528 0 . 2027E-01 306 .0 0 . 373608 50 32. .23754 0 . 2232E-01 317 . 0 0 . 387055 51 32. .16157 0 . 2462E-01 328 .0 0 . 400501 52 32 .07206 o. 2734E-01 342 .o o. 417615 53 31 .98450 0 . 2999E-01 354 .o 0 . 432282 54 31 .88046 0 . 3315E-01 366 .0 0 . 446947 55 31 .75700 0 . 3689E-01 381 .0 0 . 465278 56 31 .63576 0 . 4057E-01 395 .0 o. 482386 57 31 .49394 0 . 4487E-01 409 .0 0 . 499490 58 31 .35389 0 . 4912E-01 423 . 0 o. 516594 59 31 .14879 0 . 5534E-01 441 .0 0 . 538581 60 30 .94970 0 . 6138E-01 459 . 0 0 . .560567 154 L i s t i n g o f -0UT1 a t 13 :35 :13 on AUG 14, 19B7 f o r CCld=DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 0704B7.1 2 3 Number o f d a t a p o i n t s • 19 4 C r i t i c a l T e m p e r a t u r e • 32 .8522 5 6 Temp (K) t F r i n g e H O e l t a r h o 8 32 .84919 0 . 9 1 6 2 E - 0 4 48 .0 0 .058097 9 32. .84696 0 . 1595E-03 57 .5 0 .069717 10 32. .84483 0 . 2243E-03 64 .5 0 .078278 11 32. .84202 0 . 3099E-03 71 .0 0 .086229 12 32 .83840 0 . 4201E-03 79 .5 0 .096625 13 32. .81124 0 . 1247E-02 114 .5 0 . 139433 14 32 .77912 0 . 2225E-02 139 .0 0 .169397 15 32 .74175 0 . 3362E-02 161 .0 0 .196304 16 32 .65921 0 . 5874E-02 195 .0 0 .237884 17 32 .53859 0 . 9546E-02 232 .5 0 .283742 18 32 .42560 0 . 1299E-01 259 .5 0 .316757 19 32 .31286 0 . 1642E-01 283 .0 0 .345489 20 32 .20823 0 . 1960E-01 303 .0 0.369941 21 32 .13804 0 . 2174E-01 314 .5 0 .383999 22 32 .06094 0 . 2409E-01 326 .0 0 .398057 23 31 .91109 0 . 2865E-01 348 .0 0 .424948 24 31 .76504 0 . 3309E-01 366 .5 0 .447558 25 31 .62350 0 . 3740E-01 384 .0 O.468945 26 31 .48555 0 . 4160E-01 398 .0 0.486051 L i s t i n g o f -0UT1 a t 13 :31 :55 on AUG 14, 1987 f o r CCid -DBRU 1 H y d r o g e n f o c a l p l a n e f r i n g e d a t a , s e t 070487 .2 2 3 Number o f d a t a p o i n t s • 10 4 C r i t i c a l T e m p e r a t u r e • 3 3 . 0 0 3 0 5 6 y Temp (K) t F r i n g e * D e l t a r h o 10 33 .00168 0 . 40O0E-04 3 8 . 0 0 .045866 11 33 .00093 0 . 6272E-04 4 4 . 0 0 .053205 12 32 .99864 0 . 1321E-03 5 4 . 5 0 .066047 13 32 .99725 0 . 1742E-03 6 1 . 5 0 .074609 14 32 .99575 0 . 2197E-03 6 5 . 5 0 .079502 15 32 .99126 0 . 3557E-03 7 5 . 5 0 .091732 16 32 .98743 0 . 4718E-03 8 3 . 0 0 .100906 17 32 .97986 0 . 7011E-03 9 4 . 5 0 . 114971 18 32 .96422 0 . 1175E-02 111.5 0 . 135763 19 3 2 . 9 5 6 5 0 0 . 1409E-02 118 .0 0 . 143713 C o e x i s t e n c e C u r v e D i a m e t e r D a t a L i s t i n g o f -OUT a t 16 :45 :03 on AUG 14. 1987 f o r CCId 'DBRU 1 H y d r o g e n image p l a n e f r i n g e d a t a , s e t 250287.1 2 3 Number o f d a t a p o i n t s - 29 4 C r i t i c a l T e m p e r a t u r e * 32 .9769 K 5 6 Temp (K) t N1- •Nv r h o d f 8 32 .97614 0 . 2305E-04 1. .9 0 .002309 9 32 .97372 0 . 9643E-04 1. .3 0 . ,001557 10 32 .97113 0 . 1750E-03 0 . .9 0 . ,001053 11 32. .96719 0 . 2944E-03 0. 8 0 ,000911 12 32. .96325 0 . 4139E-03 0 . 6 0 . O00648 13 32 .95902 0 . 5422E-03 0 . .7 0. ,000754 14 32 .95036 0 . 8048E-03 - 0 . .2 - 0 . .000374 15 32. .94186 0 . 1063E-02 - 0 . .8 - 0 .001138 16 32 .93366 0 . 1311E-02 - 1 . , 1 - 0 . .001531 17 32 .92550 0 . 1559E-02 - 0 . .9 - 0 . .O01310 18 32 .91713 0 . 1812E-02 - 0 . .6 - 0 000965 19 32 .91279 0 . 1944E-02 - 0 . .6 - 0 . ,000976 20 32 .90847 0 . 2075E-02 - 1 . .0 - 0 . .001478 21 32. .90431 0 . 220 IE -02 - 1 . .5 - 0 . O021O0 22 32 .90183 0 . 2276E-02 - 2 . 0 - 0 . .002718 23 32 .89935 0 . 2352E-02 - 2 . .0 - 0 . .002725 24 32 .89681 0 . 2429E-02 - 2 , .0 - o , .002732 25 32 .89450 0 . 2499E-02 - 2 . .3 - 0 . .003104 26 32 .89199 0 . 2575E-02 - 2 , .4 - 0 . .003232 27 32 .88870 0 . 2675E-02 -2 .4 - 0 .003241 28 32 .88543 0 . 2774E-02 - 2 . .7 - 0 . .003616 29 32 .88128 0 . 2900E-02 - 2 . .6 - 0 . .003504 30 32 .87704 0 . 3028E-02 - 2 . .6 - 0 .003513 31 32 .86883 O. 3277E-02 -2 .5 - 0 . .003409 32 32 .86074 0 . 3522E-02 - 2 . .2 - 0 . .003062 33 32 .84476 0 . 4007E-02 - 2 . .0 - 0 .002852 34 32 .82876 0 . 4492E-02 -2 . 1 - 0 .003011 35 32 .81286 0 . 4974E-02 -2 .0 - 0 . .O02923 36 32 .78082 0 . 5946E-02 -1 . .5 - 0 .002376 156 L i s t i n g o f -OUT a t 16 :43 :38 on AUG 14. 1987 f o r CC1d=DBRU 1 H y d r o g e n Image p l a n e f r i n g e d a t a , s e t 250287.2 2 3 Number o f d a t a p o i n t s • 49 4 C r i t i c a l T e m p e r a t u r e • 32 .9766 K 5 6 Temp (K) t N1- Nv r h o d / 8 32 . 97613 0 . 1425E-04 0 . 3 0 . 000356 9 32. 97399 0 . 7915E-04 - 1 . 1 - 0 . 001373 10 32 . 97235 0 . 1289E-03 - 1 . 9 - 0 . O02362 11 32. 97058 0 . 1826E-03 - 2 . 5 - 0 . 003106 12 32. 96797 0 . 2617E-03 - 2 . 1 - 0 . O02630 13 32 . 96523 0 . 3448E-03 - 2 . 2 - 0 . 002766 14 32. 96261 0 . 4242E-03 - 3 . 1 - 0 . 003875 15 32. 96014 0 . 4991E-03 - 2 . 9 - 0 . 003641 16 32. 95776 0 . 5713E-03 - 3 . 2 - 0 . 004017 17 32. .95404 0 . 6841E-03 - 3 . 2 - 0 . 004028 18 32. .95017 0 . 8015E-03 - 3 . 0 - 0 . 003797 19 32. .94578 0 . 9346E-03 - 3 . 5 - 0 . 004423 20 32. .94182 O . 1055E-02 - 3 . 6 - 0 . 004559 21 32 .93683 0 . 1206E-02 - 3 . 5 - 0 . 004452 22 32. 93175 0 . 1360E-02 - 2 . 5 - 0 . 003248 23 32. .92495 0 . 1566E-02 - 3 . 4 - 0 . 004364 24 32. .91748 0 . 1793E-02 - 3 . 3 - 0 . O04262 25 32 .90925 0 . 2042E-02 - 3 . O - 0 . 003916 26 32 .9O091 0 . 2295E-02 - 3 . 0 - 0 . 003938 27 32 .89208 0 . 2563E-02 - 3 . 2 - 0 . .004205 28 32 .88238 0 . 2857E-02 - 3 . 0 - 0 003984 29 32 .87193 0 . 3174E-02 - 2 . .9 - 0 . .003887 30 32 .86146 0 . 3492E-02 - 3 . .0 -o. .004032 31 32 .85096 0 . 3810E-02 - 2 . .6 - 0 O03567 32 32 .83893 0 . 4175E-02 - 2 . .5 - 0 .003470 33 32 .81528 0 . 4892E-02 - 2 . .2 -o .003156 34 32 .79129 0 . 5619E-02 - 2 . .0 - 0 .002959 35 32 .76724 0 . 6349E-02 - 1 . .8 - 0 .002763 36 32 .73519 0 . 7321E-02 -1 .6 - 0 . .002580 37 32 .69671 0 . 8488E-02 -1 , . 1 - 0 .O02038 38 32 .65702 0 . 969 IE-02 - 0 .8 -o .001743 39 32 .61784 0 . 1088E-01 - 0 .9 - 0 .001924 40 32 .56972 0 . 1234E-01 0 . 1 - 0 .000777 41 32 .51512 0 . 1399E-01 0. .2 - 0 .000743 42 32 .45301 0 . 1588E-01 0 .9 0 .OOO016 43 32 .39117 0 . 1775E-01 1 . .5 0 .000658 44 32 .33009 0 . 1961E-01 2. .2 o .001423 45 32 .26881 0 . 2146E-01 2 .4 0 .001580 46 32 .21045 0 . 2323E-01 3 . 1 0 .002354 47 32 .14992 0 . 2507E-01 3 .8 0 .003126 48 32. .09132 o. 2685E-01 4. .5 o .003901 49 32 .03215 0 . 2864E-01 5 .0 0 .004434 50 31 .97410 o. 3040E-01 5, .5 o .004968 51 31. .91345 o. 3224E-01 5 .9 0 .O05383 52 31 .85501 o. 3401E-01 6 .7 o .006289 53 31 .78175 0 . 3623E-01 7 .8 0 .007530 54 31 .71472 0. 3827E-01 8 .4 0 .008178 55 31 .64357 o. 4042E-01 10 .4 0 .010527 56 31 .56788 0 4272E-01 11 .2 0 .011395 L i s t i n g o f -OUT a t 16:41:14 o n AUG 14, 1987 f o r CC1d»DBRU 1 H y d r o g e n Image p l a n e f r i n g e d a t a , s e t 250287 .6 2 3 Number o f d a t a p o i n t s - 6 4 C r i t i c a l T e m p e r a t u r e • 32 .9720 K 5 6 7 Temp (K ) t NI- •Nv r h o d 8 32 .97167 0. .1001E-04 0. .0 - 0 . 0 0 0 0 0 6 9 32 .97093 0 . 3245E-04 - 0 . .5 - 0 . 0 0 0 6 2 8 10 32 .96943 0 7794E-04 -1 . .6 -0 .O01986 11 32 .96783 0 . .1265E-03 - 1 . .5 - 0 . 0 0 1 8 7 5 12 32 .95941 0 3818E-03 -1 .7 - 0 . 0 0 2 1 6 3 13 32 .95113 0 . 6330E-03 - 2 . .2 -0 .0O28O6 L i s t i n g o f -OUT a t 16 :39 :59 on AUG 14, 1987 f o r CC1d*DBRU 1 H y d r o g e n Image p l a n e f r i n g e d a t a , s e t 260387.1 2 3 Number o f d a t a p o i n t s « 9 4 C r i t i c a l T e m p e r a t u r e « 32.9321 K 5 6 •» Temp (K) t N1--Nv r h o d / 8 32.92961 0 . ,756 IE -04 -1 .7 - 0 .002114 9 32 .92155 0 . 3204E-03 -1 .9 -0 .0O2399 10 32 .91328 0 . 5715E-03 -1 .9 - 0 . 0 0 2 4 3 2 11 32 .90503 O. 8220E-03 -1 .9 - 0 . 0 0 2 4 6 0 12 32 .87263 0 . 1806E-02 - 3 .5 - 0 . 0 0 4 4 8 2 13 32 .82498 0 . 3253E-02 -2 .6 - 0 .003492 14 32 .76949 0 . 4938E-02 -1 .9 -0 .002751 15 32 .71372 0 . 6631E-02 -1 .5 -O .002367 16 32 .65207 0 . .8503E-02 - 0 .9 - 0 . 001757 L i s t i n g o f -OUT a t 1 6 : 3 8 . 0 7 o n AUG 14. 1987 f o r CCId-DBRU 1 H y d r o g e n image p l a n e f r i n g e d a t a , s e t 260387.2 2 3 Number o f d a t a p o i n t s • 7 4 C r i t i c a l T e m p e r a t u r e - 32 .9166 K 5 6 7 Temp (K) 4 NI- -Nv r h o d f 8 32 .91573 0 . 2 6 4 3 E - 0 4 - 0 , . 1 -O.OOO139 9 32 .91259 0 . 1 2 1 8 E - 0 3 - 1 . ,2 - 0 . 0 0 1 5 1 0 10 32 .90925 0 . 2 2 3 3 E - 0 3 - 1 . .9 - 0 . 0 0 2 3 8 6 11 32 .90715 0 . 2 8 7 1 E - 0 3 - 2 , , 1 - 0 . 0 0 2 6 4 3 12 3 2 . 9 0 3 6 0 0 . 3 9 4 9 E - 0 3 - 1 , 8 - 0 . 0 0 2 2 9 2 13 32 .89968 0 . 5 1 4 0 E - 0 3 - 0 , 8 - 0 . 0 0 1 0 8 8 14 32 .89623 0 . 6 1 8 8 E - 0 3 - 0 .8 - 0 . 0 0 1 1 0 5 158 L i s t i n g o f -OUT a t 16 :42 :29 on AUG 14, 1987 f o r CCid=DBRU 1 2 H y d r o g e n image p l a n e f r i n g e d a t a , s e t 070487.1 3 Number o f d a t a p o i n t s • 19 4 c C r i t i c a l T e m p e r a t u r e " 32 .8522 K 6 t Temp (K) t N l - N v r h o d t 8 32 . 85029 0 . 5 8 1 4 E - 0 4 - 0 . 2 -0 .OO0267 9 32 . 84919 0 . 9 1 6 2 E - 0 4 - 1 . 0 -0 .O01255 10 32 . 84696 0 . 1 5 9 5 E - 0 3 - 2 . 0 - 0 .002492 11 32. 84483 0 . 2 2 4 3 E - 0 3 - 2 . 6 - 0 . 0 0 3 2 3 8 12 32. .84202 0 . 3 0 9 9 E - 0 3 - 3 . 0 - 0 . 0 0 3 7 4 0 13 32. 83840 0 . 4 2 0 1 E - 0 3 - 3 . 3 -O .004124 14 32. 81124 0 . 1 2 4 7 E - 0 2 - 3 . 7 - 0 . 0 0 4 7 0 9 15 32. 77912 0 . 2 2 2 5 E - 0 2 - 3 . 4 - 0 .004432 16 32. .74175 0 . 3 3 6 2 E - 0 2 - 4 . 0 - 0 . 005257 17 32. .65921 0 . 5 8 7 4 E - 0 2 - 2 . 0 - 0 . 0 0 2 9 8 3 18 32. .53859 0 . 9 5 4 6 E - 0 2 -1 . 1 - 0 .002102 19 32 .42560 0 . 1 2 9 9 E - 0 1 - 0 . 4 - 0 . 0 0 1 4 3 9 20 32. .31286 0 . 1 6 4 2 E - 0 1 0 . 8 -0 .OO0149 21 32. .20823 0 . 1 9 6 0 E - 0 1 2 . 0 0 .001158 22 32. .13804 0 . 2 1 7 4 E - 0 1 2 r 2 0 .001300 23 32. .06094 0 . 2 4 0 9 E - 0 1 2 . 8 0 .O01929 24 31 . .91109 0 .2865E-01 4 . 1 0 .003315 25 31 .76504 0 . 3 3 0 9 E - 0 1 5 .4 0 .004715 26 31 .62350 0 . 3 7 4 0 E - 0 1 7 . 0 0 .006498 B - 4 H y d r o g e n C o m p r e s s i b i l i t y D a t a L i s t i n g o f -OUT a t 14 :52 :54 on AUG 14, 1987 f o r CC1d«DBRU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 H y d r o g e n c o m p r e s s i b i l i t y d a t a . D a t a s e t t k ( f r i n g e s / m m ) k* 260387.1 8 . 5 B 4 E - 4 0 . 0 0 6 146. v a p o u r 3 . 5 6 8 E - 4 0 . 0 0 9 219 . 1 .120E-4 0 .424 10308. 070487 .1 1 . 2 4 7 E - 3 0 . 0 1 0 243 . v a p o u r 4 . 2 0 1 E - 4 0 .O66 1605. 3 . 0 9 9 E - 4 0 . 1 1 3 2747. 2 . 2 4 3 E - 4 0 .201 4887. 1 .595E-4 0 . 2 7 3 6637. 9 . 1 6 2 E - 5 0 .412 10017. 5 . 8 1 4 E - 5 0 . 4 5 3 11015. 070487 .2 3 . 5 5 7 E - 4 0 . 1 1 9 2893. v a p o u r 2 . 1 9 7 E - 4 0 .178 4328. 1 .742E-4 0 . 1 9 7 4790. 1 .321E-4 0 .304 7391 . 6 . 2 7 2 E - 5 0 . 7 9 9 19425. 6 . B 4 8 E - 5 0 . 8 9 9 21857. 070487 .3 4 . 2 1 5 E - 4 0 .066 1605. v a p o u r 2 . 2 4 5 E - 4 0 .158 3841. 1 .300E-4 0 .322 7829. 6 . 1 8 1 E - 5 1.230 29903. 260387.1 3 . 4 7 7 E - 4 0 .026 632 . l i q u i d 1 .029E-4 0 .288 7002. 070487.1 1 .247E-3 0 . 0 1 0 243 . l i q u i d 4 . 2 0 1 E - 4 0 . 0 8 0 1945. 3 . 0 9 9 E - 4 0 .096 2334. 2 . 2 4 3 E - 4 0 .184 4473. 1 .595E-4 0 . 2 7 7 6734. 9 . 1 6 2 E - 5 0 .427 10381. 5 . 8 1 4 E - 5 0 .652 15851. 070487 .2 3 . 5 5 7 E - 4 0 .056 1362 l i q u i d 2 . 1 9 7 E - 4 O.123 2990 1 .742E-4 0 . 1 4 9 3623 1 .321E-4 0 .197 4790 6 . 2 7 2 E - S 0 . 4 9 0 11913, 6 . 8 4 8 E - 5 0 . 5 8 7 14271 0 7 0 4 8 7 . 3 4 . 2 1 5 E - 4 0 .048 1167 l i q u i d 2 . 2 4 5 E - 4 0 .096 2334 1 .300E-4 0 . 1 6 8 4084 6 . 1 8 1 E - 5 0 . 4 1 7 10138 260386.1 1 .913E-5 15 .49 376593 >Tc 6 . 8 5 9 E - 5 2 . 5 3 61509 9 . 1 7 0 E - 5 2 .76 67101 9 . 8 3 8 E - 5 2 .12 51541 1 .691E-4 2 . 1 3 51785 250287 .6 4 . 8 5 0 E - 6 17.92 435456 >Tc 2 . 0 9 0 E - 8 3 .57 86751 3 . 4 0 0 E - 5 1.976 48017 5 . 1 9 0 E - S 1.142 27751 8 . 1 6 0 E - 5 0 .711 17277 1 .460E-4 0 .352 8554 160 APPENDIX C Computer Programs C-1 B I G F I T - Coexistence Curve Data Analysis Program BIGFIT performs nonlinear least-squares fits to data consisting of temper-atures and fringe numbers or temperatures and densities. It will fit up to six parameters to the order parameter or up to five parameters to the diameter. The number of parameters and their initial values are chosen interactively by the user. BIGFIT can plot the data and the fit to assist the user in the selection of starting parameter values or to allow visual evaluation of the fit. Data is read in from Unit 10. Unit 11 is an additional input file containing constants such as p c, n c , etc. Fit results and additional information are output to Unit 4. While BIGFIT is intended to be an interactive program, it can be run in batch if an additional input file assigned to Unit 5 is provided. This file would contain the commmands that would normally be entered from the terminal. With simple modifications, BIGFIT can be used to fit compressibility data. By altering the fitting function provided in the subroutine CALCR and the plotting routines in GRAF2 and GRAF3, fits to other types of data could be done. BIGFIT contains a number of subroutines whose functions are as follows: START gets the type of input data and the type of fit required. GETDAT reads in the data and converts it to Kelvin temperatures and den-sities. 161 LL reads in supplementary data from Unit 11. RHOD calculates values of the coexistence curve diameter to be used in cal-culating densities. SETLIN sets up for and calls a routine to do a linear fit to Ap* vs. T, and plots the results. NONLIN gets parameters and constants for nonlinear fits, calls fitting and plotting routines and outputs the fit results. CALCR is called by the nonlinear fitting routine. It calculates the fitting function and resuiduals. GRAF2 plots the order parameter data and the nonlinear fit to it on a sensitive log-log plot, and plots the residuals. GRAF3 plots the diameter data and fit. DUMP writes the data, fit, and residuals to the output file. INTGET reads an integer from the keyboard. AID is a rudimentary HELP routine; it contains information about data input formats. Listing of BIGFIT 162 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14. 1987 f o r CC1d=DBRU 1 C 2 C B IGF IT PURE FLUID COEXISTENCE CURVE ANALYSIS PROGRAM 3 C JOHN DEBRUYN 4 C WRITTEN MARCH 87 5 C 6 C 7 C 8 IMPLICIT R E A L * 8 ( A - H . O - Z ) 9 DIMENSION A ( 6 ) , 0 ( 6 ) . D A T L ( 5 0 0 ) , D A T V ( 5 0 0 ) , D A T A R ( 5 0 0 ) , D A T A T ( 5 0 0 ) 10 DIMENSION SIGMAY(500) 11 L0GICAL*4 I F L A G , J F L A G 12 COMMON/0/ A , 0 , D A T L . D A T V . D A T A R . D A T A T 13 COMMON / I N F O / C O R , N S , N E , N N , N P T S 14 COMMON / L L J U N K / A 1 B 1 . A B C D 2 . A C E L L 15 COMMON / D I A M / T C , D X 1 . D X 2 , D X 3 16 C 17 C 18 C 19 TC=O.ODO 20 I F L A G = . F A L S E . 21 J F L A G = . F A L S E . 22 C 23 5 CALL S T A R T ( I D A T , I T E M P , I F I T , J F L A G ) 24 C 25 CALL GETDAT( IDAT . ITEMP . I F I T , I F L A G ) 26 c 27 JFIT=IFIT+1 28 G 0 T 0 ( 2 0 , 3 0 , 1 0 ) . J F I T 29 c 30 C 1 LINEAR F IT TO N * * 1 / B VS T 31 c 32 10 CALL SETLIN(NN) 33 GOTO 40 34 c 35 c ORDER PARAMETER F IT 36 c 37 20 CALL NONLIN(NN, IF IT , IDAT) 38 GOTO 40 39 C 40 C DIAMETER F IT 41 C 42 30 CALL N O N L I N ( N N , I F I T , I D A T ) 43 C 44 40 WRITE(6 ,600) 45 NOYES=INTGET(1) 46 I F ( N O Y E S . E O . 1 ) GOTO 5 47 C 48 WRITE(6 .601) 49 C 50 600 F O R M A T ( 1 X , ' A g a i n ? ( l . O ) ' ) 51 601 F O R M A T ( / , / , 1 X , ' E n d o f r u n ' ) 52 C 53 CALL PLOTND 54 STOP 55 END 56 C 57 C 58 C 163 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 59 SUBROUTINE S T A R T ( I D A T , I T E M P , I F I T , J F L A G ) 60 C 61 C THIS SUBROUTINE GETS THE FORMAT OF THE INPUT DATA 62 C FROM THE DATA F I L E AND F IT INSTRUCTIONS FROM THE KE' 63 c 64 L0GICAL*4 JFLAG 65 c 66 I F ( J F L A G ) GOTO lO 67 IF IT=0 68 J F L A G = . T R U E . 69 WRITE(6 ,600) 70 C 71 READ(10 ,100 ) IDAT, ITEMP 72 I F ( ( I D A T . G T . 2 ) . 0 R . ( I T E M P . G T . 2 ) ) GOTO 998 73 C 74 JDAT = IDAT+ 1 75 JTEMP=ITEMP+1 76 GOTO ( 1 , 2 . 3 ) . J D A T 77 C 78 1 WRITE(6 ,610) 79 WRITE(4 ,610) 80 GOTO 4 81 2 WRITE(6 ,611) 82 WRITE(4 ,611) 83 GOTO 4 84 3 WRITE(6 ,612) 85 WRITE(4 ,612) 86 c 87 4 GOTO ( 5 , 6 . 7 ) , J T E M P 88 5 WRITE(6 ,620) 89 WRITE(4 ,620) 90 GOTO 10 91 6 WRITE(6 ,621) 92 WRITE(4,621 ) 93 GOTO 10 94 7 WRITE(6 ,622) 95 WRITE(4 ,622) 96 c 97 10 I F ( I D A T . E O . O ) GOTO 11 98 WRITE(6 ,601) 99 GOTO 12 ioo 11 WRITE(6 ,602) 101 12 IF IT= INTGET(3) 102 I F ( I F I T . E 0 . 3 ) GOTO 999 103 I F ( ( I D A T . E 0 . O ) . A N D . ( ( I F I T . N E . 0 ) . A N D . ( I F I T . N E . 2 ) ) 104 c 105 RETURN 106 C 107 998 CALL AID(O) 108 STOP 109 999 CALL AID(1 ) n o STOP 111 C 112 C FORMAT STATEMENTS 113 C 114 600 F O R M A T ( / , 1 X , ' G E N E R A L COEXISTENCE CURVE ANALYSIS 115 :10X , 'WRITTEN 0 3 / 8 7 ' ) 116 601 F O R M A T ( / . I X . ' T y p e 0 t o f i t ORDER P A R A M E T E R ' , / , 164 L i s t i n g o f B IGF IT a t 12:33:41 on AUG 14. 1987 f o r CC1d=DBRU 117 : 6 X . ' 1 D I A M E T E R ' , / , 118 : 6 X . ' 2 LINEAR FIT t o N * * 1 / b e t a ' , / , 119 : 6 X , ' 3 t o ge t H E L P . ' ) 120 602 F 0 R M A T ( I X . ' T y p e O t o f i t ORDER P A R A M E T E R ' , / . 121 : 6 X . ' 2 LINEAR FIT t o N * * 1 / b e t a ' , / . 122 :6X . ' 3 t o ge t H E L P . ' ) 123 610 F O R M A T ( / , I X , ' I n p u t f i l e c o n t a i n s FOCAL PLANE d a t a . ' ) 124 611 F O R M A T ( / , 1 X , ' I n p u t f i l e c o n t a i n s IMAGE PLANE d a t a . ' ) 125 612 F O R M A T ( / , 1 X , ' I n p u t f i l e c o n t a i n s DENSITY d a t a . ' ) 126 620 F O R M A T ( 1 X , ' I n p u t t e m p e r a t u r e s a r e In K E L V I N . ' ) 127 621 F O R M A T ( 1 X . ' I n p u t t e m p e r a t u r e s a r e m C E L S I U S . ' ) 128 622 F O R M A T ( 1 X , ' I n p u t t e m p e r a t u r e s a r e R E D U C E D . ' ) 129 100 F0RMAT(2I1) 130 C 131 END 132 C 133 C 134 C 135 C 136 SUBROUTINE G E T D A T ( I D A T . I T E M P . I F I T , I F L A G ) 137 IMPLICIT R E A L * 8 ( A - H . 0 - Z ) 138 DIMENSION A ( 6 ) . 0 ( 6 ) , D A T L ( 5 0 0 ) , D A T V ( 5 0 0 ) , D A T T ( 5 0 0 ) , D A T R ( 5 0 0 ) , D A T A R ( 5 0 0 ) 139 DIMENSION DATD(500) ,DATAT(500) 140 L0GICAL*4 IFLAG 141 COMMON/0/ A . O , D A T L , D A T V , D A T A R , D A T A T 142 COMMON / I N F O / C O R , N S , N E , N N . N P T S 143 COMMON / L L v J U N K / A IB 1, ABCD2, ACELL 144 COMMON / D I A M / T C , D X 1 , D X 2 . D X 3 145 C 146 C . . . G E T D A T READS IN DATA FROM F I L E 10 147 C . . .DATT=TEMPERATURES 148 C . . . D A T R = F R I N G E COUNT 149 C 150 IF ( IFLAG) GOTO 43 151 IFLAG=.TRUE. 152 C 153 READ(10 .100 ) NPTS,COR 1,C0R2 154 C 155 JDAT=IDAT+1 156 GOTO ( 1 0 , 2 0 , 3 0 ) , J D A T 157 C 158 C . . . F O C A L PLANE DATA 159 C 160 10 I F U T E M P . E Q . 2 ) GOTO 11 161 C 162 DO 12 1=1,NPTS 163 READ(10 ,101 ) D A T T ( I ) . D A T R ( I ) 164 12 CONTINUE 165 GOTO 41 166 C 167 C T IS REDUCED IN DATA F I L E 168 C 169 11 DO 13 1=1.NPTS 170 R E A D O 0 . 1 0 2 ) DATT( I ) ,DATR( I ) 171 13 CONTINUE 172 GOTO 41 173 C 174 C . . . I M A G E PLANE DATA 165 L i s t i n g o f B IGF IT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 175 C 176 20 I F ( I T E M P . E 0 . 2 ) GOTO 21 177 C 178 DO 22 1=1,NPTS 179 READ(10 ,103 ) DATT( I ) , D A T R ( I ) , D A T L ( I ) , D A T V ( I ) 180 DATD(I )=DATL( I ) -DATV( I ) 181 IF ( D A T R ( I ) . E Q . O . ) DATR( I )=DATL( I )+DATV( I ) 182 22 CONTINUE 183 GOTO 41 184 C 185 C T IS REDUCED IN DATA F I L E 186 C 187 21 DO 23 1=1,NPTS 188 READ(10 ,104 ) DATT( I ) ,DATR(I ) , D A T L ( I ) , D A T V ( I ) 189 DATD(I ) =DATL( I ) -DATV( I ) 190 IF ( D A T R ( I ) . E O . O . ) DATR(I ) =DATL(I)+DATV(I) 191 23 CONTINUE 192 GOTO 41 193 C 194 C . . . 195 C 196 30 I F ( I T E M P . E Q . 2 ) GOTO 31 197 C 198 DO 32 1=1,NPTS 199 READ(10 ,105 ) D A T T ( I ) . D A T V ( I ) . D A T L ( I ) , D A T D ( I ) 200 DATR(I ) =DATL( I ) -DATV( I ) 201 I F ( D A T D ( I ) . E O . O . ) DATD( I )=DATL( I )+DATV( I ) 202 32 CONTINUE 203 GOTO 41 204 C 205 C T IS REDUCED IN DATA F I L E 206 C 207 31 DO 33 1=1.NPTS 208 READ(10 ,106 ) D A T T ( I ) , D A T V ( I ) , D A T L ( I ) , D A T D ( I ) 209 DATR(I )=DATL( I ) -DATV( I ) 210 I F ( D A T D ( I ) . E Q . O . ) DATD(I )=DATL(I )+DATV(I) 211 33 CONTINUE 212 C 213 C . . . FORMAT STATEMENTS FOR DATA INPUT 214 C 215 101 F O R M A T ( 1 X . F 1 3 . 5 . 3 X . F 7 . 1 ) 216 102 F O R M A T ( 1 X . E 1 6 . 8 . 2 X . F 5 . 1 ) 217 103 F O R M A T ( 1 X . F 1 3 . 5 . 3 X , F 8 . 2 , 3 X , F 8 . 2 , 3 X , F 8 . 2 ) 218 104 F O R M A T ( 1 X , E 1 6 . 8 . 2 X , F 6 . 2 . 4 X , F 8 . 2 , 3 X , F 8 . 2 ) 219 105 F O R M A T ( 1 X , F 9 . 5 , 3 X . F 1 0 . 7 , 3 X . F 1 0 . 7 . 3 X , F 1 0 . 7 ) 220 106 F O R M A T ( 1 X , E 1 6 . 8 . 3 X . F 1 0 . 7 , 3 X . F 1 0 . 7 . 3 X , F 1 0 . 7 ) 221 C 222 C READ LORENTZ-LORENZ STUFF FROM 11 IF IT HAS BEEN 223 C 224 41 ILL=0 225 I F ( I D A T . E 0 . 2 ) GOTO 43 226 C 227 WRITE(6 .600) 228 ILL=INTGET(1) 229 I F ( I L L . N E . 1 ) GOTO 42 230 WRITE(6 .601) 231 R E A D ( 5 . 5 0 1 ) TC 232 CALL LL 166 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 233 GOTO 43 234 C 235 42 T O O . ODO 236 C 237 C 238 C 239 43 WRITE(6 ,602) NPTS 240 WRITE(4 .602) NPTS 241 C 242 WRITE(6 ,603) 243 WRITE(6 ,604) 244 READ(5 .500 ) ANS.ANE 245 C 246 IF (ANE.EQ.O . )ANE=DFLOAT(NPTS) 247 I F ( A N S . E Q . O . ) A N S - 1 . 0 D 0 248 NS=ANS 249 NE-ANE 250 WRITE(6 ,605 )NS.NE 251 WRITE(4 ,400) 252 WRITE(4 ,401 )NS,NE 253 C 254 C CORRECTION TO FRINGE NUMBER 255 C 256 I F ( I D A T . E O . O ) C0R=C0R2-O.5DO 257 I F ( I D A T . E Q . 1 ) GOTO 45 258 I F ( I D A T . E Q . 2 ) COR=O.ODO 259 GOTO 47 260 C 261 45 I F ( I F I T . N E . 1 ) GOTO 46 262 C0R=C0R2 263 C 0 R 3 = ( C 0 R 1 - 1 . 0 D 0 ) / 2 . 0 D 0 264 GOTO 47 265 C 266 46 C0R=C0R1-1.ODO 267 C 268 C TEMP CORRECTION 269 C 270 47 TC0R=273.15D0 271 I F ( I T E M P . N E . 1 ) TC0R=O.ODO 272 C 273 C . . . F I L L ARRAYS WITH DENSITY AND KELVIN 274 c 275 NN=NE-NS+1 276 F A C T L L - 1 . O D O 277 DO 50 1-1 ,NN 278 d=I+NS-1 279 DATAT(I )=DATT(d)+TCOR 280 I F ( I F I T . E O . 1 ) GOTO 52 281 c 282 c.. , .ORDER PARAM F IT 283 c 284 I F ( ( I L L . N E . 1 ) . 0 R . ( I D A T . E 0 . 2 ) ) GOTO 51 285 F A C T L L - A C E L L / ( 2 * ( A IB1+ABCD2*RH0D(DATAT(I ) ) ) ) 286 51 DATAR(I ) = (DATR(J )+COR)*FACTLL 287 GOTO 50 288 C 289 C. . . .D IAM F IT 290 C 167 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14. 19B7 f o r CC1d=DBRU 291 52 I F ( ( I L L . N E . 1 ) . O R . ( I D A T . E O . 2 ) ) GOTO 53 292 W1=0.5DO*ABCD2/ (A1B1**2) 293 W2=(DATL(J )+COR3+COR2/2.ODO)**2 + (DATV(J )+COR3-COR2/2 .ODO)* *2 294 DATAR( I )= ( (DATD(J )+COR)*ACELL - W 1 * W 2 * A C E L L * * 2 ) / ( 2 * A 1 B 1 ) 295 GOTO 50 296 53 DATAR( I )= (DATD(J)+COR)*FACTLL 297 C 298 50 CONTINUE 299 C 300 C . . . F O R M A T STATEMENTS 301 C 302 600 F O R M A T ( 1 X . / . ' D i d you s p e c i f y F I L E 11 c o n t a i n i n g LL d a t a ? ( 1 , 0 ) ' ) 303 601 F O R M A T ( 1 X , ' I n p u t TC t o u s e f o r d i a m e t e r c o r r e c t i o n . ' ) 304 602 FORMAT(1X, 'Number o f d a t a p o i n t s In f i l e = ' , 1 6 ) 305 603 F O R M A T ( 1 X , ' S t a r t Bnd end d a t a p o i n t s f o r f i t ' ) 306 604 F O R M A T * I X , 9 X , ' : ' , 9 X . ' : ' ) 307 605 F0RMAT(2I1O) 308 500 F O R M A T ( F 1 0 . 4 , F 1 0 . 4 ) 309 501 F0RMAT(F12 .6 ) 310 400 F O R M A T ( / , 1 X t ' » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' , / ) 311 401 F O R M A T ( I X , / ] ' F i t s t a r t s a t p t H',2X.14,2X,'and ends a t p t # ' . 2 X , I 4 , / ) 312 100 F 0 R M A T ( I 6 , F 4 . 1 , F 4 . 1 ) 313 C 314 C 315 RETURN 316 END 317 C 318 C 319 C 320 C 321 C 322 SUBROUTINE LL 323 C 324 C . . . R E A D S LORENZ-LORENTZ DATA FROM F ILE 11 325 C 326 IMPLICIT R E A L * 8 ( A - H . 0 - Z ) 327 COMMON / L L J U N K / A1B1 ,ABCD2 ,ACELL 328 COMMON / D I A M / T C . D X 1 , D X 2 . D X 3 329 C 330 C ALC=LCRIT . . .DLDR=SLOPE AT CRIT. . .D2LDR2=2ND DERIV AT CRIT 331 C RHOC=CRIT DENS. . .ANC=CRIT REFR INDEX 332 C DX1=C0EFFS OF DIAMETER F IT 333 C 334 R E A D O 1 , 1 1 0 ) ALC,DLDR,D2LDR2,RHOC,ANC 335 READ(11 ,111 ) DX1.DX2.DX3 336 111 F O R M A T ( F 2 0 . 1 0 , / , F 2 0 . 1 0 . / . F 2 0 . 1 0 ) 337 110 F O R M A T ( F 2 0 . 1 0 , / . F 2 0 . 1 0 , / , F 2 0 . 1 0 , / , F 2 0 . 1 0 , / . F 2 0 . 1 0 ) 338 C 339 ANSQ=ANC*ANC 340 AN1=ANS0-1.ODO 341 AN2=ANS0+2.ODO 342 AN3=3.ODO*ANS0-2.ODO 343 C 344 BB1=RHOC*DLDR/ALC 345 C 346 A 1 = A N 1 » A N 2 / ( 6 , O D O * A N C ) 347 A2=A1*AN1*AN3/ (6 .ODO*ANS0) 348 C 168 Listing of BIGFIT at 12:33:41 on AUG 14. 1987 for CC1d=DBRU 349 BB2=BB1*2.ODO*(3.ODO*AN50*ANSQ+ANSQ+2.ODO)/(AN1*AN3) 350 CC2=6.ODO*ANSQ*D2LDR2*RH0C*RH0C/(AN1*AN3*ALC) 351 DD2=BB1*BB1 352 C 353 A1B1=A1*(1.0D0+BB1) 354 ABCD2=A2*(1.ODO+BB2+CC2+DD2) 355 C 356 C 357 ALAMDA=6.328D-5 358 CELL=0.525DO 359 ACELLCALAMDA/CELL 360 C 361 C 362 RETURN 363 END 364 C 365 C 366 DOUBLE PRECISION FUNCTION RHOD(T) 367 IMPLICIT REAL*8(A-H,0-2) 368 COMMON /DIAM/ TC,DX1,DX2,DX3 369 C 370 C. . .CALC DIAMETER OF COEX CURVE USING FIT FROM PRISM DATA 371 C. . .RHOD=(RH0L+RH0V-2*RH0C)/(2*RH0C) 372 C 373 C 374 IF (TC.NE.O.O) GOTO 10 375 RH0D=O.ODO 376 RETURN 377 C 378 C 379 10 5MALLT=(TC-T)/TC 380 IF(SMALLT.LT.O.O)SMALLT=O.ODO 381 D1=DX1 382 D2=DX2*SMALLT**0.89 383 D3=DX3*SMALLT 384 C 385 RH0D=D1+D2+D3-1.ODO 386 c 387 c 388 RETURN 389 END 390 c 391 c 392 c 393 c 394 c 395 c 396 c 397 SUBROUTINE SETLIN(NN) 398 IMPLICIT REAL*8 (A-H.O-Z) 399 L0GICAL*4 LK 400 DIMENSION A(6).0(6),DATL(500),DATV(500),DATAR(5O0).DATAT(500) 401 DIMENSION Y(500).YF(500).YD(500),WT(500) 402 DIMENSION S(10).SIGMA(10).AA(10),B(10).P(10) 403 REAL*4 TP(5O0).YP(5O0),FX(10).FY(1O) 404 c 405 COMMON/0/ A.O,DATL,DATV.DATAR,DATAT 406 c 169 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14. 1987 f o r CC1d=DBRU 407 C 408 C S E T L I N SETS UP FOR AND CALLS LINEAR LSF ROUTINE 409 c F ITS TO DRHO**1 /BETA VS T 410 c 411 c 412 WRITE(6 ,600) 413 READ(5 .500 ) BETA 1 414 IF (BETA 1 . E O . 0 ) BETA 1=0.327DO 415 ABETA=1.0D0/BETA1 416 DO 20 1=1,NN 417 Y( I )=DATAR( I ) * *ABETA 418 Y P ( I ) = Y ( I ) 419 T P ( I ) = D A T A T ( I ) 420 20 CONTINUE 421 C 422 LK= .TRUE. 423 K=1 424 NWT=0 425 C 426 c. . . .CALL F ITTING ROUTINE 427 c 428 CALL D O L S F ( K , N N , D A T A T , Y , Y F , Y D , W T , N W T , S , S I G M A , A A , B , SS , LK . P) 429 c 430 c.. . OUTPUT RESULTS 431 c 432 T C = - 1 . 0 D 0 * ( P ( 1 ) / P ( 2 ) ) 433 W R I T E ( 6 , 6 0 1 ) B E T A 1 . T C 434 W R I T E ( 4 , 6 0 1 ) B E T A 1 , T C 435 c 436 c 437 WRITE(6 ,602) 438 IF ( I N T G E T O ) . N E . 1) GOTO 30 439 c 440 c.. . PLOT RESULTS 441 c 442 CALL A L A X I S ( ' T e m p e r a t u r e (K) ' . 1 5 , ' N * * ( 1 / b ) ' . 8 ) 443 CALL A L S C A L ( 0 . , 0 . . 0 . . 0 . ) 444 CALL A L G R A F ( T P , Y P , N N , - 4 ) 445 c 446 c.. DRAW A LINE THROUGH FIT 447 c 448 FX(1)=TC 449 FY (1 )=0 . 450 FX(2)=DATAT(1) 451 F Y ( 2 ) = Y F ( 1 ) 452 FX(3 )=DATAT(NN) 453 FY(3)=YF(NN) 454 c 455 CALL A L G R A F ( F X , F Y , - 3 , 0 ) 456 CALL ALDONE 457 c 458 5O0 F0RMAT(F16 .8 ) 459 600 F O R M A T ( I X , ' I n p u t v a l u e , t o u s e f o r b e t a (CR f o r 0 . 3 2 7 ) ' ) 460 601 F0RMAT(1X. 'STRAIGHT LINE FIT OF COEX CURVE DATA USING BETA= 461 : . F 6 . 4 . 2 X , ' G I V E S ' , / . ' T C « = ' , F 1 1 . 5 ) 462 602 F O R M A T ( I X , ' P L O T ? (1=YES, 0 - N O ) ' ) 463 C 464 C 170 L i s t i n g o f BIGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 465 30 RETURN 466 END 467 C 468 C 469 C 470 C 471 C 472 C 473 C 474 C 475 C 476 SUBROUTINE N0NLIN(NN,IFIT,IDAT) 477 IMPLICIT REAL*8(A-H,0-Z) 478 DIMENSION A(6),0(6),DATL(500),DATV(500),DATAR(500),DATAT(500) 479 DIMENSION P(6),V(5000),IV(70) 4BO COMMON/O/ A,0,DATL,DATV,DATAR,DATAT 481 COMMON /DIAM/ TC,DX1,DX2.DX3 482 EXTERNAL CALCR 483 C 484 C...NONLIN CONTROLS NONLINEAR FITTING I/O 485 C 486 C.. .SET INITIAL PARAMETERS AND CONSTANTS 487 C 488 M=0 489 IF(IFIT.EQ.1) GOTO 10 490 NP=6 491 WRITE(6,600) 492 GOTO 11 493 IO NP=5 494 WRITE(6,601) 495 C 496 11 IF (IFIT.NE.O) GOTO 12 497 WRITE(6,606) 498 GOTO 13 499 12 WRITE(6.607) 500 C 501 C...NUMBER OF PARAMETERS 502 C 503 13 WRITE(6,602) 504 M1=INTGET(NP) 505 IF (M1.EQ.O.AND.M.NE.O) M1=M 506 IF(M1.EQ.O) GOTO 92 507 M=M1 508 C 509 C...INPUT INITIAL VALUES 510 C 511 WRITE(6.603) 512 WRITE(6,604) 513 WRITE(6.605) 514 WRITE(6.608) 515 C 516 d=1 517 DO 20 1-1.NP 518 WRITE(6.609) I 519 READ(5,500)K.X 520 IF (K.EQ.1) GOTO 21 521 IF (K.EQ.2) GOTO 23 522 Q(I)-X 171 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 523 GOTO 20 524 21 P(J)=X 525 0(1 ) =9999 . 526 J=J+1 527 GOTO 20 528 23 IF ( 0 ( 1 ) . E O . 9 9 9 9 . ) d=d+1 529 C 530 20 CONTINUE 531 C 532 d=J-1 533 IF ( U . N E . M ) GOTO 92 534 C 535 C PLOT F IT AND RESIDUALS 536 C ROUTINE GRAF2 DOES THE WORK 537 C 538 WRITE(6 ,610) 539 I F ( I N T G E T ( 1 ) . N E . 1 ) GOTO 31 540 30 WRITE(6 ,611) 541 IRES=INTGET(1) 542 C 543 IF ( I F I T . E O . O ) CALL G R A F 2 ( N N , P , I R E S ) 544 IF ( I F I T . E 0 . 1 ) CALL G R A F 3 ( N N , P , I R E S ) 545 C 546 C . . .FIND OUT WHAT TO DO NEXT 547 C 548 31 WRITE(6 ,612) 549 N0YES=INTGET(2) 550 IF (NOYES.EO.1 )G0T0 11 551 IF (NOYES.EO.0)GOTO 50 552 C 553 C SET UP NL2SN0 554 C 555 CALL D F A L T ( I V . V ) 556 IV(14)=1 557 IV(21)=4 558 IV (15 )= -2 559 V (42 )=0 . 560 V (29 )=1 .OD-13 561 V (40 )=1 .OD-13 562 IV(17)=400 563 IV(18)=400 564 IPARM=IFIT 565 C 566 35 CALL NL2SN0(NN,M,P ,CALCR, IV ,V , IPARM,RPARM,FPARM) 567 C 568 C OUTPUT RESULTS TO SCREEN AND F ILE 4 569 C 570 DO 40 1=1,NP 571 IF ( 0(1).E O . 9 9 9 9 . 0 ) GOTO 41 572 W R I T E ( 4 , 4 0 0)1,A ( I ) 573 W R I T E ( 6 , 4 0 0)1,A ( I ) 574 GOTO 40 575 41 W R I T E ( 4 , 4 0 1)1,A ( I ) 576 W R I T E ( 6 , 4 0 1)1.A ( I ) 577 40 CONTINUE 578 C 579 C THIS VALUE OF TC IS SENT BACK TO RHOD 580 C AND USED TO CALC DIAMETER CORRECTION 172 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14. 1987 f o r CC1d=DBRU 581 C 582 I F ( I F I T . N E . O ) GOTO 42 583 TC=A(2) 584 C 585 42 CHIS0R=2.ODO*V(10) /DFL0AT(NN-M) 586 WRITE(4.402)CHIS0R 587 WRITE(6.4O2)CHIS0R 588 C 589 C 590 WRITE(6 .610) 591 I F ( I N T G E T ( 1 ) . E Q . 1 ) GOTO 30 592 GOTO 31 593 C 594 C 595 50 I F ( ( I F I T . N E . 0 ) . O R . ( I D A T . E O . 2 ) ) GOTO 51 596 WRITE(6 .613) TC 597 51 RETURN 598 C 599 C . . . E R R O R ROUTINES 600 C 601 92 WRITE(6 .620) 602 GOTO 11 603 C 604 C . . . F O R M A T STATEMENTS 605 C 606 400 FORMAT(1X, 'CONSTANT H'.I3.3X.F16.8) 607 401 FORMAT(1X, 'PARAMETER # ' , I 3 , 3 X , F 1 6 . 8 ) 608 402 F O R M A T ( / , 1 X , ' C H I * * 2 ' . 6 X . E 1 6 . 8 , / , / ) 609 500 F 0 R M A T ( I 1 , I X , F 1 6 . 8 ) 610 600 F O R M A T ( / , 1 X , ' N o n l I n e a r f i t t o o r d e r p a r a m e t e r d a t a . ' , 611 : / , 1 X , ' M a x o f 6 p a r a m e t e r s . ' ) 612 601 F O R M A T ( / , 1X, ' F i t t o d i a m e t e r d a t a . Max o f 5 p a r a m e t e r s . ' ) 613 602 FORMAT(1X, 'How many p a r a m e t e r s f o r f i t ? ' ) 614 603 F 0 R M A T ( / , 1 X , ' I n p u t 1 f o l l o w e d by i n i t i a l g u e s s i f F R E E ' ) 615 604 F O R M A T ( 1 X , ' I n p u t O f o l l o w e d by v a l u e i f F I X E D ' ) 616 605 F O R M A T ( 1 X , ' T o l e a v e a p a r a m e t e r UNCHANGED, Input 2 ' ) 617 606 F O R M A T ( / . 1 X . ' P 1 = B 0 P2=TC P3=BETA P4=B1 P5=B2 P6=DELTA' ) 618 607 F O R M A T ( / , 1 X , ' P 1 B T C P2=A0 P3=A(1-a) P4=A1 P5=A(1-a+D) ' ) 619 608 F 0 R M A T ( / , I X , ' F o r m a t 1 s ' . / . 1 X . ' 1 1234 .567890 ' ) 620 609 FORMAT ( ' PAR AM .tl' , 2X, 12 ) 621 610 F O R M A T ( / , 1 X . ' P l o t ? ( 1 , 0 ) ' ) 622 611 F O R M A T ( I X , ' R e s i d u a 1 s p l o t ? ( 1 . 0 ) ' ) 623 612 F 0 R M A T ( 1 X , ' O f o r STOP, 1 f o r NEW PARAMETERS, 2 f o r F I T ' ) 624 613 F O R M A T ( / . I X . ' R e m e m b e r , nex t r u n t h r o u g h w i l l u s e T C = ' , F 1 3 . 8 , ' t o c a l c 625 620 FORMAT( IX , 'Wrong number o f p a r a m e t e r s . R e s t a r t . . . ' ) 626 C 627 C 628 C 629 END 630 C 631 C 632 C 633 C 634 C 635 C 636 SUBROUTINE CALCR(N ,M ,P ,NF ,R , IPARM,RPARM,FPARM) 637 IMPLICIT R E A L * 8 ( A - H , 0 - Z ) 638 DIMENSION P ( M ) , R ( N ) 173 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 639 DIMENSION A ( 6 ) . 0 ( 6 ) , D A T L ( 5 0 0 ) , D A T V ( 5 0 0 ) , D A T A R ( 5 0 0 ) , D A T A T ( 5 0 0 ) , S M A L L T ( 5 0 0 ) 640 COMMON/O/ A , 0 , D A T L , D A T V , D A T A R , D A T A T 641 C 642 C . . .CALCR CALCULATES NONLINEAR FUNCTION AND RESIDUALS 643 C . . .FOR NONLINEAR FITTING PROGRAM 644 C 645 NP = 6 646 IF ( I P A R M . E O . 1 ) NP=5 647 C 648 d=1 649 DO 10 1=1,NP 650 IF ( 0 ( 1 ) . E O . 9 9 9 9 . 0 ) GOTO 11 651 A ( I ) = 0 ( I ) 652 GOTO 10 653 11 A ( I ) = P ( J ) 654 d=d+1 655 10 CONTINUE 656 C 657 C CONSTRAINTS GO HERE IF WANTED 658 C 659 IF ( IPARM.NE.O) GOTO 30 660 C 661 C . . 662 C 663 DO 20 I - I .N 664 S M A L L T ( I ) = D A B S ( ( A ( 2 ) - D A T A T ( I ) ) / A ( 2 ) ) 665 R ( I ) = A ( 1 ) * S M A L L T ( I ) * * A ( 3 ) 666 R Z = A ( 4 ) * S M A L L T ( I ) * * A ( 6 ) + A ( 5 ) * S M A L L T ( I ) * * ( 2 * A (6)) 667 R(I ) = R( I ) * (1 .ODO+RZ) 668 R ( I ) = R ( I ) - D A T A R ( I ) 669 20 CONTINUE 670 C 671 RETURN 672 C 673 C . . .DIAMETER 674 C 675 30 DO 31 1=1,N 676 S M A L L T ( I ) = D A B S ( ( A ( 1 ) - D A T A T ( I ) ) / A ( 1 ) ) 677 R ( I ) = A ( 2 ) + A ( 3 ) * S M A L L T ( I ) * * 0 . B 9 + A ( 4 ) * S M A L L T ( I ) 678 R(I )=R(I ) + A ( 5 ) * S M A L L T ( I ) * * 1 . 3 9 679 R ( I ) = R ( I ) - D A T A R ( I ) 680 31 CONTINUE 681 C 682 RETURN 683 END 684 C 685 C 686 C 687 C 688 SUBROUTINE G R A F 2 ( N N , P . I R E S ) 689 R E A L * 8 A ( 6 ) , 0 ( 6 ) , D A T L ( 5 O O ) , D A T V ( 5 O O ) . D A T A R ( 5 O 0 ) . D A T A T ( 5 0 0 ) 690 R E A L * B P ( 6 ) 691 R E A L * 4 R E S I D ( S O O ) , X T ( 5 0 0 ) . Y R ( 5 0 0 ) . V R 1 ( 5 O 0 ) . Z X ( l O ) , Z Y ( 1 0 ) 692 R E A L * 4 R D ( 5 0 0 ) , X D ( 5 0 0 ) . Y D ( 5 0 0 ) . Y D 1 ( 5 0 0 ) 693 COMMON/0/ A , 0 , D A T L . D A T V . D A T A R , D A T A T 694 C 695 C GRAF2 PLOTS DATA AND NONLINEAR F IT 696 C ON SENSIT IVE LOG-LOG PLOT 174 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 697 C A R R A Y S : D U M P . . . P L O T 698 C RD RESID . . . R E S I D U A L S 699 C XD XT . . . R E D U C E D TEMP 700 C YD YR . . . F I T TO D E N S I T Y / T * * B E T A 701 C YD 1 YR 1 . . .DENSITY D A T A / T * * B E T A 702 C 703 C 704 d=1 705 DO 10 1=1,6 706 IF ( 0 ( 1 ) . E O . 9 9 9 9 . 0 ) GOTO 11 707 A ( I ) « 0 ( I ) 708 GOTO 10 709 11 A ( I ) = P ( J ) 710 u=J+1 711 10 CONTINUE 712 C 713 C 714 DO 20 1=1,NN 715 AXT = ABS(SNGL( (A(2 ) -DATAT( I ) ) / A ( 2 ) ) ) 716 XD(I)=AXT 717 XT( I )=ALOG10(AXT) 718 X3=AXT**A(3 ) 719 C 720 YR1( I )=DATAR( I ) /X3 721 R = A ( 1 ) * A X T * * A ( 3 ) 722 RZ= A ( 4 ) * A X T * * A ( 6 ) + A ( 5 ) * A X T * * ( 2 . * A ( 6 ) ) 723 YR( I )=R* (1 .O+RZ) /X3 724 R E S I D ( I ) = ( Y R 1 ( I ) - Y R ( I ) ) / Y R ( I ) * 1 0 0 . 725 RD( I )=RESID( I ) 726 C 727 YR1( I )=AL0G10(YR1( I ) ) 728 YD1( I )=YR1( I ) 729 YR( I )=AL0G1O(YR( I ) ) 730 YD( I )=YR( I ) 731 C 732 20 CONTINUE 733 CALL P L C T R L ( ' M E T R I C , 1 ) 734 C 735 C . . . S C A L E ARRAYS AND DRAW GRAPH 736 C 737 SCAL=5 .0 738 DO 35 1=1,NN 739 X T ( I ) = ( X T ( I ) + 6 . 0 ) * S C A L 740 35 CONTINUE 741 CALL A X C T R L ( ' L O G S ' , 1 ) 742 CALL AXPL0T( 'SMALL T ; ' , 0 . 0 , 2 0 . . - 6 . O , - 2 . 0 ) 743 CALL A X C T R L ( ' L O G S ' , 0 ) 744 CALL S C A L E ( Y R 1 , N N , 2 0 . . X M I N 1 , D X 1 , 1 ) 745 DO 55 1=1.NN 746 YR( I )= (YR( I ) -XMIN1) /DX1 747 55 CONTINUE 748 CALL A X P L O T C L O G (DELTA R H O / ( T * * B E T A ) ) ; ' . 9 0 . . 2 0 . , X M I N 1 , D X 1 ) 749 DO 45 1=1.NN 750 CALL S Y M B 0 L ( X T ( I ) . Y R 1 ( I ) . 0 . 2 . 4 , 0 . 0 . - 1 ) 751 45 CONTINUE 752 CALL L I N E ( X T . Y R . N N , 1 ) 753 C 754 C DO RESIDUALS PLOT IF REQUIRED L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 755 C 756 I F ( I R E S . N E . 1 ) GOTO 12 757 C 758 CALL P L 0 T ( 2 5 . . 0 . , - 3 ) 759 CALL S C A L E ( R E S I D . N N . 2 0 . . X M I N 2 . D X 2 , 1 ) 760 CALL A X C T R L ( ' L O G S ' . 1 ) 761 CALL AXPLOT( 'SMALL T ; ' . 0 . 0 , 2 0 . , - 6 . 0 . - 2 . 0 ) 762 CALL A X C T R L ( ' L O G S ' . 0 ) 763 CALL A X P L O T ( ' % RESIDUAL; ' , 9 0 . . 2 0 . . X M I N 2 . D X 2 ) 764 DO 65 1=1,NN 765 CALL S Y M B 0 L ( X T ( I ) . R E S I D ( I ) . O . 2 . 4 , O . O . - 1 ) 766 65 CONTINUE 767 C 768 C . . .DRAW ZERO LINE 769 C 770 ZX(1 )=0 . 771 ZX(2)=20 . 772 ZY (1 )= (0 . -XMIN2 ) /DX2 773 Z Y ( 2 ) = ( 0 . - X M I N 2 ) / D X 2 774 775 C 776 CALL L I N E ( Z X . Z Y , 2 , 1 ) 777 C 778 C . . .TIDY UP PLOT AND DUMP DATA 779 C 780 12 CALL P L C T R L ( ' M E T R I C , 0 ) 781 CALL P L 0 T ( 1 2 . . 0 . , - 3 ) 782 C 783 CALL DUMP(RD,XD.YD.YD1.NN) 784 C 785 RETURN 786 END 787 C 788 C 789 C 790 C 791 C 792 SUBROUTINE G R A F 3 ( N N . P , I R E S ) 793 REAL*8 A ( 6 ) . 0 ( 6 ) . D A T L ( S O O ) , D A T V ( 5 0 0 ) , D A T A R ( 5 0 0 ) , D A T A T ( 5 0 0 ) 794 REAL»8 P ( 6 ) 795 R E A L * 4 R E S I D ( 5 0 0 ) . X T ( 5 0 0 ) , Y R ( 5 0 0 ) . Y R 1 ( 5 0 0 ) 796 R E A L M ZT( 10) .ZR( 10) 797 COMMON/0/ A . O , D A T L . D A T V , D A T A R . D A T A T 798 C 799 C . . .PLOTS DIAMETER DATA AND FIT 800 C 801 J=1 802 DO 10 1=1,6 803 IF ( 0 ( 1 ) . E O . 9 9 9 9 . 0 ) GOTO 11 804 A ( I ) = 0 ( I ) 805 GOTO 10 806 11 A ( I ) = P ( u ) 807 J = J+1 808 10 CONTINUE 809 C 810 C . . .CALC F IT POINTS 811 C B12 DO 20 I=1.NN 176 L i s t i n g o f B I G F I T a t 12:33:41 o n AUG 14, 1987 f o r CC1d=DBRU 8 1 3 X T ( I ) = A B S ( S N G L ( ( A ( 1 ) - D A T A T ( I ) ) / A ( 1 ) ) ) 8 1 4 Y R 1 ( I ) = D A T A R ( I ) 8 1 5 Y R ( I ) = A ( 2 ) + A ( 3 ) * X T ( I ) * * 0 . 8 9 + A ( 4 ) * X T ( I ) + A ( 5 ) * X T ( I ) * * 1 8 1 6 R E S I D ( I ) = ( Y R 1 ( I ) - Y R ( I ) ) 8 1 7 2 0 CONTINUE 8 1 8 C 8 1 9 C. . DRAW GRAPH 8 2 0 C 821 C A L L A L A X I S ( ' ( T - T C ) / T C ' , 9 , ' D I A M E T E R ' . 8 ) 8 2 2 C A L L A L S C A L ( 0 . . 0 . . 0 . , 0 . ) 8 2 3 C A L L A L G R A F ( X T , Y R 1 , N N , - 4 ) 8 2 4 C A L L ALGRAF.(XT , YR, -NN, 0 ) B 2 5 C A L L ALDONE 8 2 6 C 8 2 7 C. . .RESIDUALS PLOT 8 2 8 C 8 2 9 I F ( I R E S . N E . 1 ) GOTO 3 0 8 3 0 C 831 Z T ( 1 ) = 0 . 8 3 2 Z T ( 2 ) = 0 . 1 8 3 3 Z R ( 1 ) = 0 . 8 3 4 Z R ( 2 ) = 0 . 8 3 5 C A L L A L A X I S ( ' ( T - T C ) / T C , 9 . ' R E S I D U A L ' , 8 ) 8 3 6 C C A L L A L S C A L ( 0 . , 0 . 1 , 0 . . 0 . ) 8 3 7 C A L L A L G R A F ( X T , R E S I D , N N , - 4 ) 8 3 8 C A L L A L G R A F ( Z T , Z R , - 2 , 0 ) 8 3 9 C A L L ALDONE 8 4 0 C 8 4 1 C. . DUMP DATA 8 4 2 C 8 4 3 3 0 C A L L DUMP(RESID,XT,YR,YR1,NN) 8 4 4 C 8 4 5 RETURN 8 4 6 END 8 4 7 C 8 4 8 C 6 4 9 SUBROUTINE DUMP(RD,XD,YD,YD1,NN) 8 5 0 R E A L * 4 R D ( 5 O 0 ) , X D ( 5 O 0 ) , Y D ( 5 0 0 ) , Y D 1 ( 5 0 0 ) 851 C 8 5 2 C DUMPS DATA TO F I L E 4 8 5 3 C 8 5 4 W R I T E ( 6 , 6 0 0 ) 8 S 5 I F ( I N T G E T ( 1 ) . N E . 1 ) GOTO 2 0 8 5 6 C 8 5 7 W R I T E ( 4 . 4 0 0 ) 8 5 8 DO 10 1=1.NN 8 5 9 W R I T E ( 4 , 4 0 1 ) I . X D ( I ) . Y D 1 ( I ) . Y D ( I ) . R D ( I ) 8 6 0 10 CONTINUE 86 1 C 8 6 2 4 0 0 FORMAT(13X,'SMALL T ' , 9 X , ' D A T A ' , 9 X , ' F I T ' , 7 X , '% R E S I D ' / ) 8 6 3 401 F O R M A T ( 1 X , I 6 , E 1 6 . 6 , 2 X , F 1 0 . 5 . 2 X , F 1 0 . 5 , 2 X , F 1 0 5 ) 8 6 4 6 0 0 FORMAT(IX.'Dump d a t a t o u n i t 4 ? ( 1 . 0 ) ' ) 8 6 5 C 8 6 6 C 8 6 7 2 0 RETURN 8 6 8 END 8 6 9 C 8 7 0 C 177 L i s t i n g o f B IGFIT a t 12:33:41 on AUG 14, 1987 f o r CC1d=DBRU 871 C 872 C 873 INTEGER FUNCTION INTGET(MAX) 874 C 875 C READS AN INTEGER BETWEEN O AND MAX FROM KEYBOARD 876 C 877 I F ( M A X . G T . 9 ) GOTO 3 878 1 READ(5 ,50O)L 879 I F ( L . G T . M A X ) GOTO 2 880 INTGET=L 881 RETURN 882 C 883 2 WRITE(6 .600) MAX 884 GOTO 1 885 C 886 C MAX TOO BIG 887 C 888 3 WRITE(6 .601) 889 MAX = 9 890 GOTO 1 891 C 892 600 FORMAT( IX . ' INPUT A NUMBER BETWEEN O A N D ' , I X , I I ) 893 500 FORMAT(I I ) 894 601 F O R M A T ( 1 X , ' . . . W A R N I N G . . . VALUE OF MAX SENT TO INTGET TOO B I G , SET = 9 ' ) 895 C 896 END 897 C 898 C 899 C 900 C 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 10 11 C C C 400 SUBROUTINE AIO(I ) I F ( I . E O . 1 ) GOTO 10 WRITE(4.400) WRITE(6.4O0) GOTO 11 WRITE(4.410) WRITE(6.410) WRITE(4.411) WRITE(4.412) WRITE(4.413) WRITE(6.411) WRITE(6.412) WRITE(6,413) 410 411 412 , / , ' H e r e NNNNNN i s the number o f . s u b t r a c t e d from the t o t a l ' F O R M A T ( / , 1 X , ' V o u r Input f i l e appears to be In the wrong format ' . / . ' f o r t h i s program. A d e s c r i p t i o n of the proper i n p u t ' . / . ' format f o l l o w s . ' ) F 0 R M A T ( / . 2 0 X . ' F i t t i n g program HELP r o u t i n e ' ) . . . . F 0 R M A T ( / , ' A copy o f these i n s t r u c t i o n s i s b e i n g w r i t t e n onto your output f i l e 04.'././.'The f i r s t l i n e o f your inpu t f i l e must c o n t a i n a two d i g i t number In t h e ' . / . ' f i r s t two columns. T h i s number i d e n t i f i e s the type o f Input da ta as f o l l o w s : ' . / . / . 1 0 X , ' F i r s t d i g i t : 0 . . . F o c a l p l a n e d a t a ' . / . 2 4 X . ' 1 . . . I m a g e p l a n e d a t a ' , / . 2 4 X . ' 2 . . . O e n s i t y d a t a ' . / . / , 10X , 'Second d i g i t : 0 . . . T i n K e l v i n ' , / , 2 4 X , ' 1 . . . T i n C e l s i u s ' . / , 2 4 X , ' 2 . . . t In reduced t e m p e r a t u r e ' , / ) FORMAT('The second l i n e must be i n the form NNNNNNAA.ABB.B, s t a r t i n g i n column 1. da ta p o i n t s In the f i l e i n 16 format ; A A . A ' . / . ' i s 0 . f o r i - p i a n e da ta which Is to have . / . ' f r i n g e number, and i s 1. o t h e r w i s e ; BB.B i s a c o r r e c t i o n ' , ' to be added to t h e ' , / . ' t o t a l f r i n g e number f o r f - p l a n e ' . ' da ta o r to N l - N v f o r 1-piane d a t a . ' , / ) 413 FORMAT('The da ta f o l l o w s , s t a r t i n g on the t h i r d l i n e of the f i l e . The format s t a t e m e n t s ' . / , ' b e l o w a r e used to read In the d a t a . They a r e l a b e l l e d by the a p p r o p r i a t e ' , / , ' t w o - d i g i t number as d e s c r i b e d above, and f o l l o w e d by the q u a n t i t i e s to b e ' • ' r e a d . ' , / , / isX.'OO.OI ' F 0 R M A T ( 1 X . F 1 3 . 5 . 3 X . F 7 . 1 ) ' , 2 0 X , ' T . N t o t ' . / , 5 X . ' 0 2 F 0 R M A T ( 1 X . E 1 6 . 8 . 2 X . F 5 . 1 ) ' . 2 0 X . ' t . N t o t ' . / , 5 X . ' 1 0 . 1 1 F 0 R M A T (IX . F 1 3 . 5 . 3 X , F 8 . 2 . 3 X , F 8 . 2 , 3 X , F 8 . 2 ) ' . 4 X . ' T . N t o t , N I , N V , / , 5 X . ' 1 2 F O R M A T ( 1 X . E 1 6 . 8 , 2 X . F 6 . 2 , 4 X , F 8 . 2 . 3 X , F 8 . 2 ) ' . 4 X . ' t . N t o t . N I , N V . / , 5 X , ' 2 0 . 2 1 F 0 R M A T ( 1 X . F 9 . 5 . 3 X . F 1 0 . 7 , 3 X , F 1 0 . 7 . 3 X . F 1 0 . 7 ) ' , 2 X , ' T . R h o v . R h o l . R h o d ' , / , 5 X . ' 2 2 F 0 R M A T ( 1 X . E 1 6 . 8 . 3 X . F 1 p . 7 , 3 X , F 1 0 . 7 . 3 X . F 1 0 . 7 ) ' . I X . ' t . R h o v . R h o l . R h o d ' . / ) RETURN ENO oo 179 C-2 L L N E W E R and C O E X 8 6 - Prism Cell Data Analysis Programs The two programs listed in this section were used to analyze the data for L(p) and the coexisting densities obtained from the prism experiment. LLNEWER reads in data in the from of micrometer readings and cell masses, calculates the Lorenz-Lorentz coefficient, fits the results to a polynomial in the den-sity, then plots and outputs the results. A variant of this program called LLFITTON (not listed here) fits £ to a polynomial in the refractive index. COEX86 reads in micrometer and thermistor readings and calculates the den-sities of the coexisting phases as a function of temperature, using the fit to £(n) determined by LLFITTON. The coexistence curve as output by COEX86 can be analyzed by BIGFIT, described above. 180 L i s t i n g of L L N E W E R L i s t i n g o f LLNEWER a t 12 :43 :06 on AUG 14. 1987 f o r CC1d=DBRU 1 IMPLICIT R E A L * 8 ( A - H . O - Z ) 2 DIMENSION A M A S S ( 2 0 0 ) , T H F I T ( 1 0 ) , A N G A S ( 2 0 0 ) , B M I C ( 2 0 0 ) , A M I C ( 2 0 0 ) , T H E T A ( 2 0 0 ) 3 DIMENSION W T ( 2 0 0 ) . W T 2 ( 2 0 0 ) , X X ( 5 0 ) . Y Y ( 5 0 ) . Y Y N ( 5 0 ) . Y Y M ( 5 0 ) . X X M ( 5 0 ) 4 DIMENSION ZER0(200 ) ,FUNX(200 ) ,FUNY(200 ) 5 DIMENSION INC(200) 6 L0GICAL*4 LK 7 C 8 C 9 C NEW VERSION JULY 1986 INCLUDES POLYNOMIAL F IT TO LL(RHO) 10 C 11 C CONSTANTS 12 C 13 DATA A N A I R . A N S A F F / 1 . 0 0 0 2 9 . 1 . 6 6 / 14 C 15 P IETC=12 .0D0*3 .1415927D0 16 C 17 C GET DATA FROM F I L E # 4 18 C 19 R E A D ( 4 , 4 0 1 ) M 20 C 21 C GET CELL DATA 22 C 23 READ(4 ,403 ) CELLM,CELLV,WTMOL 24 403 F 0 R M A T ( F 1 0 . 5 . I X . F 8 . 4 , 1 X , F 8 . 4 ) 25 C 26 WRITE(8 ,606) CELLM.CELLV,WTMOL 27 WRITE(6 ,606) CELLM.CELLV,WTMOL 28 606 F O R M A T ( 1 X . ' C E L L MASS = ' , F 1 0 . 4 . B X . ' C E L L VOL. = ' , F 1 0 . 4 . / , 1 X . ' M O L E C U L A R 29 C 30 C 31 C PARAMETERS FOR 3RD ORDER POLY FIT TO CALIB DATA 32 C AND C E L L ANGLES 33 C 34 R E A D ( 4 , 4 0 4 ) A L P H A , T H F I T ( 1 ) , T H F I T ( 2 ) , T H F I T ( 3 ) 35 WRITE(8 ,808) A L P H A , ( T H F I T ( I ) , I = 1 , 3 ) 36 808 F O R M A T ( 1 X ' A L P H A = ' , F 8 . 4 , / , 1 X , ' M I C R O M E T E R FIT P A R A M E T E R S : ' , 3 E 2 0 . 1 0 ) 37 404 F O R M A T ( F B . 4 , 2 X . E 2 0 . 1 0 . E 2 0 . 1 0 . E 2 0 . 1 0 ) 38 PI=3. 14159265D0 39 ALPHA=ALPHA*PI /180 . 40 C 41 C 42 C 43 WRITE(8 ,601) M 44 601 FORMAT( IX, 'NO. OF P T S . •» ' . 3 X . I 4 , / ) 45 401 FORMAT(1X,14) 46 C 47 C DATA FOR AIR 48 C 49 R E A D ( 4 . 4 0 2 ) AIRM,AIRMIC,ZEROAR 50 AIRMIC=AIRMIC-ZEROAR 51 S INAIR=FUNC(AIRMIC.THFIT ,3 ) 52 THAIR=DARSIN(SINAIR) 53 COSAIR=DCOS(THAIR) 54 C 55 TANALF=DTAN(ALPHA) 56 C 57 C READ DATA 58 C 181 L i s t i n g o f LLNEWER a t 12 :43:06 on AUG 14, 1987 f o r CC1d=DBRU 59 DO 10 1=1.M 60 1 READ(4 .402 ) A M A S S f I ) , A M I C ( I ) , 2 E R 0 ( I ) . I N C ( I ) 61 I F ( 2 E R 0 ( I ) . E 0 . O . ) GOTO 2 62 B M I C ( I ) = A M I C ( I ) - 2 E R 0 ( I ) 63 GOTO 10 64 2 M=M-1 65 GOTO 1 66 10 CONTINUE 67 402 F O R M A T ( 1 X . F 8 . 4 , 1 X . F 7 . 4 . 1 X . F 7 . 4 , 2 X , I 1 ) 68 C 69 C CALC SIN THETA 70 C 71 u=M 72 C 73 DO 20 I=1.M 74 S INTH=FUNC(BMIC( I ) .THF IT ,3 ) 75 THETA(I )=DARSIN(SINTH) 76 COSTH=DCOS(THETA(I ) ) 77 C 78 C CALC INDEX OF REFR. 79 C 80 ANGASCI)=ANAIR*( 1 . ODO+C0STH-C0SAIR+ (S INTH-S INAIR) /TANALF 81 C 82 C CALC DENSITY 83 C 84 RHO(I) = (AMASS(I ) - C E L L M ) / C E L L V 85 RHO(I)=RHO(I)/WTMOL 86 C 87 IF ( I N C ( I ) . E Q . 1 ) GOTO 21 88 AN2 = ANGAS( I ) *ANGAS( I ) 89 AN21=AN2-1.ODO 90 AN22=2.0D0*AN2+1.ODO 91 FUNY( I )=P IETC*AN2*RH0( I ) / (AN21 *AN22) 92 FUNX( I )=2 .0D0*AN21/AN22 93 C 94 C CALC WEIGHTING FACTORS 95 c 96 W T ( I ) = ( ( A M A S S ( I ) - C E L L M ) / ( A M A S S ( 1 ) - C E L L M ) ) * * 2 97 c 98 c CALC LL 99 c 100 A L L ( I ) = (AN2-1 .ODO) / ( (AN2+2 .ODO) *RH0( I ) ) 101 SUM=SUM+ALL(I) 102 GOTO 20 103 c 104 21 J = J - 1 105 A L L ( I ) = 0 . 0 106 C 107 20 CONTINUE 108 AVGLL=SUM/DFLOAT(J) 109 C 110 C OUTPUT RESULTS 111 C 112 WRITE(8 ,604) 113 DO 30 I=1,M 114 WRITE(8 ,605) R H O ( I ) , A N G A S ( I ) , A L L ( I ) 115 30 CONTINUE 116 605 F 0 R M A T ( 3 ( 3 X , F 1 2 . 6 ) ) 182 L i s t i n g o f LLNEWER a t 12 :43 :06 on AUG 14, 1987 f o r CC1d=DBRU 117 604 F O R M A T ( 7 X , ' M A S S ( G ) ' , 7 X , ' M I C . R D G . ' , 5 X . ' D E N S I T Y ( G / C C ) ' , 3 X . ' T H E T A ( R A D ) 118 C 119 C 120 C F IT TO INDEX DATA 121 C 122 NWT2-0 123 NWT=0 124 C 125 DO 16 1=1,M 126 A N G A S ( I ) = ( A N G A S ( I ) - 1 . 0 D 0 ) / R H 0 ( I ) 127 16 CONTINUE 128 C 129 CALL F IT (RH0 ,ANGAS.WT2 ,YYN,XX,NWT2,M,2 ) 130 C 131 C 132 C F IT TO LL DATA 133 C 134 CALL F I T ( R H O . A L L , W T , Y Y . X X , N W T , J , 2 ) 135 C 136 C F IT TO FUNNY FUNC 137 C 138 CALL F IT (FUNX,FUNY,WT2.YYM,XXM,NWT2,d ,1 ) 139 C 140 CALL G R A F ( R H O , A L L , A N G A S , F U N X . F U N Y , X X . Y Y , Y Y N , Y Y M , X X M , M, d) 141 C 142 C 143 STOP 144 END 145 C 146 C 147 C 148 C 149 C 150 C 151 C 152 C 153 C 154 c 155 SUBROUTINE G R A F ( R H O , A L L , A N G A S . F U N X , F U N Y . X X , Y Y , Y Y N , Y Y M , X X M , M , d ) 156 R E A L * 8 R H O ( 1 ) , A L L ( 1 ) , A N G A S ( 1 ) , X X ( 1 ) ,YY (1 ) .YYN(1 ) 157 R E A L * 8 F U N X ( 1 ) , F U N Y ( 1 ) , X X M ( 1 ) , Y Y M ( 1 ) 158 DIMENSION X ( 2 O 0 ) . Y 1 ( 2 O O ) , Y 2 ( 2 0 0 ) . X X 1 ( 5 0 ) , Y Y 1 ( 5 0 ) . Y Y 2 ( 5 0 ) 159 DIMENSION Y 3 ( 2 0 0 ) , X 3 ( 2 0 0 ) , X X 3 ( 5 0 ) , Y Y 3 ( 5 0 ) 160 c 161 DO 40 I=1,M 162 X ( I )=SNGL(RHO( I ) ) 163 Y 1 ( I ) = S N G L ( A L L ( I ) ) 164 Y2 ( I )=SNGL(ANGAS( I ) ) 165 Y3 ( I )=FUNY( I ) 166 X3 ( I )=FUNX( I ) 167 40 CONTINUE 168 C 169 MM=40 170 C 171 DO 50 I « 1 . M M 172 XX1 ( I )=SNGL(XX( I ) ) 173 Y Y 1 ( I ) = S N G L ( Y Y ( I ) ) 174 YY2( I )=SNGL(YYN( I ) ) 183 L i s t i n g o f LLNEWER a t 12 :43:06 on AUG 14, 1987 f o r CC1d=DBRU 175 XX3( I )=XXM(I ) 176 YY3( I )=YYM(I ) 177 50 CONTINUE 178 C 179 C PLOT LL 180 C 181 CALL ALAXIS ( 'DENSITY ( M O L E / C C ) ' , 1 7 . ' L L ( C C / M O L E ) ' . 1 2 ) 182 CALL A L S C A L ( 0 . 0 . 0 . 0 1 5 , O . O O O . 0 . 0 ) 183 CALL A L G R A F ( X , Y 1 , d , - 4 ) 184 CALL A L G R A F ( X X 1 , Y Y 1 , - M M . O ) 185 C. 186 C PLOT INOEX 187 C 188 CALL ALAXIS ( 'DENSITY ( M O L E / C C ) ' , 1 7 , ' I N D E X ' , 5 ) 189 CALL A L S C A L ( 0 . 0 , 0 . 0 1 5 , 0 . 0 . 0 . 0 ) 190 CALL A L G R A F ( X , Y 2 . M , - 4 ) 191 CALL A L G R A F ( X X 1 , Y Y 2 , - M M . O ) 192 C 193 c PLOT FUNNY 194 c 195 CALL A L A X I S ( ' F U N X ' , 4 , ' F U N Y ' , 4 ) 196 CALL A L S C A L ( 0 . . 0 . , 0 . . 0 . ) 197 CALL A L G R A F ( X 3 , Y 3 , d . - 4 ) 198 CALL A L G R A F ( X X 3 . Y Y 3 , - M M , 0 ) 199 CALL ALDONE 200 c 201 c 202 c 203 RETURN 204 END 205 c 206 c 207 c 208 SUBROUTINE F I T ( X , Y , W T , Y Y , X X , N W T , M , K ) 209 IMPLICIT REAL*8 ( A - H . O - Z ) 210 DIMENSION Y F ( 2 0 0 ) . Y D ( 2 0 0 ) . Y Y D ( 2 0 0 ) . S ( 2 0 ) . S I G M A ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) , P ( 2 0 ) 211 DIMENSION X ( 1 ) . Y ( 1 ) , W T ( 1 ) . Y Y ( 1 ) . X X ( 1 ) 212 L0GICAL*4 LK 213 c 214 c 215 c 216 MM=40 217 LK= .TRUE. 218 c 219 CALL D 0 L S F ( K . M . X . Y , Y F , Y D , W T , N W T . S . S I G M A , A , B . S S . L K . P ) 220 c 221 KK=K+1 222 c 223 c USE F IT PARAMETERS TO CALC F IT FUNCTION 224 c 225 A I N C = X ( 1 ) / 4 0 . 226 DO 80 I -1.MM 227 X X ( I ) = ( D F L O A T ( I ) ) * A I N C 228 Y Y ( I ) « F U N C ( X X ( I ) , P . K K ) 229 80 CONTINUE 230 c 231 c 232 WRITE(8 ,801) KK 184 L i s t i n g o f LLNEWER a t 12:43:06 on AUG 14. 1987 f o r CC1d=DBRU 233 DO IO 1=1,M 234 WRITE(8 ,802) X ( I ) . Y ( I ) . Y F ( I ) , Y D ( I ) 235 10 CONTINUE 236 c 237 c 238 I F ( K . N E . 1 ) GOTO 5 239 c CALC POLARIZABIL ITY.MOL VOL 240 c 241 P 0 L = 1 . 0 D 0 / P ( 1 ) 242 P 0 L = P 0 L / 6 . 0 2 3 0 2 3 * 1 . 0 0 2 4 243 c 244 AA=1 .0D0 /P (2 ) 245 A A = - A A / 6 . 0 2 3 E 2 3 246 AA=AA**0 .3333333333*1 .OD8 247 c 248 W R I T E ( 6 , 6 1 1 ) P 0 L , A A 249 W R I T E ( 8 , 6 1 1 ) P 0 L , A A 250 61 1 F O R M A T ( / , 1X, 'POLARIZABIL ITY = ' , 2 X , F 8 . 4 . ' A * * 3 ' . / . 1 X , 251 | 'MOLECULAR RADIUS = ' , 2 X , F 8 . 4 , ' A ' ) 252 C 253 801 FORMATC'1RESULTS OF F I T : ' . / , / , 1 3 , 2 X , ' P A R A M E T E R S F IT TO 254 | / , / , 8 X , ' D E N S I T Y ' , 2 4 X . ' F I T ' , 1 0 X . ' D E V I A T I O N S ' , / ) 255 802 F 0 R M A T ( 4 ( 3 X . F 1 2 . 6 ) ) 256 C 257 5 WRITE(8 ,805) SS 258 805 FORMAT(1X. 'SUM OF SQUARES = ' , E 1 2 . 6 ) 259 C 260 C 261 WRITE(8 ,804) 262 WRITE(6 ,804) 263 DO 20 1=1,KK 264 WRITE(8 ,803) P ( I ) 265 WRITE(6 .803) P ( I ) 266 20 CONTINUE 267 C 268 804 F O R M A T ( / , 1 X , ' F I T PARAMETERS: ' ) 269 803 F 0 R M A T ( 3 X , E 1 6 . 8 ) 270 C 271 RETURN 272 END 273 C 274 C 275 C 276 C 277 C 278 C 279 C 280 FUNCTION FUNC(X.COEF.NOTERM) 281 R E A L * 8 F U N C , X , C O E F ( 1 ) 282 C 283 FUNC=O.ODO 284 C 285 DO 10 I -1 .N0TERM 286 IF ( X . E Q . O . O ) GOTO 20 287 FUNC=FUNC+COEF( I ) *X * * ( I - 1 ) 288 10 CONTINUE 289 c 290 20 RETURN 291 END 185 L i s t i n g of C O E X 8 6 L i s t i n g o f C0EX86 a t 12 :34 :07 on AUG 14, 1987 f o r CCid=DBRU 1 IMPLICIT REAL*8 ( A - H . O - Z ) 2 DIMENSION RES(100 ) ,AMICL(1O0) .AMICV(100 ) 3 DIMENSION ANV(100 ) ,ANL(100 ) 4 DIMENSION R H O V ( 1 0 0 ) , R H O L ( 1 0 0 ) , T ( 1 0 0 ) , T 2 ( 1 0 0 ) , D I A M ( 5 DIMENSION T H F I T ( 5 ) , A L ( 5 ) 6 DIMENSION A L L V ( 2 0 0 ) , A L L L ( 2 0 0 ) 7 C 8 COMMON ANAIR .ANSAFF.ALPHA,COSAIR .S INAIR 9 C 10 C CONSTANTS 11 C 12 ANAIR=1.00029 13 ANSAFF=1.66 14 C 15 C PARAMETERS FOR 3RD ORDER POLY F IT TO CALIB DATA 16 C 17 R E A D ( 3 , 3 0 1 ) T H F I T ( 1 ) , T H F I T ( 2 ) , T H F I T ( 3 ) , T H F I T ( 4 ) 18 READ(3 .302 ) THAIR 19 READ(3 ,304 ) ALPHA 20 READ(3 ,303 ) AL(1 ) . A L ( 2 ) , A L ( 3 ) 21 301 F O R M A T ( 1 X . E 2 0 . 8 . / . 1 X , E 2 0 . 8 , / f 1 X . E 2 0 . 8 . / . 1 X . E 2 0 . 8 ) 22 302 FORMAT(1X .E20 .10 ) 23 303 F O R M A T ( 1 X , F 1 2 . 6 , 1 X , F 1 2 . 6 . 1 X , F 1 2 . 6 ) 24 304 F O R M A T ( 1 X . F 1 2 . 6 ) 25 WRITE(6 .301) T H F I T ( 1 ) , T H F I T ( 2 ) , T H F I T ( 3 ) , T H F I T ( 4 ) 26 WRITE(8 ,301) T H F I T ( 1 ) , T H F I T ( 2 ) . T H F I T ( 3 ) , T H F I T ( 4 ) 27 WRITE(6 .302) THAIR 28 WRITE(8 ,302) THAIR 29 WRITE(6 .304) ALPHA 30 WRITE(8 ,304) ALPHA 31 WRITE(6 .303) AL( 1 ) . A L ( 2 ) , A L ( 3 ) 32 WRITE(8 .303) A L ( 1 ) , A L ( 2 ) . A L ( 3 ) 33 C 34 C F I L E 3 VARIES FOR DIFFERENT MICROMETER SCREWS 35 C 36 C F IT FOR GY THERMISTOR 1/T=F(LNR) 37 C 38 TEMP1=0.1349639945E-2 39 TEMP2=0.2354661076E-3 40 TEMP3=0.32B2390265E-5 41 C 42 C 43 C 44 C 45 C C E L L ANGLES 46 C 47 PI=3.14159265D0 48 ALPHA=ALPHA*PI /180 .D0 49 COSAIR=DCOS(THAIR) 50 SINAIR=DSIN(THAIR) 51 C 52 C GET DATA FROM F I L E # 4 53 c 54 R E A D ( 4 , 4 0 1 ) M 55 WRITE(6 .601) M 56 WRITE(8 .601) M 57 601 FORMAT(1X , 'NO. OF P T S . » ' . 3 X . I 4 . / ) 58 401 FORMAT( IX .14) 186 L i s t i n g o f C0EX86 a t 12 :34 :07 on AUG 14, 1987 f o r CC)d=DBRU 59 DO 10 1=1.M 60 READ(4 ,402 ) RES( I ) ,AMICV(I ) .AMICL( I ) ,AMICO 61 WRITE(6 .402) R E S ( I ) . A M I C V ( I ) , A M I C L ( I ) . A M I C O 62 AMICL( I )=AMICL( I ) -AMICO 63 AMICV(I ) = AMICV( I ) -AMICO 64 10 CONTINUE 65 402 F O R M A T ( 2 X , F 7 . 2 , 1 X . F 7 . 3 , 3 X , F 7 . 3 . 3 X . F 7 . 3 ) 66 C 67 C 68 C 69 DO 22 I=1,M 70 C 71 C CALC INDEX OF REFR 72 C 73 ANV( I )=DEX(AMICV( I ) .THF IT ) 74 C 75 A N L ( I ) = D E X ( A M I C L ( I ) . T H F I T ) 76 C 77 C CALCULATE TEMPERATURES 78 C 79 C FOR RES=GY USE F IT TO 1/T=A+BLNR+CLNR**2 80 C 81 C 82 C 83 T2( I )=TEMP1+TEMP2*DL0G(RES(I ) )+TEMP3*DL0G(RES(I 84 T 2 ( I ) = 1 . 0 D 0 / T 2 ( I ) 85 T ( I ) = T 2 ( I ) - 2 7 3 . 1 5 D O 86 C 87 C 88 A L L V ( I ) = C A L C L L ( A N V ( I ) , A L ) 89 C 90 A L L L ( I ) = C A L C L L ( A N L ( I ) , A L ) 91 C 92 C 93 C CALC DENSITIES 94 C 95 RHOV(I)=DENS(ANV(I ) , A L L V ( I ) ) 96 C 97 R H O L ( I ) = D E N S ( A N L ( I ) , A L L L ( I ) ) 98 C 99 C CALC DIAMETER ioo C ioi DIAM( I )= (RHOV( I )+RHDL( I ) ) /2 .ODO 102 C 103 C 104 22 CONTINUE 105 C 106 C 107 C OUTPUT 108 C 109 WRITE(8 .880) AMICO 110 880 F O R M A T O X . 'ZERO ORDER MIC RDG = ' . F 8 . 4 ) 111 C 112 DO 50 1=1.M 113 W R I T E ( 6 , 6 9 0 ) T 2 ( I ) . R H 0 V ( I ) , R H 0 L ( I ) . D I A M ( I ) 114 W R I T E ( 8 , 6 9 O ) T 2 ( I ) . R H 0 V ( I ) . R H 0 L ( I ) , D I A M ( I ) 115 50 CONTINUE 116 690 F O R M A T O X . F 9 . 5 , 3 ( 2 X , F 1 1 . 7 ) ) 187 L i s t i n g o f C0EX86 a t 12 :34 :07 on AUG 14, 1987 f o r CC1d=DBRU 117 C 1 18 C 1 19 CALL GRAF(T2 ,RHOV,RHOL, DIAM,M) 120 C 121 C 122 STOP 123 END 124 C 125 C 126 C 127 C SUB DENS CALCULATES DENSITIES FROM INDEX OF REFR. 128 C 129 C 130 FUNCTION D E N S ( A N , A L L ) 131 IMPLICIT REAL*8 ( A - H . O - Z ) 132 C 133 C 134 BN=AN*AN 135 CN=(BN-1 .0D0) / (BN+2 .0D0) 136 DENS=CN/ALL 137 C 138 RETURN 139 END 140 C 141 C 142 C C SUB DEX CALCULATES INDEX OF REFRACTION 143 c 144 FUNCTION DEX(AMIC,THFIT ) 145 IMPLICIT R E A L * 8 ( A - H . O - Z ) 146 DIMENSION T H F I T ( 1 ) 147 COMMON ANAIR ,ANSAFF.ALPHA.COSAIR ,S INAIR 148 c 149 TANALF=DTAN(ALPHA) 150 SINTH=THFIT(1) + THF IT (2 ) *AMIC + THFIT (3 ) *AMIC*AMIC+THFIT (4 ) *AMIC**3 151 THETA=DARSIN(SINTH) 152 COSTH=DCOS(THETA) 153 c 154 DEX=ANAIR*(1 .ODO+C0STH-C0SAIR+(SINTH-SINAIR) /TANALF) 155 c 156 c 157 RETURN 158 END 159 c 160 c 161 c 162 FUNCTION C A L C L L ( A N , A L ) 163 IMPLICIT REAL*8 ( A - H . O - Z ) 164 DIMENSION A L ( 1 ) 165 c 166 AN1=AN-1.000 167 c 168 c F I T OF LL TO N . . . 169 c NUMBERS FROM LLFITTON 170 c READ IN ABOVE FROM 3 171 c 172 c 173 CALCLL=AL(1)+AL(2) *AN1+AL(3) *AN1*AN1 174 c L i s t i n g o f C0EX86 a t 12:34:07 on AUG 14, 1987 f o r CC1d= 175 RETURN 176 END 177 C 178 C 179 C 180 SUBROUTINE GRAF(T,RHOV,RHOL.DIAM,M) 181 R E A L * 8 T ( 1 ) , R H O V ( 1 ) , R H O L ( 1 ) , D I A M ( 1 ) 182 R E A L M TT( 100) , W ( 100) , EL( 100) ,DD( 100) 183 R E A L M T 2 O 0 0 ) 184 C 185 C 186 TMAX=T(M)+0.2D0 187 C 188 DO 10 1=1,M 189 T T ( I ) = T ( I ) 190 T 2 ( I ) = T M A X - T T ( I ) 191 VV( I )=RHOV(I ) 192 EL ( I )=RHOL( I ) 193 DD(I )=DIAM(I ) 194 10 CONTINUE 195 c 196 CALL A L A X I S ( ' T E M P ' . 4 , ' D E N S ' , 4 ) 197 CALL A L S C A L ( 0 . , 0 . 0 1 5 , 2 7 0 . . 2 8 5 . ) 198 CALL A L G R A F ( V V , T T , M , - 1 ) 199 CALL A L G R A F ( E L , T T , - M , - 4 ) 200 CALL A L G R A F ( D D , T T , - M , - 3 ) 201 c 202 c 203 CALL A L S C A L ( 0 . O . O . 0 , 0 . 0 , 0 . 0 ) 204 CALL A L A X I S ( ' T E M P ' , 4 , ' D I A M ' , 4 ) 205 CALL A L G R A F ( T 2 , D D , M , - 3 ) 206 CALL ALDONE 207 RETURN 208 END 189 C - 3 Temperature Sweeping Program This program was used with a Commodore PET microcomputer to sweep the temperature of the ethylene experiment, as described in section III-2. Lines 1-150 are a machine language program that reads the temperature from the HP2804A quartz thermometer. This program is loaded into memory by lines 155-240. The rest of the program gets a sequence of sweep and wait times from the keyboard and executes them, turning on and off the motorized potentiometer in the temperature control circuit and marking the data film before each sweep by turning on and off the LEDs. The address-59457 is the output port to which the interface shown in fig. (3-6) is connected. Numbers sent to this address have the following effects: 0 - everything off 1 - LED #1 on 2 - LED #2 on 5 - motor on 6 - motor reverse 7 - reset. 190 L i s t i n g o f t h e T e m p e r a t u r e S w e e p i n g P r o g r a m 1 REM ADDRESSES: 2 REM MESSAGE 871, 878 3 REM MESSAGE OUTPUT 847 4 REM DATA INPUT 772 5 REM SEC. ADD. 15 6 REM TOTAL PROG. 634-904 7 REM DATA 905-8 REM 9 : 10 DIM NSaO0),TS<100),TW<100),TQ<100,10> 20 POKE 15,0 30 DATA A9, FB, 2D, 40, E8,8D, 40, E8,60, A9,04,0D, 40, E8,8D, 40, E8,60, A9, FD, 2D, 40, E8 31 : 40 DATA 8D,40,E8,60,A9,02,0D,40,E8,8D,40,E8,60,A9,F7,2D,21,E8,8D,21,E8,60 41 • 45 DATA A9,08,0D,21,E8,8D,21,E8,60 46 : 50 DATA A9, FF, 8D, 22, E8,60, AD, 40, E8,29,40, F0, F9, A5,01,49,FF, 8D,22, E8,A9, F7 51 : 55 DATA 2D,23,E3,8D,23,E8 56 • 60 DATA AD,40,E8,29,01,F0,F9,A9,08,0D,23,E8,8D,23,E8,20,B0,02,60 61 : 70 DATA 20,95,02,AD,40,E8,29,80,D0,F9,20,8C,02, AD,20,E8,49,FF,85,02,20,A7,02 71 : 80 DATA AD, 40,E8,29,80,F0,F9,20,9E,02,20,B0,02,60,A2,00,20,7A,02,A9,4D 81 : 90 DATA 85,01,20,B6,02,18,A5,0F,69,60,85,01,20,B6,02,EA,EA,EA,EA,EA,EA,EA,EA,EA 91 : 100 DATA EA,EA,20,8C,02,20,9E,02,20,83,02,20, DF,02,E8,A5,02,9D,89,03,C9,0A 101 : 110 DATA DO,F3,20,7A,02,20,95,02,20,A7,02,A9,5F,85,01,20,B6,02,20,83,02,60 111 : 120 DATA 20,7A,02,A9,2D,85,01,20,B6,02,EA,EA,EA,EA,EA,EA,EA,EA,EA,EA,20,83,02 121 = 130 DATA A9,54,85,01,20,B6,02,A9,32,85,01,20,B6,02,A9,58,85,01,20,B6,02 131 = 140 DATA 20,7A,02,A9,3F,85,01,20,B6,02,20,83,02,60 141 : 145 DATA EA,EA, EA,EA,EA,EA,EA,EA,EA,EA,EA,EA, EA,EA, EA,EA,EA,EA, EA,EA 146 = 150 DATA A9,00,3D,02,90,A9,00,8D,03,90,60 151 : 152 REM START PROG 153 = 154 • 155 REM LOAD GT ROUTINE 156 : 160 FOR 1=634 TO 935 170 READ A* 180 B*=RIGHT*<A*,1>:GOSUB 500 190 X=B 200 B*=LEFT*(A*,1>:G0SUB 500 210 X=X+B*16 220 POKEI/X 230 NEXT I 240 RESTORE 250 = 251 REM 252 : 255 GOSUB1000:REM SETUP 256 = 258 TF=0 191 260 I>=0 : 12=0 262 GOSUB 4000 264 PRINT"MINITIAL TEMP=",TGK0,0> 270 : 272 FOR 12=1 TO I1—1 275 FOR D=lT0NSa2) 280 GOSUB 3000:REM SWEEP 290 IF 0PO2 THEN390 300 PRINT#4,D;I2;TI*,TQ<D,I2) 390 NEXT D 395 NEXT 12 415 PRINT'TEND OF RUN..." 418 PRINT"INITIAL TEMP=",TQ<0,0> 422 FORCC=l TO I1-1 425 FORCD=lTO NSC CO 428 PRINTCD,CC,TQ<CD,CC) 430 IF 0PO2 THEN 435 432 PRINT#4,CD,CC,TQ<CD,CC> 435 NEXTCD 440 NEXTCC 445 IF 0PO2THEN 452 450 CL0SE4 451 = 452 GET fl*: IFA$="" THEN 452 454 TF=1 458 GOSUB 4000 462 PRINT"TEMP =",TU 466 GOSUB 2000 = Xl=CLOCK 468 GOSUB 2000 : X2=CL0CK 472 IF X2-X1=5 OR X2-X1=55 THEN 458 476 GET A* : IF A$=""THEN 468 480 GOTO 255 490 END 498 : 499 : 500 B=ASC<B*> 510 IF B>64 THEN B=B-55 : RETURN 520 B=B-48 530 RETURN 540 : 1000 REM SETUP 1010 • 1015 Q=59457 1020 POKE Q+2,255 1030 POKE Q,0 1040 POKE Q,? 1050 POKE Q,0 1055 PRINT"3" 1060 PRINT" tt»HELLO, WELCOME TO THE TEMPERflTUREIl»»»l»»MI SWEEPING PROGRAM" 1061 PRINT:PRINT 1065 INPUT"IS THE QUARTZ THERMOMETER THERE";V* 1066 IF LEFT*(V*, D=,'V"THEN1070 1067 TT=0 1068 GOTO 1080 1070 TT=1 1071 INPUT"OT RANGE <1,2,3>";RT* 1072 TR*="T":GOSUB 5000 1073 IHPUT"*R RANGE <1,2,3>";RT* 1074 TRS="R": GOSUB 5000 1075 REM 1080 INPUT "MOUTPUT TO SCREEN,TAPE,OR PRINTER-jO* 1090 IF LEFT*<0*,1>="S"THEN OP=0 1100 IF LEFTSCO*.1)="T"THEN 0P=1 1110 IF LEFT*<0*,1)="P"THEN 0P=2 1119 11=1 1120 INPUT"INUMBER OF T S T E P S O N S * 11 > 1125 IF HSCI1>=0 GOTO 1155 1130 INPUT"XNUMBER OF MINUTES SWEEPING";TS(I1> 1135 IF TSai>>60 OR TS<I1><0 THEN FR I NT" M1RX I MUM 60":GOTO 1130 1140 INPUT")SNUMBER OF MINUTES UfllTING'MW 1143 HR<Il>=INT<TW/60> 1145 MINCI1>=IHT<(TW/60-HRCI1> >*60+.01> 1150 11=11+1 1152 PRINT"*FOLLOWED EV 71" i 1154 GOTO1120 1155 INPUT "MINITIRL WRIT'MW 1160 GOSUB 20O0:W1=CLOCK 1162 GOSUB 2000-W2=CLOCK 1164 IF W2-W1=IW OR W1-W2=60-IW THEN1170 1166 GOTO 1160 1170 : 1175 POKE 59457,1 1180 FOR F=1T0 10000-NEXT 1190 POKE 59457,0 1200 FOR F=1T0 10000 NEXT 1205 POKE 59457,1 1207 FOR F=1T0 10000:NEXT 1210 RETURN 1990 : 1995 = 2000 CLOCK=VRL<MID$<TI$,3,2>> 2010 RETURN 2099 : 2100 C2<l>=VflLCLEFT*CTI*,2>) 2110 C2<2)=VflL<MIHf<TI*,3,2>) 2120 RETURN 2125 : 2130 = 3060 REM SWEEP 3010 GOSUB 2000 3020 Sl=CLOCK 3040 POKE Q,5:F0R V=1TO100:NEXTY 3060 PRINT-SWEEP* ";Tj;"...ON" 3070 GOSUB 2000 30SO S2=CL0CK 3090 IF S2-S1=TS<I2> OR S1-S2=60-TSCI2) THEN 3208 3100 GOTO 3070 3200 POKE Q,0:FOR Y=1T0100-NEXT V 3210 POKE Q,7:F0R V=1TO100:NEXT V 3220 POKE Q,0:FOR V=lTO100"NEXT V 3230 PR I NT "SWEEP* " i l J ; ".. .OFF" 3240 GOSUB 2100 3245 S3(1)=C2<1) 3248 S3<2)=C2(2) 3250 GOSUB 2100 3255 S4<1>=C2<1> 3260 S4<2>=C2<2> 3265 IF S4<l>-S3a>=HR(I2) RHD S4<2>-S3<2>=MIN<I2> THEN 3300 3270 IF S4<l>-S3a>=HR<I2)+l RND S3<2>-S4<2>=60-MINCI2> THEN 3300 3275 IF S3a>-S4a>=24-HR<I2> RHD S4<2>-S3<2>=MIN<I2> THEN 3300 3280 IF S3(l)-S4<l>=23-HRa2> BHD S3(2)-S4<2)=60-MINa2> THEN 3300 3290 GOTO 3250 3300 GDSUB4000:REM READ T 3308 POKE Q,l:F.OR V=1TO5000:NEXT:POKE G,0 3312 IF B/3=INT<D/3>THEN3350 3320 GOTO 3360 3350 POKE 0,2:FOR Y=1TO5000:HEXT:POKEQ,0 3360 PRINT-SWEEP* ";I>;".. .WRIT OVER." 3370 PRINT-TIME* ";TI*;TRB<6>;"TEMP= ";TQCD,I2> 3400 RETURN 349S = 3499 : 4006 REM READ T 4010 : 4015 IF TT=0 THEN TQ=0 : RETURN 4020 SVSC772) 4030 C=906 4040 R=PEEKXC> 4050 C=C+1 4060 IF A013 THEN D*=D$+CHRf(A) : GOTO4040 4065 IF TF=0 THEN 4090 4070 TU=VAL<D*> 4080 RETURN 4085 • 4090 TtKD,CC>=VAL<D*> 4100 RETURN 4998 : 4999 : 50G0 POKE 871,ASCCTR*):P0KE 878,ASC<RT$>:SVSC847):SVS<847> 5010 RETURN READV. 

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