Journal of Low Temperature Physics, Vol. 24, Nos. 3//4, 1976
Equation of State of a 3He4He Mixture Near Its LiquidVapor Critical Point* Ted Doiron, Robert P. Behringer, t and Horst Meyer Department of Physic& Duke University, Durham,
North Carolina
(Received December 8, 1975)
Measurements of the pressure coefficient (0P/0T)p x are reported for a 3He4 • . • 3 '. . He mtxture with a mole fracton X = 0.805 of He m the netghborhood of the liquidvapor critical point. These include data on 16 isochores taken over the density interval 0.5'
345 © 1976 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.
346
Ted Doiron, Robert P. Behringer, and Horst Meyer
3He4He mixtures near the liquidvapor critical point. These studies, and also more recently the calorimetric . • measurements2 on a 80.5% 3He, 19.5% 4He mixture, stimulated the proposal by Leung and Griffiths3 (LG) of an equation of state that extends the scaling theory for pure fluids to the case of a binary mixture. The predictions from the theory were then compared 3 to the experiments on the 3He4He mixtures. In the main, the numerical parameters used in the theory were taken from the properties of the pure components, 3He and 4He, and the prediction for the mixtures were found to be generally in good agreement with the experiments. This agreement is quite remarkable since in a number of ways the thermodynamic properties of a mixture at fixed concentration are quite different from those of a pure fluid.* For instance, the compressibility kr, x along the critical isochore no longer diverges strongly as Tc is approached from above. Also, Leung and Griffiths3 were able to explain quantitatively why the true asymptotic behavior of the compressibility kT,x and the specific heat C~,x at constant mole fraction, predicted from general postulates by Griffiths and Wheeler, 5 cannot be observed experimentally. Here X is the molefraction of 3He. In the original PVT experiments by Wallace and Meyer I there were difficulties in determining the shape of the dewbubble curve close to the critical temperature and above the critical density. These difficulties resulted from the density measuring technique whereby the dielectric constant of the mixture was measured and the density determined via the ClausiusMosotti equation. In the twophase region this method did not permit sampling the correct average density of the fluid and the discontinuity in (OP/OT)o,x at the bubble curve (this is the coexistence curve for P>Pc) was not clearly observed. The experiment to be described in this paper avoids this difficulty and also permits a highresolution determination of the discontinuity in (OP/OT)o.x along an isochore at the boundary of the dewbubble curve. Comparison with the predictions from the LeungGriffiths3 equation of state gives a generally good quantitative agreement, although there are some systematic deviations. Considering that the comparison is concerned with the first temperature derivative of P(T, p, X), instead of merely the pressure, makes the agreement all the more impressive. The theory also correctly foresees a number of small effects actually observed by the experiment. In Section 2 we give a brief review of the characteristic properties of a binary mixture near the liquidvapor critical point and also present an abstract of the LeungGriffiths mixture theory. 3 Section 3 describes the experimental part and in Section 4 the results are presented, discussed, and compared with the theory.
*For a recent reviewon the properties of pure fluids,see, e.g., Ref.4.
Equation of State of a 3He4He Mixture
347
2. REVIEW OF B I N A R Y MIXTURES 2.1. General
The introduction of a second component to a pure fluid produces striking changes in the properties of the system. For a complete discussion of mixtures we refer the reader to the book by Rowlinson. 7 It is convenient to present here a short review of the ,geometry of the binary mixture coexistence surface (CXS) near the liquidvapor critical point, which shows the effect of the extra degree of freedom on the measured properties to be presented and discussed later. The CXS in the pressuretemperaturemolefraction space is shown schematically in Fig. 1. This is a typical situation for a simple "wellbehaved" mixture like 3He4He. Each state of the system, when it forms a single phase, is represented by a point outside the surface. The twophase states are represented by pairs of points on the CXS, the liquid and vapor phases by points on the upper and lower surface, respectively. Two such points, A and B, are shown in Fig. 1; XA and XB are the corresponding mole fractions for a given average mole fraction X. The shaded loop is the intersection of the CXS with a plane of constant X. Hence a point inside the loop simply denotes an inhomogeneous physical
P
XB Fig. 1. Schematic presentation of the coexistence surface (CXS) in PTXspace. The hatched space is a cross section through the CXS at constant X and is b o u n d e d by the d e w  b u b b l e curve (DBC). T h e two other cross sections through the CXS are at constant temperature.
Ted Doiron, Robert P. Behringer, and Horst Meyer
348
state of the system with the average mole fraction for the total system, including both phases, equal to X. The curve itself represents the dew and bubble points for a sample of average mole fraction X. If the system is in the vapor phase below the loop and is isothermally compressed, it will form the first drop of liquid at the dew point. The collection of all dew points forms one part of the loop. The points where the first bubble forms in the liquid make up the remainder, hence the name dewbubble curve (DBC). The critical line is the locus of points on the CXS satisfying the conditions (OP/OX)TlcXS = (OT/OX)plcxs = 0. The subscript CXS is used to denote the fact that the differential paths are limited to the CXS. If the DBC's are projected in P  T space, those conditions imply that the critical line forms an envelope of the DBC's. The intersection of the critical line and a DBC is the critical point for that partiular mole fraction. It is now Useful to picture what happens along an isochore at constant X. This path is shown in Fig. 2, where the dewbubble curve is reproduced by a heavy solid line. The light solid lines represent various isochores all at the same X value and the dashed line is the projection of the critical line on the P  T plane. This is to be compared with the behavior of a pure fluid, where the dewbubble curve collapses into one line, the saturated vapo r pressure curve. One important difference between the behavior of the isochores of a pure fluid and mixture is that inside the twophase region, OP/OT)o,x is a function of both p and T, while for a pure fluid, (OP/OT)o = d P / d T is
P'4 / I
/
P
/ CP
jf
T
Fig. 2. Schematic presentation of the d e w  b u b b l e curve in the P  T plane and of the isochores for a mixture at constant X. T h e inset shows an expanded region near the critical point and the asymptotic behavior of the critical isochore.
Equation of State of a 3He4He Mixture
349
qI
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/
/ ~
T4>5>Tc>T2>V~ P
Fig. 3. Schematic behavior of the dewbubble curve (heavy line) and some isotherms (light lines) in the densitypressure plane. The density where (dp/dP)oBc = co is denoted by Pe. The isotherms inside the twophase region are not vertical, unlike in a pure fluid. One isotherm is tangent to the D B C at 07
a function of T alone. Furthermore as we shall see, along the critical isochore in a 3He4He mixture, there seems to be a discontinuity in (OP/OT)p,x at the dewbubble curve, while there is none for the pure fluids. However, the postulates of Griffiths and Wheeler 5 lead to the predictions that (aP/aT)o,x is continuous along Pc and that the sloPes of the DBC and of the critical isochore in the P  T plane become equal to Tc. This can be visualized from the inset of Fig. 2, which shows an expanded view in the immediate neighborhood of To. Furthermore, the compressibility kyx along the critical isochore is predicted to diverge weakly as Tc is approached, but the LG model indicates that the very small asymptotic temperature range is inaccessible with current experimental techniques. Thus when we speak of the "discontinuity" in (OP/OT)o,x along the critical isochore we are referring to experimental results that do not probe the expected asymptotic region. The density PD where there is experimentally no discontinuity in (aP/OT)o,x can be shown to be equal to the density PT corresponding to the highest temperature on the DBC. The thermodynamic argument, due to Griffiths, is presented in Appendix A, and our experimental findings will be discussed in Section 4. Figure 3 shows schematically the densitypressure plane for a mixturel Here the heavy lines again define: the dewbubble curve.
350
Ted Doiron, Robert P. Behringer, and Horst Meyer
t
CRITICAL LINE ~ b
p
tanh
A'
h ~
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,,5~ @
Fig. 4. Schematic presentation of the CXS in field space spanned by P, tanh (A/RT), and T, where A =/z 3/z4. For the: explanation of the three independent field variables ~', h, and ~,see text. For the 3He4He mixtures, Pc > pP > PT, where Pe is the density corresponding to the highest pressure* on the DBC. 2.2. The LeungGrittiths Model
Since we shall compare our data with the prediction of the L G model, 3 a short introduction to the main ideas of this model is necessary. The details of the theory are quite complicated and therefore only the barest outline is given here. In Appendix B we present our method of calculating (OP/OT)o. x from the model. Leung and Griffiths found it conceptually and practically advantageous to express their thermodynamic potential c0 using "fields" alone as independent variables, in the sense of Ref. 5, instead of a mixture of "fields" and "densities." They chose the independent fields ~, z, and h. A useful, though not rigorous, way to visualize these fields is by using Fig. 4, where we draw the CXS as a function of the field variables, namely the pressure, the temperature, and tanh (zX/RT), w h e r e A = P,3J~4 is the difference in the chemical potentials of the isotopes. For pure fluids, A(3He)= +oo and A(4He)=oo. The limits for tanh (A/RT) are then, respectively, +1 and  1 and therefore the use of this quantity for plotting pu.rposes is m o r e conve*In Ref. 1, this density was denoted Ps, where s stood for "symmetry" because it was believed at that time that the DBC in the pP plane was symmetric around Ps.
Equation of State of a 3He4He Mixture
351
nient than use of A. In this field space, the CXS is the surface as shown with the critical line as one border. The variable ff serves to parametrize the line of critical points, while ~is the distance of the CXS normal to the critical line a n d h is the distance above the CXS. The potential ~o ( = P/RT) is assumed to be of the form w = ~0reg(ff,~', h) + (.0sing(~ , T~ h)
(1)
where (Dregis a smooth, analytic function of its arguments and OJsingcontains in its derivatives the singularities associated with the discontinuities in density and composition at the coexistence curve. Leung and Griffiths assume O)reg: C(~) [d(ffD" +f(~)h
(2)
with c(ff), d(ff), and f(ff) polynomials. The singular part t0sing is essentially a generalization of the Schofield linear model to three dimensions, (.Osing q(~')II(l(ff), ~, h) :
(3)
where q(ff) and l(~) are polynomials and II is the linear model function (see the appendix of Ref. 3). There are 26 parameters appearing in the L G model, but 22 of these are determined by the properties of the pure fluids (for instance, the critical exponents/3 = 0.361, 3~= 1.17, pc (3He), pc(4He), etc.). The remaining four are obtained 3 from the early P V T data by Wallace and Meyer a and they do not play a major role. 3. E X P E R I M E N T A L 3.1. General
The apparatus was designed for highresolution measurements of the pressure as a function of temperature at constant density and mole fraction X. The method uses a capacitive strain gauge incorporated in a flat, circular pressure cell of radius 2 cm and height 0.8 mm. The height was chosen so as to minimize gravity effects that would induce density gradients. The apparatus, temperature control, and gas handling procedures have been described in the preceding paper 8 and we merely mention here that the pressure and temperature resolutions were l x 10 6 atm and 0.5/zK, respectively. The temperature stability over the long times needed to achieve equilibrium in the system (up to  1 h) was about 2 / z K . Twopoint and threepoint numerical differentiation of the data then led to (OP/OT)o,x, which under optimum conditions could be measured with a scatter of + 0.1% when the spacing of points was 25 mK. Close to transitions, where
Ted Doiron, Robert P. Behringer, and Horst Meyer
352
t h e temperature intervals between pressure readings were taken as small as 2 mK, the scatter was about ± 0.4%. •
3.2. Sample Preparation
The sample was the same as used by Brown and Meyer 2 in their specific heat measurements. The molefraction X was checked by a calibrated mass spectrometer and was found to be consistent with the stated value X = 0.805 ± 0.005. The gas sample was always introduced into and removed from the chamber at 3.75 K, well above the critical temperature to avoid any fractionation of the sample. A calibrated Texas Instruments quartz gauge was used to measure the initial and final pressures of a known reference volume at room temperature through which all sample transfers were made. The temperature of the volume was known to ± 0.05 K. By means of the ideal gas law, the numbers of moles transferred to and from the cell were known to 0.1%. The effective volue of the cell determined from experiments with pure 3He was Vsc = 1.580 ± 0.008 cm 3. Hence densities of the isochores could be calculated to accuracies of, respectively, 0.1% and 0.5% in the relative and absolute values. 3.3. Procedure
T h e procedure for calibrating the strain gauge has been described in Ref. 8. This device was designed to measure relative pressures and it was necessary after each run to normalize our data to the absolute pressure at a standard temperature. Before beginning our measurements, P was determined as a function of p on the "standard" isotherm T = 4.1355 K by means of the reference volume and the quartz pressure gauge. Datataking started after cooling the sample cell to 3.30 K. The cell was progressively warmed by temperature increments of between 25 and 2 mK, depending on the distance from the DBC. We always chose the last point of each isochore to be at T = 4.1355 K and used a leastsquares fit of our isothermal data to determine P(4.1355 K).
4. R E S U L T S A N D D I S C U S S I O N 4.1. General Observations
As was expected from the observations during calorimetric measurements, 2 the equilibrium times in the twophase r e g i o n were appreciably longer for the mixture than in pure 3He. After a heating period, followed by the automatic stabilization of the temperature to within 2/zK, the pressure would continue to drift upward as a function of time before settling to its
Equation of State of a aHe4He Mixture
353
equilibrium value. In the onephase region, the equilibrium times were all less than about 5 min, which is the time it took to stabilize the temperature to less than the 2 ~ K fluctuation level. As the system approached the DBC from the twophase region, the equilibrium times tended to increase beyond the time we could allot for making the measurement,  45 rain. Hence in the twophase region, the pressures could have small systematic errors within 510 mK of the transition point. For each isochore the crossing of the DBC was observed by two methods. The first one used the large difference in equilibrium times before and after the crossing. The change in equilibrium was reproducible to within 2 mK and gave a good determination of the phase diagram for p/pc > 1.05 and P/Pc<0.86. Equilibrium times over 20 min were found up to 100mK below the transition for p > 1.05 and abruptly became short on crossing the trafisition. But near the critical isochore this temperature range shrank, until for p/&0.90 the equilibrium times in the twophase and the onephase regions were indistinguishable. For densities less than P/Pc= 0.86, the temperature range of long equilibrium times in the twophase region was again quite large. This behavior is different from the one recorded in calorimetry experiments, / where uniformly long relaxation times were encountered throughout the twophase region. In those experiments, however, the temperature drift instead of the pressure drift was measured as a function of time, and the cell was not thermally coupled to a temperaturestabilized bath. The second method of determining the transition was the same as for pure 3He and consisted in finding the intersection of the one and the twophase PT curves. 8 This method worked well; however, for 0.86 < P/Pc <0.97 the isochores had a (OP/OT)p,x that was nearly continuous, leading to uncertainties of the order Of several mK in the determination of T(DBC). For one isochore, p = 0.90pc, there was no measurable change in (0P/0T)p,x and also no abrupt change in the relaxation times. There the transition could not be found. Our results for P and (c)P/aT)p,x along all the isochores, as well as the coordinates PTp characterizing the DBC, are included in a technical report available upon request. 9 We also include in this report the results shown in Figs. 8 and 9 of the present paper. 4.2. T h e C o e x i s t e n c e Curve and Critical P a r a m e t e r s
The DBC obtained from our measurements and plotted in the PT plane is shown in Fig. 5. From the experiments by Wallace and Meyer I (WM) it was concluded that the critical line is nearly linear, the slope changing less than 2% for 0.7 < X < 1.0. Assuming this linear behavior, a
Ted Doiron, Robert P. Behringer, and Horst Meyer
354
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straight line from the critical point of 3He tangent to the D B C was drawn and we obtained Tc = 3.705 ± 0.002 K,
Pc = 1037 + 1 Torr
(4a)
and furthermore, by interpolation of a plot of the D B C in the p  T and the o  P planes (see Fig. 5), we obtained Pc = 0.01445 ± 0.0004 m o l e / c m 3
(4b)
Comparison with previously reported values is restricted to Tc = 3.707 K reported by Brown and Meyer 2 for the same mixture. In Fig. 6 we show the departures from the symmetry about the densities pp and PT, the densities corresponding, respectively, to the highest pressure and temperature on the DBC, by drawing the diameter of the D B C defined by d = l dp + +0

)
(5)
Here p+ and p  are, respectively, the densities on the upper and lower branches at the same pressure on the p  P curve or the same temperature on the p  T curve. In the region p >pe, our measurements have been able to define the D B C much better than those by WM, who had to assume tentatively that the D B C is symmetric around pp and Pr. This assumption
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Fig. 6. The dewbubble curve for the 3He4Hemixture in both the pP and the pT planes, showing the experimental points and the calculations from the LeungGriffiths model (dashed lines) including the diameter defined by Eq. (5). then led to some small systematic errors in the determination of the critical p a r a m e t e r s Pc, Pc, and Tc, F r o m our experiments, as from those o f Brown and Meyer, it turns out that pc, pP, and PT are spaced quite closely. In Fig. 6 we also c o m p a r e the calculated D B C from the LeungGriffiths model. 3 The calculated critical p a r a m e t e r s Tc, Pc, and Pc, which were based on the W M data, do not quite agree with the presently measured ones, but it is quite clear tha t the calculated shape of the D B C shows the same features as the experiment. A s u m m a r y of the critical p a r a m e t e r s is presented in Table I, which also includes pp and PT. 4.3. The
(8P/OT)p,xD a t a
The experimental results are shown in Fig. 7. The onephase results are similar to those for pure 3He and 4He, except that along the critical isochore, (aP/aT)pc, x appears to be discontinuous. Figure 7 also shows that in the twophase region (aP/aT)p,x at a given T is a function of density, as discussed in Section 2. Only a few isochores are shown in full in the twophase region, the others being omitted for clarity. T h e y fall approximately parallel between those shown. It can be seen from TABLE I
Critical Parameters for the Mixture 80.5% 3He_19.5% 4He
This work Brown and Meyer2 LG model3
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Torr
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.
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the figure that the dependence of the paths on the density is not uniform, the highdensity isochores being more closely spaced. Near the phase transition, the twophase data show a slight tendency to drop, and we will consider this point later when we compare data with the LeungGriffiths equation of state) In Fig. 8 we see that (OP/OT)o,x for the onephase and twophase regions at the D B C are equal for PD = 0.0131 + 0.0003 m o l e / c m 3 or 0.92pc. This value is consistent with the value PT = 0.0130 + 0.0003 m o l e / c m 3 and the prediction that PT = PD4.4. Compressibility and
(OP/OT)o,xN e a r
t h e Critical P o i n t
The GriffithsWheeler hypothesis, discussed in Section 2, predicts that k r x along the critical isochore diverges weakly at the critical point, and that furthermore (OP/OT)o,x at the critical point tends to (dP/dT)cRs. We first investigate this latter possibility. Since the critical line is tangent to the D B C at the critical point, the GriffithsWheeler prediction is equivalent to the statement that
(OP/OT)o,x>(dP/dT)Dl3c
at
To, pc, Pc
(6)
Equation of State of a 3He4He Mixture
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In Fig. 8 we plot (dP/dT)DBC and (OP/OT)p,X in both the onephase and the twophase regions in the immediate vicinity of the DBC. At the critical density, (dP/dT)DBC 485 :i: 10Torr/K, while the data extrapolate to (OP/OT)o.X = 952 +4 T o r r / K (onephase) and 905 ± 4 T o r r / K (twophase). Of course our (OP/OT)p,×data do not come closer than about 1 mK to To, so that we do not probe the asymptotic range of T Tc < 10 11 K estimated by Leung and Griffiths. Assuming the DBC to be smooth, we can visualize the behavior of (OP/OT)pc,x extremely close to the critical point as shown by the inset in Fig. 8 (compare with the inset of Fig. 2). We now evaluate the compressibility along the DBC by considering a path in the P  T plane at constant X, in the onephase region, asymptotically close to the DBC. We have
(~_~) DBC =
OP
DBC
(7)
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Ted Doiron, Robert P. Behringer, and Horst Meyer I00
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Fig. 9. The compressibility p2kr,X in the onephase region asymptotically close to the DBC plotted against the density Ap = ~ Pc)/Pc, and compared with data of Ref. 1 (WM)for an 80% He mixture, From this relation we calculate kw,x using our measured (dP/dT)DBC and p  T curves. The quantity (dp/dT)DBC is always nonzero, so that the situation where kTx, ~ o~ is synonymous with Eq. (6). In Fig. 9 we plot P2kT, x as a function of Ap = (ppc)/p~ on both the liquid and vapor sides of the DBC, in analogy with similar plots for pure fluids. For comparison we show the data by WM taken on the vapor side alone, and the agreement is excellent. Here again, as expected from the L G model, the asymptotic region where kT, x diverges weakly at T~ is not reached. Another observation is that the p2kT, x curves on each side of the D B C are practically parallel. 4.5. Comparison with the LeungGrifliths Model
Due to the fact that (OP/OT)p, x is a function of second derivatives of the potential to with respect to the field variables % ~,"and h, our data provide a sensitive test of any proposed equation of state. In Fig. 10 we compare our results for eight isochores with values calculated from the L G equation of state, using their tabulated numerical parameters. We have shifted the modelcalculated temperatures by 16 mK so that the experimental Tc and the calculated Tc coincide. Clearly the agreemen t is best in the onephase region, but there are systematic deviations, which become especially pronounced in the twophase region. Nevertheless, considering that this theory is labeled only approximate by its authors, the agreement on the whole is
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satisfactory. We also note that the two extreme isochores P/Pc = 1.23 and 0.71 are outside the density range in which experimental data are usually compared with the linear model, so that the agreement here is also gratifying. An important small effect found is the drop in (OP/OT)po,x near the critical point in the twophase region. Model calculations for temperatures closer to Tc than the data show the same form as in Fig. 8 (inset), namely a deep cusp in (aP/OT)pc,x. For instance, at Tc  T = 1 × 10 9 K, (OP/OT)po,x = 810 T o r t / K , which is still a long way from the asymptotic value dP/dT= 450 T o r r / K at To. It can be seen in Fig. 10 that the data have the same rounding of (OP/OT)pc,x near To, signaling the beginning of the cusp. Hence we conclude that in spite of the approximate nature of the L G equation of state, a number of small effects perceived in (OP/OT)p,x measurements are predicted by the theory. 5. CONCLUSIONS We have presented an account of highresolution measurements of
(OP/OT)p,x in a mixture of 80.5% 3He and 19.5% 4He near the liquidvapor critical point. While the behavior of (OP/OT)p,x is somewhat similar to that of
360
Ted Doiron, Robert P. Behringer, and Horst Meyer
a pure fluid, there are important differences. In the mixture, (OP/aT)o.x in the twophase region depends on density. Also, on the critical isochore, (OP/OT)pc.x appears from extrapolation of the data to be discontinuous at To. The isochore p/pc = 0.92, where (OP/OT)pox shows a smooth behavior across the dewbubble curve (DBC), is found also to cut the dew curve at its highest temperature. This result is in agreement with a general thermodynamic argument. Furthermore, we have shown that there is good agreement for the measured dewbubble curve (OP/OT)o,x in the onephase region and fair agreement for (OP/OT)o,x in the twophase region with the predictions from the LeungGriffiths equation of state for 3He4He mixtures. We have also shown that (OP/OT)oo,x fails to exhibit the asymptotic behavior predicted by the GriffithsWheeler postulates, according to which this quantity should have a sharp cusp, but not a discontinuity, at To, and also should become equal to dP/dT along the DBC. Leung and Griffiths predict the range of this asymptotic behavior to be so small as to be inaccessible to experiments. APPENDIX
A. DEMONSTRATION
THAT
p . = Or
We have demonstrated elsewhere 1° on general thermodynamic grounds that pp >_OD >Or for a binary mixture along the dewbubble curve. This demonstration has been outdated by a more elegant argument by Griffiths, 1~ who showed that OD = Or and who kindly suggested that we might include his demonstration here. We shall begin by reviewing the "standard argument" we use to relate various quantities across the CXS. Let C(Y, Z) be a function which is continuous, but whose derivatives are discontinuous along a curve in the (Y, Z) plane separating regions I and II. Taking a derivative along the curve, we have
[OC\ I
(OC] 1
"OC" II
"OC" 'I
d C = {~y ) z d Y + \ Oz /y dZ = (ffffy)z d y + (~ffz) dZ
(A,)
where (OC/Oy)lzrefers to the derivative at the curve in region I, etc. Hence
[(OC]I_(oCtlI ] \0y/z
" [(0C) I
\3y /zJ d Y =  k ; / y '
"0C "II"
(0ffz)yJ dz
(A2)
and thus
oc
_dz (o%
(A3)
We begin our argument by noting that (oe/op)T,x > o
(A4)
Equation of State of a 3He4He Mixture
361
in both the onephase and twophase regions from convexity arguments. We now refer to Fig. 3. If we identify the average molar density p of the mixture at fixed temperature with the function C, and the DBC as the curve in question, we obtain from Eq. (A4)
(0o ,
P,T
(A5)
From the Maxwell relation
1 (00 0a p2\~'p,T=(~)X,
T
we obtain
=2d and using again the "standard argument," we obtain =P
tdP)?[0X),
(A7)
Now we note that (OA/OX)e.r is zero in the twophase region, and by stability arguments is nonnegative in the onephase region. Hence
a (Oo/OP)7,x ~ 0
(A8)
But Eq. (A8) with Eq. (A4) implies
6(OP/Op)x,T ~ 0
(A9)
Finally, by the "standard argument"
\~p / x , T =  \ ~ p x \OT/o,x
(A10)
But by Eq. (A9) the lefthand side is always nonnegative; thus 6(OP/OT)p,x always has the opposite sign from (dT/dp)x. For the 3He4He mixtures, (dT/dO)x changes sign at the point we have labeled pT, the density corresponding to the highest temperature on the DBC. Hence PD, the density for which 6(OP/OT)o,x was found to be zero, must be equal to PT" APPENDIX
B. C A L C U L A T I O N OF (OP/OT)o,x F R O M L E U N G  G R I F F I T H S E Q U A T I O N OF S T A T E
THE
Since we are interested in the variation of quantities along isochores at constant X, the procedure fs to start from the given variables (T, p, X) and
362
Ted Doiron, Robert P. Behringer, and Horst Meyer
calculate the correspondig model fields 0, ~, h). From the model field values any quantity of interest could be calculated. In the onephase region, the fields are found by a NewtonRaphson iteration procedure on the equations, 1X
~  ~ ( 1  ~)O0, ~, h ) / p = 0
(B1)
p ~0h = 0
(B2)
~+BB~(~') = 0
(B3)
which are Eqs. (2.25), (2.23), and (2.8) of the Ref. 3, and where B = (RT) 1
(B4)
P = RTw(.c, ~, h)
(B5)
Wh = (Oo~/Oh)~,~
(B6)
In the twophase region, h = 0, so only two equations are needed to obtain the field variables (7, ~). But because the density and concentration of the liquid differ from those of the vapor, Eqs. (B1) and (B2) are no longer adequate since X and p are not singlevalued. To obtain a usable equation we let q be the fraction of the available volume occupied by the liquid, and use the conservation of particles of each isotope tO obtain X,o~q + X~pv(1  q) = X o &q +p~(1  q ) = P
(B7) (B8)
We now need to obtain 0, ~) from a given (X, P, T). If we eliminate q, we obtain p ( X , pt  Xvp~) + Xp(O~  P,) + O,p~ (X~  X,) = 0
(B9)
From Eq. (B9), where each of the quantities is singlevNued, and from Eq. (B3), (P, X, T) is converted to (~, 1, h = 0), Once a set of (P, X, T) values is converted to (~, ~, h), the pressure can be easily calculated from Eq. (B5). For comparison with the (OP/OT)o, x data, the pressure at a set of temperatures along each isochore is calculated, and the slopes are found by numerical differentiation.
ACKNOWLEDGMENTS
The authors are very grateful to Prof. R. B. Griiliths for detailed constructive criticism of the manuscript and for providing the demonstration shown in Appendix A.
Equation of State of a 3He4He Mixture
363
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
B. Wallace and H. Meyer, Phys. Rev. A 5, 953 (1972). G. R. Brown and H. Meyer, Phys. Rev. A 6, 1578 (1972). S. S. Leung and R. B. Griffiths, Phys. Rev. A 8, 2670 (1973). J . M . H . Levelt Sengers, Thermodynamic Properties near the Critical State, Chapter 14 in Pure and Applied Chemistry (Butterworth, London, 1975); Physica 73, 73 (1974). R. B. Griffiths and J. C. Wheeler, Phys. Rev. A 2, 1047 (1970). H. Kierstead, Phys. Rev. A 3, 329 (1971). J. S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, Belfast, 1969). R. P. Behringer, T. Doiron, and H. Meyer, J. Low Temo. Phys. 24, 315, (1976). R. P. Behringer, T. Doiron, and H. Meyer, Technical Report, Duke University, October 1975 (unpublished). R. P. Behringer, Ph.D. thesis, Duke University, June 1975 (unpublished). R. B. Griffiths, private communication.