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Calculation of the thermodynamic properties of liquid 3He4He mixtures for temperatures below 150 mK and 3He concentrations between 0.1 and 8% at zero pressure
Physica 12813(1985) 197-200 North-Holland, Amsterdam
C A L C U L A T I O N OF THE T H E R M O D Y N A M I C PROPERTIES OF LIQUID 3He--4He MIXTURES FOR T E M P E R A T U R E S B E L O W 150 m K AND 3He C O N C E N T R A T I O N S B E T W E E N 0.1 A N D 8% AT ZERO PRESSURE J.G.M. K U E R T E N , C.A.M. C A S T E L I J N S , A.T.A.M. D E W A E L E and H.M. G I J S M A N Eindhoven University of Technology, Post Box 513, 5600 MB Eindhoven, The Netherlands Received 4 October 1984 We performed calculations of thermodynamic quantities of dilute liquid 3He-4He mixtures, starting from experimental values of the specific heat and the osmotic pressure. The calculations are confined to temperatures below 150 mK and 3He concentrations between 0.1 and 8% at zero pressure. Contrary to previous calculations performed by Radebaugh, our results are in good agreement with the experimental data on both the osmotic pressure and the enthalpy in dilute 3He--4HeII mixtures.
1. Introduction In 1967 Radebaugh calculated the thermodynamic properties of 3He-4He mixtures at low temperatures [1]. There are, however, discrepancies between his calculated results and the measured values of important quantities, such as the osmotic pressure [2, 3] and the enthalpy of the 3He quasiparticles [4]. T h e r e f o r e we recalculated the thermodynamic quantities, making use of experimental data not available in 1967. The calculations are restricted to liquid properties at zero pressure, temperatures below 150 mK, and 3He concentrations ranging from 0.1 to 8%.
2. Calculation scheme The G i b b s - D u h e m equation for the mixture is xdl~3+(1-x)
dl~4=-SmdT+
Vmdp,
(1)
where x is the molar 3He concentration, Sm is the molar entropy and Vm the molar volume of the mixture; ~3 and/.t 4 are the chemical potentials of the 3He and 4He c o m p o n e n t s per mole, respectively. In our t e m p e r a t u r e region the molar volume
Vm(T, x ) can be considered as being t e m p e r a t u r e independent; the concentration dependence is obtained from experiments [3]. T h e value of S m can be calculated from m e a s u r e m e n t s as follows: experiments [5,6] show that the contribution of the 3He component to the specific heat of the mixture per mole 3He is equal to the specific heat at constant volume, Cvr, of an ideal Fermi gas at the same quasiparticle density. This Col is a known function [1, 7] of the reduced t e m p e r a t u r e t = T/Tf, where T r is the Fermi t e m p e r a t u r e of the quasiparticle gas. In the calculation of Tf a concentration-independent quasiparticle mass of 2.46 times the 3He atomic mass was taken. This value at the same time fits the specific heat data of Anderson [5] and GreywaU [6] and the data of the osmotic pressure within 2%. It also leads to a good result for the T - x dependence of the phase separation curve [8]. The entropy Sf and the chemical potential /zf of the ideal Fermi gas, both per mole, follow from C a, using thermodynamics. We can write the entropy of the mixture as S m = xSf + (1 - x ) S ° ,
(2)
where S o is the molar entropy of pure 4He. In our t e m p e r a t u r e region this quantity may be neglected.
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J.G.M. Kuerten et al. I Thermodynamic properties of liquid 3He-4He mixtures
198
Furthermore we consider the situation in which the pressure is constant. Together with eq. (2) the Gibbs-Duhem equation then simplifies to
T/T1
17(T, x) = h’(0, x) +
dT, x2 V,,,(T, 0) dx I
G(t)
dt.
”
1-X d/+= - -dj+-
00) S,dT.
X
( )
The osmotic pressure at finite temperature was calculated with C,, obtained as described above, and n(O, x) fitted to the data of Landau et al. [2]. The 4He chemical potential p4 follows from (8) and (10). From (3) pLj was obtained by integration over x and T. The zero point of ps was chosen at T = 0 on the phase separation curve (x, = 6.6%) [S]. The Gibbs free energy per mole mixture, G,, is given by
-s
-zx=
and
G,,,=x/++(l-x)F~.
where
is the it follows,
entropy of (2) that
4He com-
($$ =xp)T.
(6)
Since the specific heat C,, is a function of the reduced temperature t only, the derivative X$/%X and the specific heat are related according to = --- 1 dT, C”‘ . T, dx With the definition ]21: P~CC
Xl=
~4(T,o)-
of the osmotic pressure
(7) Zl
V,CCO).WT,X),
All other thermodynamic quantities of the mixture can now be calculated [9]. The enthalpy of the quasiparticle gas, w3, which plays an important role in flow experiments, was calculated according to its definition: q(T,
x) = ~0,
x) + TS,(T, x) .
(12)
This quantity should not be confused with the partial 3He enthalpy. In order to compute the phase separation curve, the experimental data on the specific heat of pure 3He of Greywall [lo] were used to calculate pi, the chemical potential of pure 3He. The dependence of the concentration x, of the saturated solution on T follows from P~(T, X,(V)=
d(T).
(13)
3. Results
it follows that --1 dT, c V,,,(T, 0) T, dx “f ’
(II)
X2
(9)
where we again neglected the contribution due to the entropy of pure 4He. Integration yields
Some results of our computations are given in the T2-x diagram (fig. 1) in which lines of constant w (which we will call isenthalps for short) and isotones (lines of constant osmotic pressure) as well as the phase separation curve are given. For low temperatures and x > 0.02 the isenthalps can in good approximation be written as
J.G.M. Kuerten et al. / Thermodynamic properties of liquid 3He-4He mixtures
199
Table I T h e values of the osmotic pressure and the enthalpy of the quasiparticle gas as functions of the temperature and the 3He concentration The osmotic pressure of the mixture, in pascal, as a function of T, in kelvin, and the molar 3He concentration T~
0.001
0.010
0.020
0.030
0.040
0.050
0.060
0.066
0.070
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
3.0 4.5 7.1 9.9 12.8 15.8 18.7 21.7 24.7 27.6 30.6 33.6 36.6 39.6 42.6 45~6
126.5 130.9 143.1 160.9 182.3 205.9 231.0 257.0 283.7 310.9 338.6 366.5 394.8 423.2 451.8 480.5
379.1 384.6 400.7 426.2 459.2 497.6 540.2 585.8 633.7 683.4 734.4 786.5 839.5 893.2 947.6 1002.4
707.4 713.7 732.4 762.6 803.0 851.8 907.2 967.9 1032.7 1101.0 1172.0 1245.2 1320.1 1396.6 1474.2 1552.9
1087.5 1094.4 1115.0 1148.7 1194.6 1251.1 1316.4 1389.1 1467.8 1551.5 1639.6 1731.1 1825.6 1922.4 2021.4 2122.0
1502.7 1510.1 1532.3 1568.9 1619.1 1681.6 1754.9 1837.5 1927.9 2025.0 2127.7 2235.2 2346.9 2462.1 2580.3 2701.1
1940.2 1948.1 1971.7 2010.7 2064.5 2132.0 2212.0 2302.9 2403.3 2511.9 2627.5 2749.2 2876.2 3007.9 3143.7 3282.9
2209.0 2217.2 2241.5 2281.9 2337.5 2407.7 2491.2 2586.5 2692.3 2807.0 2929.7 3059.2 3194.7 3335.6 3481.0 3630.6
2389.6 2397.9 2422.8 2463.9 2520.8 2592.7 2678.3 2776.3 2885.4 3004.1 3131.1 3265.6 3406.5 3553.2 3704.9 3861.0
T h e enthalpy of the quasiparticle gas in joule per mole quasiparticles as a function of T, in kelvin, and the molar 3He concentration TN~ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
0.001
0.010
0.020
0.030
0.040
0.050
0.060
0.066
0.070
-2.365 -2.259 -2.079 -1.884 -1.684 -1.482 -1.278 -1.073 -0.868 -0.662 -0.456 -0.250 -0.044 0.163 0.369 0.576
-1.674 -1.644 -1.560 -1.437 -1.290 -1.127 -0.955 -0.776 -0.592 -0.405 -0.215 -0.023 0.172 0.367 0.564 0.761
-1.208 -1.189 -1.134 -1.046 -0.933 -0.801 -0.655 -0.499 -0.334 -0.164 0.012 0.190 0.372 0.556 0.743 0.931
-0.852 -0.838 -0.795 -0.726 -0.634 -0.523 -0.396 -0.258 -0.110 0.046 0.208 0.375 0.546 0.721 0.898 1.078
-0.562 -0.550 -0.515 -0.457 -0.379 -0.283 -0.171 -0.047 0.088 0.231 0.381 0.537 0.699 0.864 1.033 L205
-0.318 -0.308 -0.278 -0.228 -0.160 -0.074 0.026 0.138 0.262 0.394 0.534 0.680 0.833 0.990 1.151 1.316
-0.110 -0.101 -0.075 -0.030 0.031 0.107 0.198 0.301 0.415 0.538 0.669 0.807 0.951 1.100 1.254 1.412
0.000 0.008 0.033 0.075 0.132 0.204 0.290 0.388 0.497 0.615 0.742 0.875 1.014 1.159 1.309 1.463
0.068 0.076 0.100 0.140 0.195 0.265 0.348 0.443 0.549 0.664 0.787 0.917 1.054 1.196 1.343 1.494
Table II Comparison of the p a r a m e t e r fl and two values of the osmotic pressure between this work, R a d e b a u g h ' s calculations, and the experimental data
/3 (K 2) H(0, x0) (Pa) H(0.142, 0.0628) (Pa)
This work
R a d e b a u g h [1]
Exp.
Ref.
0.209 2209 3330
0.14 1629 2998
0.21 2240 3324
[41 [2, 3] [2]
200
J.G.M. Kuerten et al. / Thermodynamic properties of liquid 3He--4He mixtures
.025 ~./ I i"-"1
tween our results and the results of Radebaugh are listed in table II. The main origin of the discrepancy between the two calculations is the fact that Radebaugh incorrectly extrapolated /z3(0, x), obtained from the phase separation curve, to concentrations below 6.4%. However, we were in the position to use the measurements of the osmotic pressure. This approach leads to a significantly better agreement with the experiments.
00 liI !:i .015 ¢'4
.010
00s
. \.
Acknowledgments
\-. ~-. .02
.or,
.06
.08
x
Fig. 1. The T2-x diagram with isenthalps ( - - - - ) , isotones ( ) and the phase separation curve ( ) resulting from the calculations described in this paper (H in Pa and /-~ in J/tool 3He).
T 2 +/3x
=
constant,
where/3 is a constant. In table I the osmotic pressure and the enthalpy of the quasiparticle gas are tabulated as functions of the temperature and the 3He concentration. More detailed results will be published later.
4. Discussion Some examples showing the differences be-
We would like to thank Messrs. R.C. Kommeren and J.M. Upperman for their valuable contributions.
Rderences [1] R. Radebaugh, N.B.S. Technical Note 362 (1967). [2] J. Landau, J.T. Tough, N.R. Brubaker and D.O. Edwards, Phys. Rev. A2 (1970) 2472. [3] A. Ghozlan and E. Varoquaux, Ann. de Phys. 3 (1979) 239. [4] A.T.A.M. de Waele, J.C.M. Keltjens, C.A.M. Castelijns and H.M. Gijsman, Phys. Rev. B28 (1983) 5350. [5] A.C. Anderson, D.O. Edwards, W.R. Roach, R.E. Sarwinski and J.C. Wheatley, Phys. Rev. Lett. 17 (1966) 367. [6] D.S. Greywall, Phys. Rev. B20 (1979) 2643. [7] E.C. Stoner, Phil. Mug. 25 (1938) 899. [8] G.W. Watson, J.D. Reppy and R.C. Richardson, Phys. Rev. 188 (1969) 384. [9] See, for example, E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1967). [10] D.S. Greywall, Phys. Rev. B27 (1983) 2747.
C A L C U L A T I O N OF THE T H E R M O D Y N A M I C PROPERTIES OF LIQUID 3He--4He MIXTURES FOR T E M P E R A T U R E S B E L O W 150 m K AND 3He C O N C E N T R A T I O N S B E T W E E N 0.1 A N D 8% AT ZERO PRESSURE J.G.M. K U E R T E N , C.A.M. C A S T E L I J N S , A.T.A.M. D E W A E L E and H.M. G I J S M A N Eindhoven University of Technology, Post Box 513, 5600 MB Eindhoven, The Netherlands Received 4 October 1984 We performed calculations of thermodynamic quantities of dilute liquid 3He-4He mixtures, starting from experimental values of the specific heat and the osmotic pressure. The calculations are confined to temperatures below 150 mK and 3He concentrations between 0.1 and 8% at zero pressure. Contrary to previous calculations performed by Radebaugh, our results are in good agreement with the experimental data on both the osmotic pressure and the enthalpy in dilute 3He--4HeII mixtures.
1. Introduction In 1967 Radebaugh calculated the thermodynamic properties of 3He-4He mixtures at low temperatures [1]. There are, however, discrepancies between his calculated results and the measured values of important quantities, such as the osmotic pressure [2, 3] and the enthalpy of the 3He quasiparticles [4]. T h e r e f o r e we recalculated the thermodynamic quantities, making use of experimental data not available in 1967. The calculations are restricted to liquid properties at zero pressure, temperatures below 150 mK, and 3He concentrations ranging from 0.1 to 8%.
2. Calculation scheme The G i b b s - D u h e m equation for the mixture is xdl~3+(1-x)
dl~4=-SmdT+
Vmdp,
(1)
where x is the molar 3He concentration, Sm is the molar entropy and Vm the molar volume of the mixture; ~3 and/.t 4 are the chemical potentials of the 3He and 4He c o m p o n e n t s per mole, respectively. In our t e m p e r a t u r e region the molar volume
Vm(T, x ) can be considered as being t e m p e r a t u r e independent; the concentration dependence is obtained from experiments [3]. T h e value of S m can be calculated from m e a s u r e m e n t s as follows: experiments [5,6] show that the contribution of the 3He component to the specific heat of the mixture per mole 3He is equal to the specific heat at constant volume, Cvr, of an ideal Fermi gas at the same quasiparticle density. This Col is a known function [1, 7] of the reduced t e m p e r a t u r e t = T/Tf, where T r is the Fermi t e m p e r a t u r e of the quasiparticle gas. In the calculation of Tf a concentration-independent quasiparticle mass of 2.46 times the 3He atomic mass was taken. This value at the same time fits the specific heat data of Anderson [5] and GreywaU [6] and the data of the osmotic pressure within 2%. It also leads to a good result for the T - x dependence of the phase separation curve [8]. The entropy Sf and the chemical potential /zf of the ideal Fermi gas, both per mole, follow from C a, using thermodynamics. We can write the entropy of the mixture as S m = xSf + (1 - x ) S ° ,
(2)
where S o is the molar entropy of pure 4He. In our t e m p e r a t u r e region this quantity may be neglected.
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J.G.M. Kuerten et al. I Thermodynamic properties of liquid 3He-4He mixtures
198
Furthermore we consider the situation in which the pressure is constant. Together with eq. (2) the Gibbs-Duhem equation then simplifies to
T/T1
17(T, x) = h’(0, x) +
dT, x2 V,,,(T, 0) dx I
G(t)
dt.
”
1-X d/+= - -dj+-
00) S,dT.
X
( )
The osmotic pressure at finite temperature was calculated with C,, obtained as described above, and n(O, x) fitted to the data of Landau et al. [2]. The 4He chemical potential p4 follows from (8) and (10). From (3) pLj was obtained by integration over x and T. The zero point of ps was chosen at T = 0 on the phase separation curve (x, = 6.6%) [S]. The Gibbs free energy per mole mixture, G,, is given by
-s
-zx=
and
G,,,=x/++(l-x)F~.
where
is the it follows,
entropy of (2) that
4He com-
($$ =xp)T.
(6)
Since the specific heat C,, is a function of the reduced temperature t only, the derivative X$/%X and the specific heat are related according to = --- 1 dT, C”‘ . T, dx With the definition ]21: P~CC
Xl=
~4(T,o)-
of the osmotic pressure
(7) Zl
V,CCO).WT,X),
All other thermodynamic quantities of the mixture can now be calculated [9]. The enthalpy of the quasiparticle gas, w3, which plays an important role in flow experiments, was calculated according to its definition: q(T,
x) = ~0,
x) + TS,(T, x) .
(12)
This quantity should not be confused with the partial 3He enthalpy. In order to compute the phase separation curve, the experimental data on the specific heat of pure 3He of Greywall [lo] were used to calculate pi, the chemical potential of pure 3He. The dependence of the concentration x, of the saturated solution on T follows from P~(T, X,(V)=
d(T).
(13)
3. Results
it follows that --1 dT, c V,,,(T, 0) T, dx “f ’
(II)
X2
(9)
where we again neglected the contribution due to the entropy of pure 4He. Integration yields
Some results of our computations are given in the T2-x diagram (fig. 1) in which lines of constant w (which we will call isenthalps for short) and isotones (lines of constant osmotic pressure) as well as the phase separation curve are given. For low temperatures and x > 0.02 the isenthalps can in good approximation be written as
J.G.M. Kuerten et al. / Thermodynamic properties of liquid 3He-4He mixtures
199
Table I T h e values of the osmotic pressure and the enthalpy of the quasiparticle gas as functions of the temperature and the 3He concentration The osmotic pressure of the mixture, in pascal, as a function of T, in kelvin, and the molar 3He concentration T~
0.001
0.010
0.020
0.030
0.040
0.050
0.060
0.066
0.070
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
3.0 4.5 7.1 9.9 12.8 15.8 18.7 21.7 24.7 27.6 30.6 33.6 36.6 39.6 42.6 45~6
126.5 130.9 143.1 160.9 182.3 205.9 231.0 257.0 283.7 310.9 338.6 366.5 394.8 423.2 451.8 480.5
379.1 384.6 400.7 426.2 459.2 497.6 540.2 585.8 633.7 683.4 734.4 786.5 839.5 893.2 947.6 1002.4
707.4 713.7 732.4 762.6 803.0 851.8 907.2 967.9 1032.7 1101.0 1172.0 1245.2 1320.1 1396.6 1474.2 1552.9
1087.5 1094.4 1115.0 1148.7 1194.6 1251.1 1316.4 1389.1 1467.8 1551.5 1639.6 1731.1 1825.6 1922.4 2021.4 2122.0
1502.7 1510.1 1532.3 1568.9 1619.1 1681.6 1754.9 1837.5 1927.9 2025.0 2127.7 2235.2 2346.9 2462.1 2580.3 2701.1
1940.2 1948.1 1971.7 2010.7 2064.5 2132.0 2212.0 2302.9 2403.3 2511.9 2627.5 2749.2 2876.2 3007.9 3143.7 3282.9
2209.0 2217.2 2241.5 2281.9 2337.5 2407.7 2491.2 2586.5 2692.3 2807.0 2929.7 3059.2 3194.7 3335.6 3481.0 3630.6
2389.6 2397.9 2422.8 2463.9 2520.8 2592.7 2678.3 2776.3 2885.4 3004.1 3131.1 3265.6 3406.5 3553.2 3704.9 3861.0
T h e enthalpy of the quasiparticle gas in joule per mole quasiparticles as a function of T, in kelvin, and the molar 3He concentration TN~ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
0.001
0.010
0.020
0.030
0.040
0.050
0.060
0.066
0.070
-2.365 -2.259 -2.079 -1.884 -1.684 -1.482 -1.278 -1.073 -0.868 -0.662 -0.456 -0.250 -0.044 0.163 0.369 0.576
-1.674 -1.644 -1.560 -1.437 -1.290 -1.127 -0.955 -0.776 -0.592 -0.405 -0.215 -0.023 0.172 0.367 0.564 0.761
-1.208 -1.189 -1.134 -1.046 -0.933 -0.801 -0.655 -0.499 -0.334 -0.164 0.012 0.190 0.372 0.556 0.743 0.931
-0.852 -0.838 -0.795 -0.726 -0.634 -0.523 -0.396 -0.258 -0.110 0.046 0.208 0.375 0.546 0.721 0.898 1.078
-0.562 -0.550 -0.515 -0.457 -0.379 -0.283 -0.171 -0.047 0.088 0.231 0.381 0.537 0.699 0.864 1.033 L205
-0.318 -0.308 -0.278 -0.228 -0.160 -0.074 0.026 0.138 0.262 0.394 0.534 0.680 0.833 0.990 1.151 1.316
-0.110 -0.101 -0.075 -0.030 0.031 0.107 0.198 0.301 0.415 0.538 0.669 0.807 0.951 1.100 1.254 1.412
0.000 0.008 0.033 0.075 0.132 0.204 0.290 0.388 0.497 0.615 0.742 0.875 1.014 1.159 1.309 1.463
0.068 0.076 0.100 0.140 0.195 0.265 0.348 0.443 0.549 0.664 0.787 0.917 1.054 1.196 1.343 1.494
Table II Comparison of the p a r a m e t e r fl and two values of the osmotic pressure between this work, R a d e b a u g h ' s calculations, and the experimental data
/3 (K 2) H(0, x0) (Pa) H(0.142, 0.0628) (Pa)
This work
R a d e b a u g h [1]
Exp.
Ref.
0.209 2209 3330
0.14 1629 2998
0.21 2240 3324
[41 [2, 3] [2]
200
J.G.M. Kuerten et al. / Thermodynamic properties of liquid 3He--4He mixtures
.025 ~./ I i"-"1
tween our results and the results of Radebaugh are listed in table II. The main origin of the discrepancy between the two calculations is the fact that Radebaugh incorrectly extrapolated /z3(0, x), obtained from the phase separation curve, to concentrations below 6.4%. However, we were in the position to use the measurements of the osmotic pressure. This approach leads to a significantly better agreement with the experiments.
00 liI !:i .015 ¢'4
.010
00s
. \.
Acknowledgments
\-. ~-. .02
.or,
.06
.08
x
Fig. 1. The T2-x diagram with isenthalps ( - - - - ) , isotones ( ) and the phase separation curve ( ) resulting from the calculations described in this paper (H in Pa and /-~ in J/tool 3He).
T 2 +/3x
=
constant,
where/3 is a constant. In table I the osmotic pressure and the enthalpy of the quasiparticle gas are tabulated as functions of the temperature and the 3He concentration. More detailed results will be published later.
4. Discussion Some examples showing the differences be-
We would like to thank Messrs. R.C. Kommeren and J.M. Upperman for their valuable contributions.
Rderences [1] R. Radebaugh, N.B.S. Technical Note 362 (1967). [2] J. Landau, J.T. Tough, N.R. Brubaker and D.O. Edwards, Phys. Rev. A2 (1970) 2472. [3] A. Ghozlan and E. Varoquaux, Ann. de Phys. 3 (1979) 239. [4] A.T.A.M. de Waele, J.C.M. Keltjens, C.A.M. Castelijns and H.M. Gijsman, Phys. Rev. B28 (1983) 5350. [5] A.C. Anderson, D.O. Edwards, W.R. Roach, R.E. Sarwinski and J.C. Wheatley, Phys. Rev. Lett. 17 (1966) 367. [6] D.S. Greywall, Phys. Rev. B20 (1979) 2643. [7] E.C. Stoner, Phil. Mug. 25 (1938) 899. [8] G.W. Watson, J.D. Reppy and R.C. Richardson, Phys. Rev. 188 (1969) 384. [9] See, for example, E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1967). [10] D.S. Greywall, Phys. Rev. B27 (1983) 2747.