Russell J. Donnelly
Saturated Vapor Pressure
Calculated Thermodynamic Properties of Superfluid Helium-4
Figure 8. The relative change in molar volume given by the expression in table I. The nonlinear scale is chosen to emphasize the locus of maximum molar volume, or zero thermal expansion.
Figure 9. The P-V-T surface of helium II. Solid lines come from the expression in table I; open circles, Elwell and Meyer; open triangles, Boghosian and Meyer ; solid circles, Abraham et al. ; solid triangles, Kerr and Taylor . The deviation between calculation and the data may be seen as a difference perpendicular to the P-T plane. The experimental data have been numerically interpolated to 5 atm intervals in pressure.
Figure 10. The velocity of first sound as a function of temperature for different pressures calculated from eq (25). Data points from Heiserman et al. .
4.2.b. The Isothermal Compressibility
isothermal compressibility, which is defined as
obtained by noting that V(T, P)
= V(0, P) + V(T,
P) from table I. The derivative V(T, P)=V(O,
P) + V(T,
P) is calculated separately for the two terms, resulting in more
reliable results than differentiating the total molar volume directly.
The results are in table 4.
4.2.c. The Grüneisen Constant
Another quantity which appears in expressions for the ultrasonic attenuation and dispersion in helium II is the Grüneisen constant, defined as
is listed in table III below; the values at T= 0 K are in. agreement with
the T= 0. 1 K data of Abraham et al.  at all pressures, to within
It is listed in table III below; the values at T= 0 K are in. agreement with the T= 0. 1 K data of Abraham et al.  at all pressures, to within 0.25%.
4.2.d. The Coefficient of Thermal Expansion
thermal expansion coefficient is a temperature derivative of the equation
is, however, directly calculated from the integral expression of Roberts
and Donnelly given in table I.
for the vapor pressure, where the systematic work of Van Degift exists,
the experimental situation on P
is quite unsatisfactory and the data often in mutual conflict. A systematic
study over the entire T-P plane would be of great benefit in
reducing the deviations listed above.
Figure 11. The thermal expansion coefficient as a function of pressure and temperature. The solid lines are calculated from the expression in table I; circles, Elwell and Meyer ; solid triangles, Boghosian and Meyer ; open triangles, Van DeGrift ; open diamonds, Kerr and Taylor.
In this section we describe our use of the Landau theory . and the effective spectrum to compute the thermodynamic properties of helium II.
4.3.a. The Entropy
The entropy is the fundamental quantity used to find e effective spectrum. Deviations, then, reflect imperfections in the data itself -as well as the effective spectrum. The temperature averaged deviations S = (Scalc.-Smeas)/Smeas are as follows:
The entropy is listed in table 9 and plotted in figure 12.
4.3.b. The Helmholtz Free Energy
I shows that the Helmholtz free energy F consists of a ground
state part Fo(V) and an excitation part FE given by the double
integral over the spectrum. Fo(V) can be determined by integrating the expression dFo=-PdV.. The results give Fo at T = 0
K to within an additive constant Lo [Lo
= F (0, 0) = (0,
0)] where Lo is the latent heat
of vaporization extrapolated to zero temperature, and is approximately
14.90 J g-1. L is a measure of the energy required
to separate the atoms of the liquid to infinity.
The Helmholtz free energy is not a directly accessible quantity, and no comparison with experimental data can be readily made. However, table IV comes from the equation of state of Abraham et al.  and should be quite accurate. The excitation free energy is tabulated in table 6 and plotted in figure 13. The derivative (n / q) in the expression for F in table I is taken at constant volume. We approximated it by constant pressure. In order to check the accuracy of our calculation, we computed - (F / T)at constant pressure and compared it to S. Except at the highest temperatures and pressures, the error in our procedure is generally less than 1% and never more than 2.2%.
Figure 12. The entropy of helium II as a function of pressure and temperature calculated from equation (2). The data for 0.3 T 1.6 is from Wiebes , and for 1.6 T 2.05 is from Van den Meijdenberg et al. .
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