<<Previous

Page1

Page2

Page3

Page4

Page5

Page6

Page7

Next>>

Table 1 : : 2 : : 3 : : 4 : : 5

The Calculated Thermodynamic Properties of Superfluid Helium-4
James S. Brooks and Russell J. Donnelly

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[33]; open triangles, Boghosian and Meyer [32]; solid circles, Abraham et al. [31]; solid triangles, Kerr and Taylor [34]. 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. [21].

4.2.b. The Isothermal Compressibility

The isothermal compressibility, which is defined as

is 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.

The first term k_T(0)can be compared directly with the data of table V of Boghosian and Meyer [32]. The deviations _K (k_eale - k_meas)/k_meas range from + 4.1% to -5.0%. However, we believe that our data, derived from the later measurements of Abraham et al. [31] are considerably more reliable. Since the contribution from the temperature dependent part of V is so small, the deviations quoted are the dominant ones at all temperatures up to 1.25 K, the highest reported by Boghosian and Meyer [32].

Elwell and Meyer [33] report measurements Of k_T in the range above 1.2 K. Comparing with their tables IV and V, we find the pressure averaged deviations k to be + 0, - 2.4% at 1.3 K; and + 0, - 2.0% at 1.8 K.

Our computed values of the compressibility are not sufficiently accurate near the lambda line to exhibit the peak structure observed by Grilly [42].

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

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. [31] at all pressures, to within 0.25%.

TABLE III. The Grüneisen constant UG
T(K)			P(atm)
	0	5	10	15	20	25
0.0	2.843	2.608	2.460	2.356	2.276	2.212
0.5	2.842	2.608	2.460	2.356	2.275	2.213
1.0	2.842	2.624	2.466	2.339	2.270	2.194

4.2.d. The Coefficient of Thermal Expansion

The thermal expansion coefficient is a temperature derivative of the equation of state:

It is, however, directly calculated from the integral expression of Roberts and Donnelly given in table I.

We experienced great difficulty with _P because it is a very small quantity which passes through zero near 1 K. Our calculations, listed in table 5 and illustrated in figure 11, are sufficiently accurate to give a good account of the locus of zero expansion indicated by the dashed line in figure 2. We have also managed to keep the deviations from growing rapidly above 1.6 K as they did in our earlier work [18, 19]. The temperature. averaged deviations _P = (_Pcalculated-_Pmeasured)/ _Pmeasured follow:

P(atm)	0	5	10	15	20	25
	14	16	34	0.8	1.1	1.6
	16	3.2	8	9	10	11

Except 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.

The density and expansion coefficient have been investigated recently, but only at the SVP. The reference is J. J. Niemela and R. J. Donnelly, J. Low Temperature Physics 98 1-16, (1995).

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 [33]; solid triangles, Boghosian and Meyer [32]; open triangles, Van DeGrift [20]; open diamonds, Kerr and Taylor[34].

4.3. The Fundamental Thermodynamic Functions

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 = (S_calc.-S_meas)/S_meas are as follows:

P(atm)	0	5	10	15	20	25
	0.7	1.5	1.2	0.5	0.6	0.2
	3.9	2.6	2.4	2.5	2.8	2.0

The entropy is listed in table 9 and plotted in figure 12.

4.3.b. The Helmholtz Free Energy

Table I shows that the Helmholtz free energy F consists of a ground state part F_o(V) and an excitation part F_E given by the double integral over the spectrum. F_o(V) can be determined by integrating the expression dF_o=-PdV.. The results give F_o at T = 0 K to within an additive constant L_o [L_o = F (0, 0) = (0, 0)] where L_o 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.

Table IV. The ground state Helmholtz free energy Fo(P)=[F (0, P) -F (0,0)]J g-1
P(atm)	0	2.5	5	7.5	10	12.5
F0(V) (J.g-1)	0	0.0232554	0.082368	0.16671	0.26980	0.38737
P(atm)		15	17.5	20	22.5	25
F0(V) (J.g-1)		0.51645	0.65490	0.80111	0.95385	1.1122

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. [31] 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 [36], and for 1.6 T 2.05 is from Van den Meijdenberg et al. [35].

<<Previous Page1 Page2 Page3 Page4 Page5 Page6 Page7 Next>>