3. Vapor Pressure

An expression for the 1958 helium temperature scale from McCarty [8] was used to predict the vapor pressure of helium between .8 and 2.172 K. The expression is

where P is in μm Hg = 133.322 Pa and T is in kelvin. The range of validity for eq (1)

is 0.5 to 2.172 K. The coefficients to eq
The coefficients to eq (1) are given in table 1. The

vapor pressures in table 7 below 0.8 K are calculated via the thermodynamic

conditions for the coexistence of two phases in equilibrium assuming the ideal
gas law for gaseous helium and using the Gibbs free energy (G) of Brooks and
Donnelly [3] for the liquid phase. The agreement between the vapor pressures
calculated from eq (1) and those calculated as described above between 2.172 and
0.5 K is good, with typical differences of 0.1 percent or less. Table 2 gives a
detailed comparison between the 1958 helium scale and pressures calculated via
the phase rule using the Brooks and Donnelly [3] data for the saturated liquid
and the PVT surface of McCarty [8] for the saturated vapor. The agreement (as
shown in table 2) between the 1958 scale and this work is a little surprising
since the real gas contribution at 2.0 K on the vapor side amounts to a 3.8
percent of the pressure (i.e., the difference between using an ideal gas equation
of state and the one from McCarty). At 1.5 K this real gas contribution has
fallen to 1.1 percent and at 0.8 K it has disappeared entirely. Below 1 K the
value of the Gibbs energy of the saturated liquid has become so small that it has
little effect on the solution and the vapor pressure is determined almost
entirely by the choice of Lo, the latent heat of vaporization at 0 K. A value
of 59.62 joules/mole was used in the 1958 helium vapor pressure scale and a value
of 59.60202 joules/mole was used here. In using the Brooks and Donnelly [3]
tables for this calculation, both the tabulated G and the tabulated H-TS were
used for comparative purposes.
purposes. The results are similar but the procedure of
using the tabulated G produced results slightly closer to the 58 scale. The G
obtained by forming H-TS agrees quite well with the tabulated G down to 0.5 K but
begins to differ from the tabulated G below 0.5 K and the two values actually
have opposite signs at 0.2 K and below. Since the values of G are so small at
these temperatures the resulting vapor pressure is the same for either the
the value of G.

·

or

Table 2. Comparison of 1958 He Vapor-Pressure-Temperature

Scale and Values From This Work.

4. Density of the Saturated Liquid and Gaseous Phases

The density of the saturated liquid phase was taken from Ker and Taylor [5]

Table 3. Coefficients for Eq. 2 (Saturated Liquid)

where V is in liters/mole, T is in kelvin and the &,, l2, lz, and T, are given

in table 3.

λ

The density of the saturated gaseous state was calculated using the equation of state from McCarty [8] between 0.8 and 2.172 K. Below 0.8 K the density of the saturated gas was calculated using the vapor pressure (as described in the previous section) and the ideal gas equation of state.

5. The Mathematical Model of the Equation of State

It is desirable to model the equation of state of a fluid with a single mathematical function. A single function insures thermodynamic consistency and eliminates potential mathematical continuity problems going from one function to another. Unfortunately a single function for the helium IV equation of state was not found; instead it was necessary to divide the surface into three regions: one from 0.0 to 0.8 K (Region I), one from 0.8 to 1.2 K (Region II), and from 1.2 to the lambda line (Region III). Although mathematical continuity is not achieved at the boundaries, the fit is good enough at the boundaries of each

region to minimize the thermodynamic inconsistencies. For the region of 0 to

- Fs, 10274/4 - Fs,20275/5 - FS, 3D2T6/6

2Fs, 4 (Ap)D2T3/3 - 2Fs,5 (AP)D274/4

2FS,6(Ap)D2+5/5 - 2Fs.7(4)D2+6/6 - 2Fs.g(Ap)D2t7/7

S,8

- 3Fs, 9(Ap)202+3/3 - 3Fs, 10(4) 202+4/4 - 3Fs, 11 (Ap) 20275/5

(40)

2-3
Τ

(5)

3Fs,12(4p)2d2t6/6 - 3Fs,13(Ap)2d2t7/7 - 4Fs, 14(Ap)3d2t3/3

- 4Fs, 15 (Ap)30275/5 - 4Fs,16 (Ap) 30277/7 - 4Fs, 17 (Ap) 302+99

- 4Fs,18 (40)302+11/11 - 5F5, 19(Ap)402+3/3

where P is the pressure in bar, D is the density in moles/liter, T is temperature

in kelvin and Ap = D -36.27877.

The coefficients for Region I are given in

table 4.