Appendix C
Virial Coefficients for Quantum Gases

T his appendix presents the relationship between virial coefficients B and C with B′ and C′. In addition, it presents second and third virial coefficient data for quantum gases (hydrogen, helium, and neon) as a function of temperature.

Virial Equation as a Power Series in Density or Pressure

The compressibility factor of a gas can be expressed by an expansion using either the density or the pressure as the independent variable:

(C-1)

(C-2)

where, for a pure gas, virial coefficients B, C, D,… and B′, C′, D′,… are functions only of temperature. Equation (C-1) is the Leiden form and Eq. (C-2) is the Berlin form of the virial equation in recognition of early workers in these cities who first used these equations.

We now show how the coefficients of one series are related to those of the other. First, we multiply both equations by RTρ and obtain

(C-1a)

and

(C-2a)

Next, we substitute Eq. (C-1a) into Eq. (C-2a):

(C-3)

Equations (C-3) and (C-1a) are both power series in ρ. If we compare like terms in the two equations, we obtain from terms in ρ²:

(C-4)

or

(C-5)

From terms in ρ³:

(C-6)

or

(C-7)

Similarly, from terms in ρ⁴ we obtain

(C-8)

Equations (C-5), (C-7), and (C-8) are exact if we compare the two infinite series given by Eqs. (C-1) and (C-2). In other words, these relations are correct only if we evaluate the coefficients in both series from isothermal experimental compressibility factor (z) data according to

(C-9)

(C-10)

(C-11)

In practice, it is not possible to evaluate virial coefficients from experimental data very near zero density or zero pressure because experimental measurements at these conditions are insufficiently accurate, especially for the second and third derivatives. As a result, Eqs. (C-5), (C-7), and (C-8) are only approximations when they are used to convert experimental virial coefficients from one series to those of the other. To illustrate the approximate nature of these equations, we consider two methods of data reduction discussed by Scott and Dunlap (1962).

Suppose we want to evaluate B from low-pressure volumetric measurements of some gas at constant temperature. We can evaluate B either by fitting data to the truncated virial equation

(C-12)

or else by fitting to the truncated virial equation

(C-13)

In either case, we obtain B by fitting the experimental data over a finite region of density or pressure. If we use Eq. (C-12), we are, in effect, assuming that over the density range used, C = 0. If we use Eq. (C-13), we are, in effect, assuming that over the pressure range used, C′ = 0. Since B is generally not zero, we can see from Eq. (C-7) that if one of these assumptions is valid, then the other one is not. Therefore, it is not surprising that the value of B that we obtain by reduction of actual data depends on the method used and we find that Eq. (C-5) cannot, in practice, be satisfied exactly.

Scott and Dunlap made accurate volumetric measurements at low pressures for n-butane at 29.88°C. When they used Eq. (C-12) for data reduction they found

B = (-715 ± 5) cm³ mol^-1

However, when Eq. (C-13) was used, they obtained

B = (-715 ± 6) cm³ mol^-1

We can see from these results that even if experimental uncertainties are considered, Eq. (C-5) is only an approximation.

A preferable procedure for determining B from experimental data is to use for data reduction either one of the virial equations truncated after the third virial coefficient. When this was done, Scott and Dunlap found for the same data, using the density series,

B = (-691 ± 26) cm³ mol^-1

C = (-10 ± 5)×10⁵ (cm³ mol^-1)²

With this method of data reduction, the uncertainties in B are larger than before but agreement between the two methods is now very much improved. The results obtained for C are of little value but for obtaining B, it is preferable to include even a rough estimate for C than to include none at all.

A similar investigation was made by Lichtenthaler and Strein (1971), who concluded that Eq. (C-5) can be used for many typical systems provided that the second virial coefficients have been determined from highly accurate P-V-T data at low pressures, generally well below 1 bar.

Virial Coefficients for Hydrogen, Helium, and Neon

Because of their small masses, the properties of hydrogen, helium, and neon cannot be described by classical statistical mechanics. As discussed elsewhere (Hirschfelder et al., 1954), it is possible to write an expression for the second virial coefficient of light gases based on quantum mechanics:

(C-14)

where

and

Λ* is the reduced de Broglie wavelength; it depends on Planck’s constant h, molecular mass m, and intermolecular potential parameters ε and σ.

Figure C-1 gives reduced experimental second virial coefficients of hydrogen and helium at very low temperatures. The parameters used for data reduction are:

For comparison, Fig. C-1 also shows reduced second virial coefficients calculated from the Lennard-Jones potential using classical statistical mechanics.

The results shown in Fig. C-1 are useful for estimating second virial cross-coefficients B₁₂ whenever component 1 or 2 (or both) is a light (quantum) gas. To estimate B₁₂ we use the customary combining rules

(C-15)

and

(C-16)

Figure C-1 Reduced second virial coefficients for helium and hydrogen at low temperatures.

and in addition we have

(C-17)

where

(C-18)

For details, see Hirschfelder et al. (1954) and Prausnitz and Myers (1963).

Experimental Results. Dymond and Smith (1980) give a compilation of experimental data for the virial coefficients of pure gases and mixtures. For hydrogen they found that different sets for the second virial coefficient, reported by various authors, all agree within estimated error limits. Table C-1 gives virial coefficients obtained from a smooth curve through all data. This table also gives third virial coefficients for hydrogen reported by Michels et al. (1959).

Table C-1 Second and third virial coefficients for hydrogen.

Goodwin and coworkers (1964) established empirical correlations for second and third virial coefficients of hydrogen over a wide temperature range. For the second virial coefficient, their results are given in the form

(C-19)

where

Equation (C-19) covers the temperature range 15 to 423 K and the mean deviation from experimental results is ±0.066 cm³ mol^-1.

The third virial coefficient (in cm⁶ mol^-2) is given by

(C-20)

Equation (C-20) covers the temperature range 24 to 423 K and the mean deviation from experimental results is ±17.4 cm⁶ mol^-2.

For helium, agreement between data of various authors for the second virial coefficient is good above 15 K. Below this temperature, there is a significant discrepancy between the results of different sources. Dymond and Smith recommend the virial coefficients given in Table C-2. Third virial coefficients are those reported by Keesom (1942).

Table C-2 Second and third virial coefficients for helium.

For high temperatures (300 to 1000 K), virial coefficients of helium have been calculated using results from molecular-beam experiments. Second and third virial coefficients, reported by Harrison (1964) are given by

(C-21)

and

(C-22)

where the units are the same as those in Table C-2.

There is good agreement between data of various authors for the second virial coefficient of neon. Dymond and Smith (1980) obtained the virial coefficients given in Table C-3 from a smooth curve drawn through all data. Third virial coefficients are those reported by Crommelin et al. (1919) and by Michels et al. (1960).

Table C-3 Second and third virial coefficients for neon.

For very low temperatures, estimates of B for neon can be made using Fig. C-1 with parameters ε/k = 34.9 K, b₀ = 27.1 cm³ mol^-1, and Λ* = 0.593.

References

Crommelin, C. A., J. P. Martinez, and H. Kammerlingh Onnes, 1919, Commun. Phys. Lab. Univ. Leiden, 151a.

Dymond, J. H. and E. B. Smith, 1980, The Virial Coefficients of Pure Gases and Mixtures. Oxford: Clarendon Press.

Goodwin, R. D., D. E. Diller, H. M. Roder, and L. A. Weber, 1964, J. Res. Natl. Bur. Std., 68A: 121.

Harrison, E. F., 1964, AIAA J., 2: 1854.

Hirschfelder, J. O., C. F. Curtiss, and R. B. Bird, 1954, Molecular Theory of Gases and Liquids. New York: John Wiley & Sons.

Keesom, W. H., 1942, Helium. Amsterdam: Elsevier.

Lichtenthaler, R. N. and K. Strein, 1971, Ber. Bunsenges. Phys. Chem., 75: 489.

Michels, A., T. Wassenaar, and P. Louwerse, 1960, Physica, 26: 539.

Michels, A., W. de Graaff, T. Wassenaar, J. M. H. Levelt, and P. Louwerse, 1959, Physica, 25: 25.

Prausnitz, J. M. and A. L. Myers, 1963, AIChE J., 9: 5.

Scott, R. L. and R. D. Dunlap, 1962, J. Phys. Chem., 66: 639.