The Virial Expansion

The virial expansion, also called the virial equation of state, is the most interesting and versatile of the equations of state for gases.. The virial expansion is a power series in powers of the variable, n/V, and has the form,

(1)            .

The coefficient, B(T), is a function of temperature and is called the "second virial coefficient. C(T) is called the third virial coefficient, and so on. The expansion is, in principle, an infinite series, and as such should be valid for all isotropic substances. In practice, however, terms above the third virial coefficient are rarely used in chemical thermodynamics.

Notice that we have set the quantity pV/nRT equal to Z. This quantity (Z) is called the "compression factor." It is a useful measure of the deviation of a real gas from an ideal gas. For an ideal gas the compression factor is equal to 1.

The Boyle Temperature

The second virial coefficient, B(T), is an increasing function of temperature throughout most of the useful temperature range. (It does decrease slightly at very high temperatures.) B is negative at low temperatures, passes through zero at the so-called "Boyle temperature," and then becomes positive. The temperature at which B(T) = 0 is called the Boyle temperature because the gas obeys Boyle's law to high accuracy at this temperature. We can see this by noting that at the Boyle temperature the virial expansion looks like,

(2)             .

If the density is not too high the C term is very small so that the system obeys Boyle's law.

Alternate form of the virial expansion.

An equivalent form of the virial expansion is an infinite series in powers of the pressure.

(3)             .

The new virial coefficients, B', C', . . . , can be calculated from the original virial coeffients, B, C, . . . . To do this we equate the two virial expansions,

(4)             .

Then we solve the original virial expansion for p,

(5)             ,

and substitute this expression for p into the right-hand-side of equation (4),

(6a)

(6b)

Both sides of Equation (6b) are power series in n/V. (We have omitted third and higher powers of n/Vbecause the second power is as high as we are going here.) Since the two power series must be equal, the coefficients of each power of n/V must be the same on both sides. The coefficient of (n/V)0 on each side is 1, which gives the reassuring but not very interesting result, 1 = 1. Equating the coefficient of (n/V) 1 on each side gives B = B'RT and equating the coefficients of (n/V)2 gives

(7)            .

These equations are easily solved to give B' and C' in terms of B, C, and R.

(8)             .

Useful exercises would be:

1. Extend the two virial expansions to the D and D' terms respectively and find the expression for D' in terms of B, C, and D.

2. Find B' and C' in terms of the van der Waals a and b constants. (You were asked, in the homework to find the virial coefficients B and C in terms of a and b so you already have these.)

The word "virial" is related to the Latin word for force. Clausius (whose name we will see frequently) named a certain function of the force between molecules "the virial of force." This name was subsequently taken over for the virial expansion because the terms in that expansion can be calculated from the forces between the molecules.

The virial expansion is important for several reasons, among them: It can, in principle, be made as accurate as desired by keeping more terms. Also, it has a sound theoretical basis. The virial coefficients can be calculated from a theoretical model of the intermolecular potential energy of the gas molecules

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