Virial equation of state

The virial equation of state is a mathematical expression that models the behaviour of a real gas.

Recognising van der Waals’ work, Heike Kamerlingh, another Dutch physicist, attempted to establish a more extensive equation of state and devised the virial equation of state in 1882:

pV_m=RT\left ( 1+\frac{B}{V_m}+\frac{C}{V_m^{\: 2}}+... \right )\; \; \; \; \; \; \; \; 16

The word virial is latin meaning force or energy. The RHS of eq16 suggests that the virial equation is derived using a power series.

In fact, the equation can be perceived as a Maclaurin series of van der Waals’ equation. To illustrate this, let’s multiply the right side of eq10 from the previous section by \frac{x}{x} where x=\frac{1}{V_m} to give:

p=\frac{RTx}{1-bx}-ax^2\; \; \; \; \; \; \; \; 17

A Maclaurin series for the polynomial p(x)=c_0+c_1x+c_2x^2+c_3x^3+... is in the form:

p(x)=p(0)+\frac{p'(0)}{1!}x+\frac{p''(0)}{2!}x+\frac{p'''(0)}{3!}x+...\; \; \; \; \; \; \; \; 18

From eq17,

p(0)=0\; \; \; \; \; \; 19

p'(0)=\frac{RT(1-bx)+bRTx}{(1-bx)^2}-2ax=RT\; \; \; \; \; \; \; 20

p''(0)=\frac{2b\left [ RT(1-bx)+bRTx \right ](1-bx)}{(1-bx)^4}-2a=2(bRT-a)\; \; \; \; \; \; \; 21

p'''(0)=\frac{6b^2RT}{(1-bx)^4}=6b^2RT\; \; \; \; \; \; \; 22

Substituting eq19 through eq22 in eq18

p(x)=RTx\left [ 1+\left ( b-\frac{a}{RT} \right )x+b^2x^2+... \right ]\; \; \; \; \; \; \; \; 23

Substitute x=\frac{1}{V_m} back in eq23

p(x)=\frac{RT}{V_m}\left [ 1+\frac{\left ( b-\frac{a}{RT} \right )}{V_m}+\frac{b^2}{V_m^{\: 2}}+... \right ]\; \; \; \; \; \; \; \; 24

Eq24 is the same as eq16 if B=\left ( b-\frac{a}{RT} \right ) and C = b2. If so, the coefficient C (and perhaps other higher coefficients) is not a function of temperature, which limits the accuracy of the equation in predicting the pressure of the real gas. Kamerlingh therefore decided to use experimental data to determine the coefficients B, C and so on, instead of using the van der Waals factors, resulting in eq16 as the final form rather than eq24.

Comparing eq16 and eq2, the compression factor is equal to the expansion part of the virial equation

Z=1+\frac{B}{V_m}+\frac{C}{V_m^{\: 2}}+...

If all the virial coefficients are equal to zero, Z = 1 and the virial equation becomes the ideal gas equation. For large Vm (i.e. at low pressures), all the terms with virial coefficients tend to zero with the virial equation transforming into the ideal gas equation again. The virial equation can also be expressed in terms of powers of pressure:

pV_m=RT\left ( 1+B'p+C'p^2+... \right )\; \; \; \; \; \; \; \; 25

The pressure version of the virial equation is constructed in a way that is consistent with the volume version, such that the equation becomes the ideal gas equation when all the virial coefficients are zero.

In summary, the virial equation’s accuracy in modeling the behaviour of real gases increases with increasing number of expansion terms. However, it may be cumbersome to determine the virial coefficients that are needed to make the equation work.

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