Virial expansion

The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the number density. The virial expansion, introduced in 1901 by Heike Kamerlingh Onnes, is a generalization of the ideal gas law. He wrote that for a gas containing $N$ atoms or molecules,

{\frac {p}{k_{B}T}}=\rho +B_{2}(T)\rho ^{2}+B_{3}(T)\rho ^{3}+\cdots ,

where $p$ is the pressure, $k_{B}$ is the Boltzmann constant, $T$ is the absolute temperature, and $\rho \equiv N/V$ is the number density of the gas. Note that for a gas containing a fraction $n$ of $N_{A}$ (Avogadro's number) molecules, truncation of the virial expansion after the first term leads to $pV=nN_{A}k_{B}T=nRT$ , which is the ideal gas law.

Writing $\beta =(k_{B}T)^{-1}$ , the virial expansion can be written as

{\frac {\beta p}{\rho }}=1+\sum _{i=1}^{\infty }B_{i+1}(T)\rho ^{i}

.

The virial coefficients $B_{i}(T)$ are characteristic of the interactions between the particles in the system and in general depend on the temperature $T$ . Virial expansion can also be applied to aqueous ionic solutions, as shown by Harold Friedman.

Comparison with Van der Waals equation

The Van der Waals equation can be used to derive the approximation $B_{2}(T)\approx {\frac {b}{N_{A}}}-{\frac {a}{N_{A}^{2}k_{\rm {B}}T}}$ with the Van der Waals constants a and b.

Derivation of the approximation

Starting with

\left(p+{\frac {n^{2}a}{V^{2}}}\right)\left(V-nb\right)=nRT

;^[1] where

V

is the volume and

n

is the amount of substance of the gas (in moles).

Using

n={\frac {N}{N_{A}}}

and

R=k_{\rm {B}}N_{\rm {A}}

, the equation can be written in the form:

\left(p+{\frac {N^{2}a}{N_{A}^{2}V^{2}}}\right)\left(V-{\frac {bN}{N_{A}}}\right)=Nk_{\rm {B}}T

Hence

{\frac {p+{\frac {N^{2}a}{N_{A}^{2}V^{2}}}}{k_{\rm {B}}T}}={\frac {N}{V-{\frac {bN}{N_{A}}}}}={\frac {\frac {N}{V}}{1-{\frac {bN}{VN_{A}}}}}

and with the number density

\rho ={\frac {N}{V}}

one can write:

{\frac {p+{\frac {\rho ^{2}a}{N_{A}^{2}}}}{k_{\rm {B}}T}}={\frac {\rho }{1-{\frac {b\rho }{N_{A}}}}}

and therefore

{\frac {p}{k_{\rm {B}}T}}={\frac {\rho }{1-{\frac {b\rho }{N_{A}}}}}-{\frac {\rho ^{2}a}{N_{A}^{2}k_{\rm {B}}T}}

And when

{\frac {b\rho }{N_{A}}}\ll 1

one can approximate

{\frac {\rho }{1-{\frac {b\rho }{N_{A}}}}}\approx \rho \left(1+{\frac {b\rho }{N_{A}}}\right)=\rho +{\frac {b\rho ^{2}}{N_{A}}}

Hence:

{\frac {p}{k_{\rm {B}}T}}=\rho +\rho ^{2}\cdot B_{2}(T)

with

B_{2}(T)\approx {\frac {b}{N_{A}}}-{\frac {a}{N_{A}^{2}k_{\rm {B}}T}}

And when $T_{B}={\frac {a}{b\cdot k_{\rm {B}}N_{A}}}={\frac {a}{bR}}$ then $B_{2}(T_{B})\approx 0$ , see Boyle temperature.

According to Van der Waals constants (data page) the constants for hydrogen gas are for example a = 0.2476 L²bar/mol² and b = 0.02661 L/mol and therefore the estimation of the Boyle temperature for hydrogen is $T_{B}={\frac {0.2476\cdot 10^{-1}{\rm {\frac {m^{6}Pa}{mol^{2}}}}}{0.02661\cdot 10^{-3}{\rm {{\frac {m^{3}}{mol}}\cdot 8.3145{\rm {\frac {J}{mol\cdot K}}}}}}}=112\ {\rm {K}}$ . (The real value for hydrogen is 110 K.^[2] In nitrogen the difference is bigger.)

Notes and references

^ Chang, Raymond (2014). Physical Chemistry for the Chemical Sciences. University Science Books. p. 14. ISBN 978-1-891389-69-6.
^ Real Gases and the Virial Equation Table 1.2

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[1] Chang, Raymond (2014). Physical Chemistry for the Chemical Sciences. University Science Books. p. 14. ISBN 978-1-891389-69-6.

[2] Real Gases and the Virial Equation Table 1.2

[1]

[2]

Comparison with Van der Waals equation

See also

Notes and references