Virial decomposition

virial series

The series on the right-hand side of the equation of state of a gas:

$$ \frac{Pv }{kT } = \ 1 + \sum _ {1 \leq i \leq \infty } \frac{B _ {i + 1 } ( T) }{v ^ {i} } , $$

where $ P $ is the pressure, $ T $ is the temperature, $ v $ is the specific volume, and $ k $ is the Boltzmann constant. The term of the series which contains the $ k $- th virial coefficient $ B _ {k} $ describes the deviation of the gas from ideal behaviour due to the interaction in groups of $ k $ molecules. $ B _ {k} $ can be expressed in terms of irreducible repeated integrals $ b _ {k} $:

$$ B _ {k} = { \frac{k - 1 }{k} } \sum \frac{( k - 2 + \sum n _ {j} ) ! }{( k - 1) ! } (- 1) ^ {\sum n _ {j} } \prod _ { j } \frac{( jb _ {j} ) ^ {n _ {j} } }{n _ {j} ! } , $$

summed over all natural numbers $ n _ {j} $, $ j \geq 2 $, subject to the condition

$$ \sum _ {2 \leq j \leq k } ( j - 1) n _ {j} = k - 1. $$

In particular,

$$ B _ {2} = - b _ {2} , \ B _ {3} = 4b _ {2} ^ {2} - 2b _ {3} ; $$

$$ b _ {2} = \frac{1}{2 ! V } \int\limits \int\limits f _ {12} d ^ {3} q _ {1} d ^ {3} q _ {2} , $$

$$ b _ {3} = \frac{1}{3 ! V } \times $$

$$ \times \int\limits \int\limits \int\limits ( f _ {31} f _ {21} + f _ {32} f _ {31} + f _ {32} f _ {21} + f _ {21} f _ {32} f _ {31} ) \ $$

$$ d ^ {3} q _ {1} d ^ {3} q _ {2} d ^ {3} q _ {3} , $$

where

$$ f _ {ij} = \mathop{\rm exp} \left [ - \frac{\Phi ( | q _ {i} - q _ {j} | ) }{kT } \right ] - 1, $$

$ V $ is the volume of the gas, the integration extends over the total volume occupied by the gas, and $ \Phi $ is the interaction potential. There is a rule for writing down $ b _ {j} $ for any $ j $ in terms of $ f _ {ij} $. The expression obtained after simplification is:

$$ B _ {3} = - { \frac{1}{3} } \int\limits \int\limits f _ {12} f _ {13} f _ {23} d ^ {3} q _ {1} d ^ {3} q _ {2} . $$

In practice, only the first few virial coefficients can be calculated.

Power series in $ v ^ {-} 1 $, with coefficients expressed in terms of $ b _ {j} $, can be used to represent equilibrium correlation functions for $ s $ particles; a corollary of this fact is that the equation of state can be obtained in a simple manner [3].

There exists a quantum-mechanical analogue of the virial decomposition.

References

[1]	J.E. Mayer, M. Goeppert-Mayer, "Statistical mechanics" , Wiley (1940)
[2]	R. Feynman, "Statistical mechanics" , M.I.T. (1972)
[3]	N.N. Bogolyubov, "Problems of a dynamical theory in statistical physics" , North-Holland (1962) (Translated from Russian)
[4]	G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963)