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) |
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