Statistical Theory of the Dielectric Constant of an Imperfect Gas

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Abstract
By means of the linked‐diagram expansion of the grand partition function of a molecular gas in an electrostatic field, an expression for the polarization P(R) of the gas is obtained. Spatial variation of the external electric field E0(R) requires an explicit treatment of long‐range cooperative interactions between ``clusters'' of molecules. For fields that vary appreciably over microscopic dimensions, an integral relation is found relating the polarization P(R) to the electric field E(R). For fields varying negligibly over microscopic regions, an expression for the dielectric constant K of the gas is obtained: (K − 1)/(K + 2) = (4/3) π Σ∞m=1 nmαm(θ). This generalization of the Clausius‐Mossotti formula involves the density n and the temperature‐dependent polarizabilities of m‐molecule linked clusters: α1(θ) is the effective polarizability of a single (possibly polar) molecule; α2(θ) is the diagonal element of the scalar tensor α2(θ) = ∫ d3R exp [−βφ(θ, R)][½α2(θ, R) − α1(θ)]. In this expression α2(θ, R) is the effective polarizability tensor of two molecules with fixed positions of their centers of mass; φ is the free energy of the two‐molecule system, relative to infinite separation.