Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 15.3-1

THE CANONICAL FORMALISM; STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION

Physics — Statistical Mechanics · two-state-model-gruneisen

Problem

In the two-state model system of this section suppose the excited state energy

\varepsilon

of an ``atom'' depends on its average distance from its neighboring atoms so that

\varepsilon = \frac{a}{\bar{v}^\gamma}, \qquad \bar{v} \equiv \frac{V}{\tilde{N}}

where

a

and

\gamma

are positive constants. This assumption, applied to a somewhat more sophisticated model of a solid, was introduced by Gruneisen, and

\gamma

is the ``Gruneisen parameter.'' Calculate the pressure

P

as a function of

\bar{v}

and

T

. Answer:

P = \frac{\gamma a}{\bar{v}^{\gamma+1}}\left(e^{a/k_BT\bar{v}^\gamma} + 1\right)^{-1}