# Ideal dielectric gas in a capacitor

## Homework Statement

Ideal dielectric gas is in a container closed by a movable piston and in thermal contact with its surroundings, so is kept at constant tempertaure $$T_0$$ and pressure $$p_0$$. Inside there is a capacitor with fixed voltage and total electric field $$E$$. The gas has permittivity $$\epsilon(n,T) = 1 + n\alpha(T)$$, where $$n$$ is the density of gas in the capacitor and $$\alpha$$ is some function of temperature.

Find equilibrium value of $$n$$.

## Homework Equations

Capacitor energy $$U = \frac{1}{2}\epsilon E^2 V_{cap}$$

## The Attempt at a Solution

With constant pressure and temperature, the quantity minimized at equilibrium is Gibbs free energy, so at equilibrium chemical potentials $$\mu_1$$ and $$\mu_2$$ of the gas inside and outside the capacitor must be equal. While we can find $$\mu_2$$ easily, since this is an ideal gas, I'm not sure about $$\mu_2$$.

## Answers and Replies

For the gas in the capacitor you should add to the Gibbs free energy a term describing the electric field:
$$dG' = dG + \frac{1}{4\pi}V_{\text{cap}}\vec{E}d\vec{D}$$
This additional term is proportional to dn and makes a contribution to the chemical potential.