Ideal dielectric gas in a capacitor

[neworder1]

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.

 

[Maxim Zh]

For the gas in the capacitor you should add to the Gibbs free energy a term describing the electric field:
<br /> dG&#039; = dG + \frac{1}{4\pi}V_{\text{cap}}\vec{E}d\vec{D}<br />
This additional term is proportional to dn and makes a contribution to the chemical potential.