Answer to Question #210982 in Molecular Physics | Thermodynamics for Mrityunjoy Kundu

Question

derive the formula for compressiblility of an ideal Fermi gas when the degeneracy is strong. Calculate the compressibility of Na crystal which

has one free electron per atom. (atomic weight of Na is 23, and density is 0.97 g/cm3)

Answer 1

$p=-(\frac{\partial U}{\partial V})_{N,S}=-\frac 35 N(\frac{\partial e_F}{\partial V})_{N,S}=-\frac 35 N (-\frac 23\frac{e_F}{V})=\frac 23 \frac UV,\implies$

$U=\frac 32 pV,$

$B=-V (\frac{\partial p}{\partial V})_{T},$

$p=\frac 23 \frac UV\sim V^{-\frac 53},\implies$

$(\frac{\partial p}{\partial V})_{T}=-\frac 53 \frac pV=-\frac{10}{9}\frac{U}{V^2},\implies$

$B=\frac{10}{9}\frac UV.$