Question

Calculate the partition function, free energy, entropy, CV and Cp of N linear 

harmonic oscillators

Answer 1

We know that Maxwell distribution function

n_i= g_i$e^{-\alpha }e^{-\beta\epsilon}$

n_i=g_i$/e^{(\alpha+\beta\epsilon)}$

Total no of partical

N=$\Sigma$ n_i

N=$\Sigma$ g_i$e^{-\alpha}e^{-\beta\epsilon}$

N=A$\Sigma$g_i$e^{-\beta\epsilon}$

$\frac{N}{A}=\Sigma$ g_i$e^{\Sigma-\beta\epsilon}$ Where g_i degenracy

Z=$\Sigma e^{-\beta\epsilon}$

Where n D harmonic oscillator energy

$\epsilon$ =E=$(n+1/2)$

Free energy

F= -K_B$TlnZ$

F=-K_BT$\ln\frac{k(B)T}{hw}$

Entropy

S=K_B$\frac{d}{dT}(T\ln Z)$

C_p=C_v=$\frac{d}{dT}(\overline E)$ =K_B

$\overline E =KT^2\frac{d}{dT}\ln Z$