Three Energies System

For a system with 3 possible energies $0$, $ε$, and $2ε$ having the $\epsilon$ level double degenerate, please find the partition function, the average energy, its entropy and specific heat.

There are possible 4 states, one with energy 0, two with energies $ε$ and one state with energy $2ε$ . The probability of one state having the energy $ε$ is $=1/Z*exp⁡(-ε/k T)$ . Therefore the partition function Z is found from the condition that the sum of all probabilities is 1:

$Z=∑_i exp⁡(-ε_i/k T)=∑_i exp⁡(-β*ε_i)$

$Z=exp⁡(-0/k T)+exp⁡(-ε/k T)+exp⁡(-ε/k T)+exp⁡(-2ε/k T)=$

$=1+2*exp⁡(-ε/k T)+exp⁡(-2ε/k T)$

 $Z=(1+exp⁡(-ε/k T) )^2=[1+exp⁡(-βε) ]^2$

The value of the average total energy is:

$E =<ε>=∑_i ε_i*P_i= (∑_i ε_i*exp⁡(-βε_i ))/Z=-(1/Z) ∑_i d/dβ*exp⁡(-βε_i )=$

$=-1/Z*d/dβ*∑_iexp⁡(βε_i )=-(1/Z)*d Z/dβ=-d(ln⁡Z )/dβ=k T^2*(d(ln⁡ Z))/d T$

For entropy one has

$S=-k*∑_i P_i*ln⁡(P_i )=-k*∑_i P_i*ln⁡(1/Z)+k*∑_i P_i*ε_i/k T=k*ln⁡Z+1/T*<ε>$

$S=k*ln⁡Z+E/T=k*ln⁡ (Z)+k E/β$

The heat capacity is found from the equation:

$d E=d<U>=C*d T$ so that $C=d E/d T$

The results are


$S=2k*ln⁡[ 1+exp⁡(-βε) ]-2k/β*(exp⁡(1-βε))/(1+exp⁡(-βε))$

$C=(2*exp⁡(2-2ε/k T))/(k T^2*[exp⁡(-ε/k T)+1]^2 )-$

$-(2*exp⁡(2-2ε/k T))/(k T^2*[exp⁡(-ε/k T)+1] )=$

$=(2*e^{2-2βε})/(βT*(e^{-βε}+1)^2 )-(2*e^{2-2βε})/(βT*(e^{-βε}+1))$

The computations of the derivatives have been done online.