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## Homework Statement

For a single molecule, derive the internal energy U = 3/2k

_{B}T

In terms of the partition function Z, F = -k

_{B}TlnZ

Where Z = V(aT)

^{3/2}

## Homework Equations

Thermodynamic identity: δF = -SδT - pδV

p = k

_{B}T/V

S = k

_{B}[ln(Z) + 3/2]

## The Attempt at a Solution

U = F + TS

δU = δF + δT.S + δS.T

= -SδT - pδV + δT.S + δS.T

δU = -pδV + TδS

(δS/δU)

_{V}= 1/T

However, don't have a variable U in S to differentiate with respect to.