Hello

From the expression of the partition function of a fermi dirac ideal gas

ln(Z)=αN + ∑ ln(1+exp(-α-βE

show that

S= k ∑ [ n

**1. The problem statement, all variables and given/known data**From the expression of the partition function of a fermi dirac ideal gas

ln(Z)=αN + ∑ ln(1+exp(-α-βE

_{r}))show that

S= k ∑ [ n

_{r}ln(n_{r})+(1-<n_{r}>)ln(1-<n_{r}>)
**2. Relevant equations**

S=k( lnZ+β<E>)

<nr>=-1/β ∂ln(Z)/∂E_{r}

<E>=-∂ln(Z)/∂β

**3. The attempt at a solution**

I tried to start with

S=k( lnZ+β<E>)

But I don’t know how we get to introduce the nr in that ?

Thanks

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