Correct option for n dependence of free energy f per unit

[pallab]

Homework Statement

The equation of state of an ideal gas is p = nkT, where p is the thermodynamic
pressure and n = N / V is the thermodynamic variable for the number of particles per
unit volume. The n dependence of the free energy f per unit volume of the ideal gas is
obtained by the following expression , where c is temperature-dependent constant k is Boltzmann constant.
(a) nkT[In(n)+c]
(b) 2nkT[n ln(n)+c.]
(c) 3/2 nkT
(d) 3nkT

Homework Equations

∂f/∂n=μ
pV=NkT
p=NkT/V

The Attempt at a Solution

internal energy U=U(S,V,N)
∴μ=∂U/∂N
and ∂μ/∂V=∂²U/∂N∂V= -∂p/∂N
∂μ/∂V=-kT/V
∴μ=-kTlnV+c

 

[Yosty22]

If you know these are free particles (i.e. potential energy term in the Hamiltonian is 0) then the best start would be to calculate the N particle partition function, it's usually called Z or Q_N. Once you have the partition function, the Helmholtz free energy is given by: A(N,T,V) = -kTln(Z). You can then use the laws of logarithms as well as the Stirling Approximation (to estimate the term ln(N!)).

 

[pallab]

If you know these are free particles (i.e. potential energy term in the Hamiltonian is 0) then the best start would be to calculate the N particle partition function, it's usually called Z or Q_N. Once you have the partition function, the Helmholtz free energy is given by: A(N,T,V) = -kTln(Z). You can then use the laws of logarithms as well as the Stirling Approximation (to estimate the term ln(N!)).

thank you.