# Correct option for n dependence of free energy f per unit

## 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

∂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

## Answers and Replies

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 QN. 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 QN. 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.