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

1. Apr 4, 2017

### pallab

1. The problem statement, all variables and given/known data
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

2. Relevant equations
∂f/∂n=μ
pV=NkT
p=NkT/V
3. 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

2. Apr 5, 2017

### 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 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!)).

3. Apr 18, 2017

thank you.