Correct option for n dependence of free energy f per unit

AI Thread Summary
The discussion centers on determining the n dependence of the free energy per unit volume for an ideal gas, with various expressions presented as potential solutions. The equation of state for an ideal gas is given as p = nkT, where n represents the number of particles per unit volume. Key equations include the relationship between free energy and chemical potential, and the use of the partition function to derive Helmholtz free energy. The Stirling Approximation is suggested for simplifying calculations involving factorial terms. The conversation emphasizes the importance of understanding thermodynamic principles and partition functions in solving the problem.
pallab
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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
 
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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!)).
 
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Yosty22 said:
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.
 
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