The expression for the entropy an chemical potential for a ideal gas.

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Answer 1

Question

Chapter 6.7, Problem 45P

To determine

The expression for the entropy an chemical potential for a ideal gas.

Expert Solution & Answer

Explanation of Solution

Given:

The equilibrium temperature of an ideal gas is T .

Formula used:

Write the expression for the Helmholtz free energy of an ideal gas.

F=−NkT[lnV−lnN−lnvQ+1]+Fint .......... (1)

Here, F is the Helmholtz free energy, V is the total volume of the ideal gas, N is the total number of molecules, k is the Boltzmann constant, T is the equilibrium temperature, vQ is the quantum volume of the gas and Fint is the internal free energy of the system.

Write the expression for quantum volume of an ideal gas.

vQ=( h 2 2πmkT)3/2 .......... (2)

Here, h is the Plank’s constant.

Write the expression for entropy of the system.

S=−( ∂F ∂T)N,V .......... (3)

Write the expression for the chemical potential of a system.

μ=( ∂F ∂N)T,V .......... (4)

Write the expression for internal energy of an ideal gas.

F = −NkT lnZ

Here, μ is the chemical potential of the system.

Calculation:

Apply logarithm to the both side of equation (2).

lnvQ=ln( h 2 2πmkT)3/2=32[−ln(T)+ln( h 2 2πmkT)]

Substitute (32[−ln(T)+ln( h 2 2πmkT)]) for lnvQ in equation (1).

F=−NkT[lnV−lnN−32[−ln(T)+ln( h 2 2πmkT )]+1]+FintF=−NKT[lnV−lnN+32ln(T)−32ln( h 2 2πmkT)+1]+Fint

Substitute (−NKT[lnV−lnN+32ln(T)−32ln( h 2 2πmkT)+1]+Fint) for F in equation (3).

S=∂∂T(NKT[lnV−lnN+ 3 2ln( T)− 3 2ln( h 2 2πmkT )+1]+F int)=NK[lnV−lnN+32ln(T)−32ln( h 2 2πmkT)+1]−∂F int∂T+32Nk

Substitute (ln vQ) for (32ln(T)−32ln( h 2 2πmkT)) in the above equation.

S=NK[lnV−lnN+ln vQ+52]−∂F int∂TS=Nk[ln( V N v Q )+52]−∂F int∂T

Substitute (−NkT[lnV−lnN−lnvQ+1]+Fint) for F in equation (4).

μ=∂∂N(−NkT[lnV−lnN−ln v Q+1]+F int)μ=−kT[lnV−lnN−lnvQ+1]+∂F int∂Nμ=−kT ln(V N v Q )+∂F int∂N

Substitute ( −NkT lnZ) for Fint in the above equation.

μ=−kT ln(V N v Q )+∂∂N( −NkT lnZ)μ=−kT ln( VZ N v Q )

Conclusion:

Thus, the entropy for an ideal gas is Nk[ln(VN v Q)+52]−∂Fint∂T and the chemical potential of the system is −kT ln(VZNvQ)

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