Average Energy of Grand Canonical Ensemble

[phys_student1]

Hello,

The entropy of the Grand Canonical Ensemble (GCE) is:

S = K_B ln Z_G + ($\bar{E}$/T) - μ_o$\bar{N}$/T

Helmholtz function is:

F = $\bar{E}$ - TS = $\bar{E}$ - TK_B ln Z_G - $\bar{E}$ + μ_o$\bar{N}$
= -TK_B ln Z_G + μ_o$\bar{N}$

But

$\partial$F/$\partial$T = -S (From thermodynamics).

Then,

-TK_B $\partial$ln Z_G/$\partial$T - K_Bln Z_G = -k_B ln Z_G - $\bar{E}$/T + μ_o$\bar{N}$/T

This gives:

$\bar{E}$ = k_BT² $\partial$ln Z_G/$\partial$T + μ_o$\bar{N}$

This is not the correct answer. The correct answer does not have the μ_o$\bar{N}$ term, what's wrong ?

 

[phys_student1]

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