Average Energy of Grand Canonical Ensemble

AI Thread Summary
The discussion focuses on the calculation of average energy in the Grand Canonical Ensemble (GCE) and the derivation of related thermodynamic equations. The entropy and Helmholtz function are defined, leading to a relationship involving the partition function ZG. A discrepancy arises when the derived average energy expression includes an unwanted term, μoN. Participants are exploring the source of this error in the calculations and seeking clarification on the correct formulation. The conversation emphasizes the need for accuracy in thermodynamic relationships within the GCE framework.
phys_student1
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Hello,

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

S = KB ln ZG + (\bar{E}/T) - μo\bar{N}/T

Helmholtz function is:

F = \bar{E} - TS = \bar{E} - TKB ln ZG - \bar{E} + μo\bar{N}
= -TKB ln ZG + μo\bar{N}

But

\partialF/\partialT = -S (From thermodynamics).

Then,

-TKB \partialln ZG/\partialT - KBln ZG = -kB ln ZG - \bar{E}/T + μo\bar{N}/T

This gives:

\bar{E} = kBT2 \partialln ZG/\partialT + μo\bar{N}

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