- #1
phys_student1
- 106
- 0
Hello,
The entropy of the Grand Canonical Ensemble (GCE) is:
S = KB ln ZG + ([itex]\bar{E}[/itex]/T) - μo[itex]\bar{N}[/itex]/T
Helmholtz function is:
F = [itex]\bar{E}[/itex] - TS = [itex]\bar{E}[/itex] - TKB ln ZG - [itex]\bar{E}[/itex] + μo[itex]\bar{N}[/itex]
= -TKB ln ZG + μo[itex]\bar{N}[/itex]
But
[itex]\partial[/itex]F/[itex]\partial[/itex]T = -S (From thermodynamics).
Then,
-TKB [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T - KBln ZG = -kB ln ZG - [itex]\bar{E}[/itex]/T + μo[itex]\bar{N}[/itex]/T
This gives:
[itex]\bar{E}[/itex] = kBT2 [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T + μo[itex]\bar{N}[/itex]
This is not the correct answer. The correct answer does not have the μo[itex]\bar{N}[/itex] term, what's wrong ?
The entropy of the Grand Canonical Ensemble (GCE) is:
S = KB ln ZG + ([itex]\bar{E}[/itex]/T) - μo[itex]\bar{N}[/itex]/T
Helmholtz function is:
F = [itex]\bar{E}[/itex] - TS = [itex]\bar{E}[/itex] - TKB ln ZG - [itex]\bar{E}[/itex] + μo[itex]\bar{N}[/itex]
= -TKB ln ZG + μo[itex]\bar{N}[/itex]
But
[itex]\partial[/itex]F/[itex]\partial[/itex]T = -S (From thermodynamics).
Then,
-TKB [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T - KBln ZG = -kB ln ZG - [itex]\bar{E}[/itex]/T + μo[itex]\bar{N}[/itex]/T
This gives:
[itex]\bar{E}[/itex] = kBT2 [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T + μo[itex]\bar{N}[/itex]
This is not the correct answer. The correct answer does not have the μo[itex]\bar{N}[/itex] term, what's wrong ?