Why Does the Entropy Formula for a Fermionic Gas Include a Degeneracy Factor?

[Kara386]

Homework Statement

The question asked me to show that the entropy of a fermionic gas is
##S = -k_B \Sigma_i (1-f_i)\ln(1-f_i) +f_i\ln(f_i)##

Using the Fermi-Dirac distribution so ##f_i = \frac{n_i}{g_i}##.

Homework Equations

The Attempt at a Solution

The number of microstates ##\Omega## is
##\Omega =## Π##\frac{g_i!}{n_i!(g_i-n_i)!}## and using ##S = k_B ln(\Omega)## I've arrived at the expression:
##S = -k_B \Sigma_i g_i (1-f_i)\ln(1-f_i) +f_i\ln(f_i)##
So there's an unwanted factor of ##g_i##. I'm told it is meant to be there, and I need to explain why it can be set to 1. Something to do with i being the state index and a quantum state can't be degenerate. Could someone explain that to me? Any help is much appreciated! :)

 

[Oxvillian]

Hi Kara386

First observation here is that you need some big brackets in your equations - for example in your first equation, the [itex]k_B[/itex] should be multiplying everything, not just the first term.

The business with the [itex]g_i[/itex] is just bookkeeping. In their formula they're using the index [itex]i[/itex] to label different states, and in your formula you're using the index [itex]i[/itex] to label different energies. Your formula includes a [itex]g_i[/itex] factor in it to allow for the possibility that there might be lots of states with the same energy.

 

[Kara386]

Hi Kara386

First observation here is that you need some big brackets in your equations - for example in your first equation, the [itex]k_B[/itex] should be multiplying everything, not just the first term.

The business with the [itex]g_i[/itex] is just bookkeeping. In their formula they're using the index [itex]i[/itex] to label different states, and in your formula you're using the index [itex]i[/itex] to label different energies. Your formula includes a [itex]g_i[/itex] factor in it to allow for the possibility that there might be lots of states with the same energy.

Thanks! I thought that might be it, because although originally the sum was over multiple states taken to be at the same energy, I think the introduction of the average ##f_i## allows the sum to be over individual states.

Thanks for your help!