An equation for chemical potential from the Boltzmann entropy equation

Question

I am trying to derive an equation for chemical potential from Boltzmann entropy and this is what I have come up with so far:

$$S = kln{left(frac{N!}{N_mathrm{up}!,N_mathrm{down}!}right)} label{eq:1} tag{1}$$

and the derivation of chemical potential from the 1st law of thermodynamics,

$$mu = -Tleft(frac{partial S}{partial N}right)_{UV} tag{2}$$

I have used Stirling's approximation and properties of logs and have come up with an answer, but after searching for my answer it is not online anywhere, so it feels like I might be going in the wrong direction.

From my computation of entropy which I found to be,

$$S = k_bleft( Ncdotln{N} - sum(N_icdotln{N_i}) right) tag{3}$$

I got chemical potential to be

$$mu = -Tk_bleft( ln{N} +1 - sum{(ln{N_i} + 1)} right) tag{4}$$

Is this a derivation for entropy that anyone has ever seen before? Is it even possible to take the equation $eqref{eq:1}$ and derive an equation for chemical potential?

Any answer or push in the right direction would be greatly appreciated.

Themis · Answer

Start with 
$$
   S(N_1,N_2) = k logfrac{(N_1+N_2)!}{N_1! N_2!} 
$$
Write the chemical potential as
begin{align}
   mu &= - kTleft(frac{partial S}{partial N_1}right)_{N_2}
       &= -kT frac{S(N_1+1,N_2)-S(N_1,N_2)}{(N_1+1)-N_1} 
       & = -kT log 
      frac{(N_1+N_2+1)!}{(N_1+N_2)!}
      frac{N_1!}{(N_1+1)!}
      frac{N_2!}{N_2!}
   & = -kT frac{N_1+N_2+1}{N_1+1}
end{align}
Make the approximation
$$
   frac{N_1+N_2+1}{N_1+1} approx frac{N_1+N_2}{N_1} = frac{1}{x_1}
$$
and write the final result as
$$
boxed{
vphantomint
   mu = kT log x_i
}
$$
with 
$$
   x_i = frac{N_i}{N} = frac{N_i}{sum_i N_i}
$$

An equation for chemical potential from the Boltzmann entropy equation

One Answer

Add your own answers!

Ask a Question