# Entropy of Ideal Mixture Proof

1. Sep 16, 2013

### superspartan9

1. The problem statement, all variables and given/known data

Explain why, for an ideal mixture, the mixing entropy is given by
ΔSmixing = k ln( Binomial Coefficient ( N, NA )
where N is the total number of molecules and NA is the number of molecules of type A. Use Stirling's Approximation to show that this expression is the same as the result from the previous problem when both N and NA are large.

2. Relevant equations

ΔS = Nk ln(Vf/Vi)
Previous Solution: ΔSmixing = -Nk[ x ln (x) + (1-x) ln (1-x) ]
Stirling Approx: N! ≈ NNe-N√(2∏N)

3. The attempt at a solution
I originally thought it might mean mathematically derive, but it looks like that is more related to the second part of the problem. I have no idea how to get from the equations they give me to the binomial coefficient part of the solution. I believe I can do the second part if someone can provide some assistance to the explanation.

2. Sep 17, 2013

### BruceW

for the first part, yes, I think you are just meant to explain. (not do anything mathematical). So, you've used the equation S = k ln(something) before, I'm guessing. What is that 'something' ? and why does it make sense that in this case, that 'something' is Binomial coefficient (N,NA) ?