The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well. For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by

$\mathrm{\Delta }{S}_{\text{mixing}}=k\ln \left(\begin{array}{c}N\\ N_A\end{array}\right)$

where $N$is the total number of molecules and $N_A$is the number of molecules of type $A$. Use Stirling's approximation to show that this expression is the same as the result of the previous problem when both $N$and $N_A$are large.

Let's say we start with a set of $N$identical molecules. This system's entropy is a number $S_0$, which may or may not be straightforward to determine. Assume that we suddenly transform $N_A$of these molecules to a new species at some point in the future (which has similar properties to the original species as mentioned).The amount of different ways we may select to locate these $N_A$molecules among the $N$sites accessible will increase the entropy.

The entropy of mixing is:

$\mathrm{\Delta }{S}_{\text{mixing}}=k\ln \left(\begin{array}{c}N\\ N_A\end{array}\right)$

The Coefficient binomial is

$\begin{array}{c}\mathrm{\Delta }{S}_{\text{mixing}}=k\ln \left(\frac{N!}{N_A!\left(N-N_A\right)!}\right)\\ \end{array}$

where,$N_A=(1-x)N$

$\mathrm{\Delta }{S}_{\text{mixing}}=k\ln \left(\frac{N!}{\left(\right(1-x\left)N\right)!\left(xN\right)!}\right)$

By using Stirling's approximation$n!\approx \sqrt{2\pi n{n}^n}{e}^{-n}$,the factorial is

$\begin{array}{l}N!\approx \sqrt{2\pi N}{N}^N{e}^{-N}\\ xN!\approx \sqrt{2\pi xN}(xN{)}^{xN}{e}^{-xN}\\ (1-x)N!\approx \sqrt{2\pi (1-x)N}\left(\right(1-x)N{)}^{(1-x)N}{e}^{(x-1)N}\end{array}$

Substitutiong values,we get

$\begin{array}{l}\mathrm{\Delta }{S}_{\text{mixing}}\approx k\ln \left(\frac{\sqrt{2\pi N}{N}^N{e}^{-N}}{\sqrt{2\pi (1-x)N}\left(\right(1-x\left)N{)}^{(1-x)N}{e}^{(x-1)N}\sqrt{2\pi xN}\right(xN{)}^{xN}{e}^{-xN}}\right)\\ \mathrm{\Delta }{S}_{\text{mixing}}\approx k\ln \left(\frac{\sqrt{2\pi N}{N}^N}{\sqrt{2\pi (1-x)N}\left(\right(1-x\left)N{)}^{(1-x)N}\sqrt{2\pi xN}\right(xN{)}^{xN}}\right)\\ \mathrm{\Delta }{S}_{\text{mixing}}\approx k\ln \left(\frac{N^N}{\sqrt{2\pi Nx(1-x)}\left(\right(1-x\left)N{)}^{(1-x)N}\right(xN{)}^{xN}}\right)\end{array}$

Where,$\ln \left(ab\right)=\ln \left(a\right)+\ln \left(b\right);\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

we get,

$\mathrm{\Delta }{S}_{\text{mixing}}\approx k\left[N\ln N-\frac{1}{2}\ln \left(2\pi Nx\right(1-x\left)\right)-\left(\right(1-x\left)N\right)\ln \left(\right(1-x\left)N\right)-xN\ln \left(xN\right)\right]$

Taking third term part

$\begin{array}{l}-\left(\right(1-x\left)N\right)\ln \left(\right(1-x\left)N\right)=-\left(\right(1-x\left)N\right)(\ln (1-x)+\ln (N\left)\right)\\ -\left(\right(1-x\left)N\right)\ln \left(\right(1-x\left)N\right)=-N\ln (1-x)+xN\ln (1-x)-N\ln \left(N\right)+xN\ln \left(N\right)\end{array}$

Taking fourth term part

$-xN\ln \left(xN\right)=-xN\ln x-xN\ln N$

The entropy mixing by

$\begin{array}{l}\mathrm{\Delta }{S}_{\text{mixing}}\approx k\left[-\frac{1}{2}\ln \left(2\pi Nx\right(1-x\left)\right)-(1-x)N\ln (1-x)-xN\ln x\right]\\ \mathrm{\Delta }{S}_{\text{mixing}}\approx -Nk\left[\right(1-x)\ln (1-x)+x\ln x]\end{array}$

The first component in the second line has been omitted since it is insignificant in comparison to the following two terms for big$N$.