Find an expression for the entropy of the two-dimensional ideal gas considered in Problem $2.26$. Express your result in terms of $U$,$A$and$N$.

Entropy is described as a measure of the degree of unpredictability in a system, or in other words, the growth in disorder.Entropy is a measure of disarray that has an impact on many facets of our existence. In reality, it's akin to a tax imposed by nature. If problem is not addressed, it will worsen over time. Energy dissipates, and systems disintegrate. We consider something to be more entropic if it is more disordered.

The entropy substance as

$S=k\ln \left(\mathrm{\Omega }\right)$

where,

localid="1650262053991" $\Omega$is the number of microstates substance accessible.

The localid="1650262058146" $2-d$ideal gas multipilicity is

$\mathrm{\Omega }=\frac{(\pi A{)}^N}{(N!{)}^2{h}^{2N}}(2mU{)}^N$

where,localid="1650262061679" $A$is the area gas.

By using Stirling's approximation,

$\begin{array}{l}n!\approx \sqrt{2\pi n}{n}^n{e}^{-n}\\ N!\approx \sqrt{2\pi N}{N}^N{e}^{-N}\\ (N!{)}^2\approx 2\pi N^{2N+1}{e}^{-2N}\end{array}$

Substituting we get,

$\begin{array}{r}\mathrm{\Omega }\approx \frac{(2m\pi UA{)}^N}{2\pi N^{2N+1}{e}^{-2N}{h}^{2N}}\\ \end{array}$

Where localid="1650262075329" $N$is the large,the couple of factors away is

$\begin{array}{l}\mathrm{\Omega }\approx \frac{(2m\pi UA{)}^N}{N^{2N}{e}^{-2N}{h}^{2N}}\\ \mathrm{\Omega }\approx \frac{(2m\pi UA{)}^N}{(Nh{)}^{2N}}{e}^{2N}\\ \mathrm{\Omega }\approx {\left[\frac{\left(2m\pi UA\right)}{(Nh{)}^2}{e}^2\right]}^N\end{array}$

Taking logarithm on both sides,

$\ln \left(ab\right)=\ln \left(a\right)+\ln \left(b\right)\text{and}\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

is taking into account,so

$\begin{array}{l}\ln \left(\mathrm{\Omega }\right)\approx N\ln \left[\frac{\left(2m\pi UA\right)}{(Nh{)}^2}{e}^2\right]\\ \ln \left(\mathrm{\Omega }\right)\approx N\ln \left[\frac{\left(2m\pi UA\right)}{(Nh{)}^2}\right]+N\ln \left(e^2\right)\\ \ln \left(\mathrm{\Omega }\right)\approx N\left[\ln \left(\frac{2m\pi UA}{(Nh{)}^2}\right)+2\right]\end{array}$

The entrpy of ideal gas gives as,

$S=Nk\left[\ln \left(\frac{2m\pi UA}{(Nh{)}^2}\right)+2\right].$