Using the classical statistical mechanics of a gas of molecules with negligible interactions, derive the $i$ deal gas law. Explain briefly to what extent this law is independent of the molecule's internal structure.

Calculate the entropy $S$ of a monatomic gas of low density, with negligible interactions. Deduce the equation relating the pressure $P$ and volume $V$ of the gas on a curve in the $P V$-plane along which $S$ is constant.

[You may use $\int_{-\infty}^{\infty} e^{-\alpha x^{2}} d x=\left(\frac{\pi}{\alpha}\right)^{\frac{1}{2}}$ for $\left.\alpha>0 .\right]$