Question

$2$ moles of a diatomic gas undergoes a thermodynamic process $\frac{P T^{2}}{ V}=$ constant. The molar heat capacity of the gas is $\frac{n R}{2}.$ Find $n.$

Answer 1

$\frac{P T^{2}}{ V}=k\Rightarrow \frac{n R T}{V}\cdot \frac{T^{2}}{ V}=k\Rightarrow T^{3}=\frac{k V^{2}}{n R}$
Now, Differentiating,
$3T^{2}\frac{d T}{d V}=\frac{2 k V}{n R}\Rightarrow \frac{d V}{d T}=\frac{3 n R T^{2}}{2 k V}=\frac{3}{2}\frac{n R}{P}$
Molar Heat capacity,
$C=C_{V}+\frac{P}{n}\frac{d V}{d T}=\frac{5 R}{2}+\frac{P}{n}\cdot \frac{3}{2}\frac{n R}{P}=\frac{8 R}{2}$