Thank you for reporting, we will resolve it shortly
Q.
The molar heat capacity in a process of a diatomic gas if
it does a work of $\frac{Q}{4}$ when a heat of $Q$ is supplied to it is
Thermodynamics
Solution:
$d U=C_{V} d T=\left(\frac{5}{2} R\right) d T$ or $d T=\frac{2(d U)}{5 R}$ ...(i)
From first law of thermodynamics
$d U=d Q-d W=Q-\frac{Q}{4}=\frac{3 Q}{4} .$
Now molar heat capacity $C=\frac{d Q}{d T}=\frac{Q}{\frac{2(d U)}{5 R}}$
$=\frac{5 R Q}{2\left(\frac{3 Q}{4}\right)}=\frac{10}{3} R .$