Question

The molar heat capacity of 1 mole of a diatomic gas, if it does a work of $Q/4$ in a process when heat $Q$ is supplied to it, is -

• A. $\frac{2}{5}R$
• B. $\frac{5}{2}R$
• C. $\frac{10}{3}R$
• D. $\frac{6}{7}R$

Answer 1

The correct option is C $\frac{10}{3}R$

Given ,
Number of moles of diatomic gas, $\left(n\right)=1$
So, we can write that
$\Delta U=C_v\Delta T=\left(\frac{5}{2}R\right)\Delta T$
$\Rightarrow \Delta T=\frac{2\left(\Delta U\right)}{5R}......\left(1\right)$
From first law of thermodynamics, we can say that
$\Delta U=\Delta Q-\Delta W$
From the data given in the question
$\Delta U=Q-\frac{Q}{4}=\frac{3Q}{4}.....\left(2\right)$
Now, from the definition of molar heat capacity,
$C=\frac{1}{n}\frac{\Delta Q}{\Delta T}$ where $n=1$ here.
From $\left(1\right)$ and $\left(2\right)$,
$C=\frac{Q}{\frac{2\left(\Delta U\right)}{5R}}=\frac{5RQ}{2\left(\frac{3Q}{4}\right)}=\frac{10}{3}R$
Hence, option (c) is the correct answer.