Question

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

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

Answer 1

The correct option is A

$\frac{10}{3}R$

Given work done,
$W=\frac{Q}{4}$
Where, $Q$ is heat supplied,
As we know Molar heat capacity at constant volume,
$C_V=\frac{5}{2}R$
Internal energy,
$dU=C_{V}dT$
$\Rightarrow dU=\left(\frac{5}{2}R\right)dT$
$\Rightarrow dT=\frac{2\left(dU\right)}{5R}\dots \left(i\right)$
From first law of thermodynamics,
$dU=dQ-dW=Q-\frac{Q}{4}$
$dU=\frac{3Q}{4}$
Now molar heat capacity,
$C=\frac{dQ}{dT}=\frac{Q}{\frac{2\left(du\right)}{5R}}$
Substituting the value of $dU$,
$C=\frac{5RQ}{2\left(\frac{3Q}{4}\right)}$
$C=\frac{10}{3}R$