Question

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

• A. $\frac{5}{3}R$
• B. $\frac{7}{3}R$
• C. $\frac{8}{3}R$
• D. $\frac{10}{3}R$

Answer 1

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

We know that,

$dU=C_{v}dT$

For a diatomic gas $C_v=\frac{5}{2}R$

$dU=\frac{5}{2}RdT$

$dT=\frac{2}{5R}dU......\left(1\right)$

Now, we have from the first law of thermodynamics,

$dQ=dU+dW$

$dU=dQ-dW$

Given that $dQ=Q$ and $dW=\frac{Q}{4}$

$dU=Q-\frac{Q}{4}=\frac{3Q}{4}......\left(2\right)$

So, we have the molar heat capacity as

$C=\frac{dQ}{dT}$

$C=\frac{Q\times 5R}{2dU}$ $[$From $\left(1\right)$$]$

$C=\frac{Q\times 5R}{2\times \left(\frac{3Q}{4}\right)}$ $[$From $\left(2\right)$$]$

$\Rightarrow C=\frac{10R}{3}$