Question

An amount $Q$ of heat is added to a monoatomic ideal gas in a process in which the gas performs a work $\frac{Q}{2}$ on its surrounding. The molar heat capacity for the process is $($ $R$ is universal gas constant$)$

• A. $5R$
• B. $R$
• C. $4R$
• D. $3R$

Answer 1

The correct option is D $3R$

Given,
Amount of heat given to the gas $=Q$,

So, change in heat, $\Delta Q=Q$,

Work done by the gas, $\Delta W=\frac{Q}{2}$

From first law of thermodynamics,

$\Delta Q=\Delta U+\Delta W$

Substituting the know value, we have
$\Rightarrow Q=\Delta U+\frac{Q}{2}$

$\Rightarrow \Delta U=\frac{Q}{2}.......\left(1\right)$

We know that, change in internal energy, for monoatomic gas,
$\Delta U=\frac{3nR\Delta T}{2}$

Change in heat energy, $Q=nC\Delta T$

Substituting the value of $\Delta U$ and $Q$ in equation $\left(1\right)$, we get

$\frac{3nR\Delta T}{2}=\frac{nC\Delta T}{2}$

$\therefore C=3R$