Question

An amount $Q$ of heat is added to a monoatomic ideal gas in a process in which the gas performs a work of $\frac{Q}{2}$ on its surrounding. Find equation of the process that monoatomic gas is undergoing.

• A. $P{V}^{\frac{2}{3}}=\text{constant}$
• B. $P{V}^{\frac{4}{3}}=\text{constant}$
• C. $P{V}^{\frac{1}{3}}=\text{constant}$
• D. $P{V}^{\frac{3}{5}}=\text{constant}$

Answer 1

The correct option is C $P{V}^{\frac{1}{3}}=\text{constant}$

Given $\Delta W=\frac{Q}{2}$
From first law of thermodynamics
$\Delta Q=\Delta U+\Delta W$
$\Rightarrow \Delta U=\frac{Q}{2}$
$\therefore dU=dW$
$\Rightarrow n{C}_{V}dT=PdV$
$\Rightarrow \left(\frac{nR}{\gamma -1}\right)dT=PdV$
We know that for monoatomic gas $\gamma =\frac{5}{3}$,
using $PV=nRT$, we get, $PdV+VdP=nRdT$
$\Rightarrow \frac{3}{2}(PdV+VdP)=PdV(\because \gamma =\frac{5}{3})$
$\Rightarrow PdV+3VdP=0$
$\Rightarrow \frac{dV}{V}+\frac{3dP}{P}=0$
$\Rightarrow V{P}^3=\text{constant}$
$\Rightarrow P{V}^{\frac{1}{3}}=\text{constant}$