For a monoatomic gas, the work done at constant pressure is ' \(\mathrm{W}\) '. The heat supplied at constant volume for the same rise in temperature of the gas is

At constant pressure, \(\mathrm{W}=\mathrm{P} . \Delta \mathrm{V}\)
For an ideal gas, \(\mathrm{PV}=\mathrm{nRT}\)
\(
\begin{aligned}
& \therefore \mathrm{P} \Delta \mathrm{V}=\mathrm{n} \mathrm{R} \Delta \mathrm{T} \\
& \mathrm{W}=\mathrm{n} \mathrm{R} \Delta \mathrm{T}
\end{aligned}
\)
Heat supplied at constant volume,
\(
\begin{aligned}
& \mathrm{Q}=\Delta \mathrm{U}=\mathrm{nC}_{\mathrm{v}} \Delta \mathrm{T}=\mathrm{n} \frac{3 \mathrm{R}}{2} \Delta \mathrm{T} \\
& \quad\left(\text { Formonoatomic gas } \mathrm{C}_{\mathrm{v}}=\frac{3}{2} \mathrm{R}\right) \\
& \therefore \mathrm{Q}=\frac{3 \mathrm{~W}}{2}
\end{aligned}
\)