$743 \mathrm{J}$ of heat energy is needed to raise the temperature of $5 \mathrm{moles}$ of an ideal gas by $2 \mathrm{K}$ at constant pressure. How much heat energy is needed to raise the temperature of the same mass of the gas by $2 \mathrm{K}$ at constant volume? Given, molar gas constant $\mathrm{R}=8.3 \mathrm{J} {\mathrm{K}}^{-1} {\mathrm{mol}}^{-1}$.

Heat energy needed to raise the temperature of n moles of a gas through $∆\mathrm{T}$ kelvin at constant pressure is

$\mathrm{Q} = {\mathrm{nC}}_{\mathrm{p}} ∆\mathrm{T}= 743 \mathrm{J} \left(\mathrm{given}\right)$ 

Heat energy needed to raise the temperature of n moles of a gas through $∆\mathrm{T}$ kelvin at constant volume is:

$$\begin{gathered} {\mathrm{Q}}^{\mathrm{\prime }}={\mathrm{nC}}_{\mathrm{v}}\mathrm{\Delta T}=\mathrm{n}\left({\mathrm{C}}_{\mathrm{p}}-\mathrm{R}\right)\mathrm{\Delta T} \left(\because {\mathrm{C}}_{\mathrm{p}}-{\mathrm{C}}_{\mathrm{v}}=\mathrm{R}\right) \\ ={\mathrm{nC}}_{\mathrm{p}}\mathrm{\Delta T}-\mathrm{nR\Delta T} \\ =743-5\times 8.3\times 2 \\ = 743-83=660 \mathrm{J} \end{gathered}$$