The $P-V$ diagram of $2\,g$ of helium gas for a certain process $A$ $\to$ $B$ is shown in the figure. What is the heat given to the gas during the process $A$ $\to$ $B$?
Diffcult
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$\Delta \mathrm{U}=\mathrm{nC}_{\mathrm{v}} \mathrm{dT}$
$\Delta \mathrm{Q}=\Delta \mathrm{W}+\Delta \mathrm{V}$
$\Delta \mathrm{W}=\frac{1}{2} \times\left(2 \mathrm{P}_{0}+\mathrm{P}_{0}\right) \times \mathrm{V}_{0}=\frac{3}{2} \mathrm{P}_{0} \mathrm{V}_{0}$
$\frac{\mathrm{PV}}{\mathrm{T}}=\mathrm{const}=\mathrm{nR}$
$\frac{P_{0} V_{0}}{T_{0}}=\frac{2 P_{0} 2 V_{0}}{T^{\prime}}$
$\mathrm{T}^{\prime}=4 \mathrm{T}_{\mathrm{o}}$
change in temperature $=3 \mathrm{T}_{0}$
$\Delta \mathrm{U}=\mathrm{n} \times \frac{3}{2} \mathrm{R} \times 3 \mathrm{T}_{0}=\frac{9}{2}\left(\mathrm{nRT}_{0}\right)=\frac{9}{2} \mathrm{P}_{0} \mathrm{V}_{0}$
$\Delta \mathrm{Q}=6 \mathrm{P}_{0} \mathrm{V}_{0}$
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