Question

A sample of an ideal gas initially having internal energy $U_{1}$ is allowed to expand adiabatically performing work $W .$ Heat $Q$ is then supplied to it, keeping the volume constant at its new value, until the pressure raised to its original value. The internal energy is then $U_{3}$ (see figure). Find the increase in internal energy $\left(U_{3}-U_{1}\right) ?$

• A. Q + W
• B. Q - W
• C. $\gamma W-Q$
• D. $Q-\gamma W$

Answer 1

Answer: (B)

Process $1 \rightarrow 2$ is adiabatic
$Q_{12}=\Delta U_{12}+W_{12}$
$\Rightarrow 0=\left(U_{2}-U_{1}\right)+W$
$U_{2}-U_{1}=-W$
Process $2 \rightarrow 3$ is isochoric
$Q_{23}=\Delta U_{23}+W_{23}$
$\Rightarrow Q=U_{3}-U_{2}$
$W_{23}=0 \therefore V=$ constant
$\Rightarrow U_{3}-U_{2}=Q$
Now $\left(U_{3}-U_{2}\right)+\left(U_{2}-U_{1}\right)=U_{3}-U_{1}$
$\Rightarrow Q+(-W)=U_{3}-U_{1}$
$\Rightarrow U_{3}=Q+U_{1}-W$
$\Rightarrow U_{3}-U_{1}=Q-W$