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 $1\to 2$ is adiabatic
$Q_{12}=\Delta U_{12}+W_{12}$
$\Rightarrow 0=\left(U_2-U_1\right)+W$
$U_2-U_1=-W$
Process $2\to 3$ 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 $\left(U_3-U_2\right)+\left(U_2-U_1\right)=U_3-U_1$
$\Rightarrow Q+(-W)=U_3-U_1$
$\Rightarrow U_3=Q+U_1-W$
$\Rightarrow U_3-U_1=Q-W$