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 $(U_3-U_1)$.

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

Answer 1

The correct option is B $Q-W$

Find work done during process $1\to 2$ .

Process $1\to 2$ is adiabatic,

$Q_{12}=\Delta U_{12}+W_{12}$

$\Rightarrow 0=(U_2-U_1)+W$

$U_2-U_1=-W\dots \left(i\right)$

Find heat supplied during process$2\to 3$.

Process$2\to 3$ is isochoric, so work done will be zero.

$W_{23}=0$

$Q_{23}=\Delta U_{23}+W_{23}$

$\Rightarrow Q=U_3-U_2\dots \left(ii\right)$

Find increase in internal energy $(U_3-U_1)$.

Now $(U_3-U_2)+(U_2-U_1)=U_3-U_1$

Using equation $\left(i\right)$ and $\left(ii\right)$

$\Rightarrow Q+(-W)=U_3-U_1$

$\Rightarrow U_3-U_1=Q-W$

Final answer:(b)