Check whether the following statements are true or false:
$1.$ The change in internal energy $\Delta U = 0$ in a cyclic process.
$2.$ In an adiabatic process,temperature remains constant.
$3.$ The internal energy of a system during an isothermal process decreases.

$A$ cylinder made of perfectly non-conducting material,closed at both ends,is divided into two equal parts by a heat-proof piston. Both parts of the cylinder contain the same mass of a gas at a temperature $t_0 = 27^{\circ}C$ and pressure $P_0 = 1 \text{ atm}$. If the gas in one of the parts is slowly heated to $t = 57^{\circ}C$ while the temperature of the first part is maintained at $t_0$,the distance moved by the piston from the middle of the cylinder will be $x \text{ cm}$. Find $x$ (total length of the cylinder $= 84 \text{ cm}$).

The specific heat at constant pressure of a real gas obeying $PV^2=RT$ equation is:

In Column-$I$ devices and in Column-$II$ efficiency are given. Match them appropriately:

Column-$I$	Column-$II$
$(a)$ Heat engine	$(i)$ $\eta = \frac{Q_2}{Q_1 - Q_2}$
$(b)$ Heat pump	$(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$
	$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

Five moles of an ideal gas has pressure $p_0$, volume $V_0$, and temperature $T_0$. The gas is expanded to volume $3V_0$ along a path such that the pressure $p$ changes as a function of volume $V$ as $p = p_0(V/V_0)$. The pressure is then reduced to $p_0$ while maintaining constant volume. Finally, the gas undergoes an isobaric compression until the volume and temperature return to $V_0$ and $T_0$, respectively. The total work done by the gas during the entire process is:

Write the equation for the work done during the compression of an ideal gas.