$A$ soap bubble of radius $r$ contains a monoatomic ideal gas. The gas is heated in such a manner that the bubble remains in mechanical equilibrium. Assuming that the soap material of the bubble has no heat capacity,the molar heat capacity of the gas in the process will be (Neglect atmospheric pressure). (in $R$)

$A$ 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 $= 84 \text{ cm}$).

Match the following $:-$

Column-$I$	Column-$II$
$(i)$ Adiabatic process	$(a)$ Constant temperature
$(ii)$ Isolated system	$(b)$ No exchange of energy and matter
$(iii)$ Isothermal change	$(c)$ First law of thermodynamics
$(iv)$ Law of conservation of energy	$(d)$ No transfer of heat only

Three samples of the same gas $A, B$ and $C$ $(\gamma = 3/2)$ have initially equal volume. Now the volume of each sample is doubled. The process is adiabatic for $A$,isobaric for $B$,and isothermal for $C$. If the final pressures are equal for all three samples,the ratio of their initial pressures is:

One mole of a gas expands obeying the relation as shown in the $P-V$ diagram. The maximum temperature in this process is equal to

Heat is applied to a rigid diatomic gas at constant pressure. The ratio $\Delta Q : \Delta U : \Delta W$ is