$A$ metal ball immersed in alcohol weighs $W_1$ at $0^{\circ}C$ and $W_2$ at $59^{\circ}C$. The coefficient of cubical expansion of the metal is less than that of alcohol. Assuming that the density of metal is large compared to that of alcohol,it can be shown that

Explain why:
$(a)$ Two bodies at different temperatures $T_1$ and $T_2$,if brought in thermal contact,do not necessarily settle to the mean temperature $(T_1 + T_2) / 2$.
$(b)$ The coolant in a chemical or a nuclear plant (i.e.,the liquid used to prevent the different parts of a plant from getting too hot) should have high specific heat.
$(c)$ Air pressure in a car tyre increases during driving.
$(d)$ The climate of a harbour town is more temperate than that of a town in a desert at the same latitude.

The heat energy required to convert $10 \,kg$ of ice at $-10^{\circ} C$ into water at $0^{\circ} C$ is (specific heat capacity of ice $=0.5 \,cal \,g^{-1} \,^{\circ}C^{-1}$ and latent heat of fusion of ice $=80 \,cal \,g^{-1}$)

$M \ kg$ of water at $t^{\circ} C$ is divided into two parts so that one part of mass $m \ kg$ when converted into ice at $0^{\circ} C$ would release enough heat to vaporize the other part. Then $\frac{m}{M}$ is equal to (Specific heat of water $= 1 \ cal \ g^{-1} {}^{\circ} C^{-1}$,Latent heat of fusion of ice $= 80 \ cal \ g^{-1}$,Latent heat of steam $= 540 \ cal \ g^{-1}$)

An iron rod of heat capacity $C$ is heated to temperature $8T_0$. It is then put in a cylindrical vessel of adiabatic walls having two moles of air which can be treated as a diatomic ideal gas at temperature $T_0$ and closed by a movable piston which is also adiabatic. The atmospheric pressure is $P_0$. The cylinder with the piston combined have a heat capacity $2C$. Find the equilibrium temperature. (Assume the temperature of air to be uniform and equal to the vessel at all times).

$10 \, g$ of ice at $0^{\circ} C$ is kept in a calorimeter of water equivalent $10 \, g$. How much heat (in $cal$) should be supplied to the apparatus to evaporate the water thus formed? (Neglect loss of heat)