A solid cube of mass $m$ at a temperature $\theta_0$ is heated at a constant rate. It becomes liquid at temperature $\theta_1$ and vapour at temperature $\theta_2$. Let $s_1$ and $s_2$ be specific heats in its solid and liquid states respectively. If $L_f$ and $L_v$ are latent heats of fusion and vaporisation respectively, then the minimum heat energy supplied to the cube until it vaporises is

One mole of an ideal monoatomic gas is taken round the cyclic process MNOM. The work done by the gas is

$$100 \mathrm{~g}$$ of ice at $$0^{\circ} \mathrm{C}$$ is mixed with $$100 \mathrm{~g}$$ of water at $$100^{\circ} \mathrm{C}$$. The final temperature of the mixture is

[Take, $$L_f=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}$$ and $$S_w=4.2 \times 10^3 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1} \text { ] }$$

The $$p$$-$$V$$ diagram of a Carnot's engine is shown in the graph below. The engine uses 1 mole of an ideal gas as working substance. From the graph, the area enclosed by the $$p$$-$$V$$ diagram is [The heat supplied to the gas is 8000 J]