In ideal gas of $27^{\circ} \mathrm{C}$ is compressed adiabatically to $(8 / 27)$ of its original volume. If $\gamma=\frac{5}{3}$, the rise in temperature of a gas is

A cylindrical rod is having temperatures $\theta_1$ and $\theta_2$ at its ends. The rate of heat flow is $\mathrm{Q} J / \mathrm{S}$. All the linear dimensions of the rod are doubled by keeping the temperature constant. The new rate of flow of heat is

A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with frictionless piston. The gas is allowed to expand adiabatically to a temperature $T_2$ by releasing the piston suddenly. $L_1$ and $L_2$ are the lengths of the gas columns before and after the expansion respectively. The ratio $T_2 / T_1$ is

In an ideal gas at temperature $T$, the average force that a molecule applies on the walls of a closed container depends on $T$ as $\mathrm{T}^{\mathrm{x}}$. The value of $x$ is