In an adiabatic change, the pressure $P$ and temperature $T$ of a monoatomic gas are related by the relation $P \propto {T^C}$, where $C$ equals

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in$P-V$ diagram. The relation between the ratio $\frac{V_a}{V_d}$ and the ratio $\frac{V_b}{V_c}$ is:

A sample of an ideal gas is contained in a cylinder. The volume of the gas is suddenly decreased. A student makes the following statements to explain the change in pressure of the gas
$I.$ The average kinetic energy of the gas atoms increases
$II.$ The atoms of the gas hit the walls of the cylinder more frequently
$III.$ Temperature of the gas remains unchanged
Which of these statements is true?

An ideal gas is compressed to half its initial volume by means of several processes. Which of the process results in the maximum work done on the gas? 

The pressure and volume of an ideal gas are related as $\mathrm{PV}^{3 / 2}=\mathrm{K}$ (Constant). The work done when the gas is taken from state $A\left(P_1, V_1, T_1\right)$ to state $\mathrm{B}\left(\mathrm{P}_2, \mathrm{~V}_2, \mathrm{~T}_2\right)$ is :