The equation of state for a gas is given by $PV = nRT + \alpha V$, where $n$ is the number of moles and $\alpha $ is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are $T_o$ and $P_o$ respectively. The work done by the gas when its temperature doubles isobarically will be

A mixture of gases at $STP$ for which $\gamma=1.5$ is suddenly compressed to $\frac{1}{9}$ th of its original volume. The final temperature of mixture is ………. $^{\circ} C$

During the adiabatic expansion of $2$ moles of a gas, the internal energy of the gas is found to decrease by $2$ joules, the work done during the process on the gas will be equal to ……. $J$

Monoatomic, diatomic and triatomic gases whose initial volume and pressure are same, are compressed till their volume becomes half the initial volume.

In the following $P-V$ diagram two adiabatics cut two isothermals at temperatures $T_1$ and $T_2$ (fig.). The value of $\frac{{{V_a}}}{{{V_d}}}$ will be