Write equation for work done for compression for an ideal gas.

A sample of gas at temperature $T$ is adiabatically expanded to double its volume. The work done by the gas in the process is $\left(\right.$ given, $\left.\gamma=\frac{3}{2}\right)$ :

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 ………. $^{\circ} C$

A monoatomic gas at pressure $P$ and volume $V$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be $…..P$

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 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 heat capacity $2C$ . Find the equilibrium temperature . (Assume temperature of air to be uniform and equal to vessel at all times) .