One mole of an ideal gas at an initial temperature of $T \ K$ does $6R \ J$ of work adiabatically. If the ratio of specific heats of this gas at constant pressure and at constant volume is $5/3$,the final temperature of the gas will be:

During an adiabatic process,the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is:

When air of the atmosphere rises up,it cools. Why?

An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. To reduce the root mean square velocity of molecules two times, the gas should be expanded (in $\times$)

Given below are two statements:
Statement-$I$: When $\mu$ amount of an ideal gas undergoes adiabatic change from state $(P_1, V_1, T_1)$ to state $(P_2, V_2, T_2)$,the work done is $W = \frac{\mu R(T_2 - T_1)}{1 - \gamma}$,where $\gamma = \frac{C_P}{C_V}$ and $R$ is the universal gas constant.
Statement-$II$: In the above case,when work is done on the gas,the temperature of the gas would rise.
Choose the correct answer from the options given below:

$A$ cylinder with a movable piston contains $3$ moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator,and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume?