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 increases if the gas is compressed to half its original volume? 

The work of $146\,kJ$ is performed in order to compress one kilo mole of a gas adiabatically and in this process the temperature of the gas increases by $7\,^oC$ . The gas is $(R = 8.3\, J\, mol^{-1}\, K^{-1})$

Areversible adiabatic path on a $P-V$ diagram for an ideal gas passes through stateAwhere $P=0$.$7\times 10^5 \,\,N/ m^{-2}$ and $v = 0.0049 \,\,m^3$. The ratio of specific heat of the gas is $1.4$. The slope of path at $A$ is :

A van der Waal's gas obeys the equation of state $\left(p+\frac{n^2 a}{V^2}\right)(V-n b)=n R T$. Its internal energy is given by $U=C T-\frac{n^2 a}{V}$. The equation of a quasistatic adiabat for this gas is given by

Starting at temperature $300\; \mathrm{K},$ one mole of an ideal diatomic gas $(\gamma=1.4)$ is first compressed adiabatically from volume $\mathrm{V}_{1}$ to $\mathrm{V}_{2}=\frac{\mathrm{V}_{1}}{16} .$ It is then allowed to expand isobarically to volume $2 \mathrm{V}_{2} \cdot$ If all the processes are the quasi-static then the final temperature of the gas (in $\left. \mathrm{K}\right)$ is (to the nearest integer)