At $STP$,the speed of a sound wave in a gas is $330 \ m/s$ and the density of the gas is $1.3 \ kg/m^3$. Calculate the degrees of freedom $(f)$ of the gas.

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ $3$ Translational degrees of freedom	$(I)$ Monoatomic gases
$(B)$ $3$ Translational,$2$ rotational degrees of freedom	$(III)$ Rigid diatomic gases
$(C)$ $3$ Translational,$2$ rotational and $1$ vibrational degrees of freedom	$(IV)$ Non-rigid diatomic gases
$(D)$ $3$ Translational,$3$ rotational and more than one vibrational degrees of freedom	$(II)$ Polyatomic gases

Choose the correct answer from the options given below:

$500 \text{ g}$ of a diatomic gas is enclosed at a pressure of $10^5 \text{ N m}^{-2}$. The density of the gas is $5 \text{ kg m}^{-3}$. The energy of one mole of the gas due to its thermal motion is [consider the gas molecule as a rigid rotator].

If the kinetic energy of a monoatomic gas molecule is $\frac{3}{2}PV$,then the kinetic energy of a diatomic gas molecule is:

Write the degrees of freedom for:
$(i)$ Monoatomic gas
$(ii)$ Diatomic gas

An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and $f$ number of vibrational modes. If the ratio of heat capacities $C_p/C_v$ of the gas is $8/7$, then the value of $f$ is: