A gas has $n$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $1:4$, then

$\mathrm{A}$. The r.m.s. velocity of gas molecules in two vessels will be the same.

$\mathrm{B}$. The ratio of pressure in these vessels will be $1:4$.

$\mathrm{C}$. The ratio of pressure will be $1:1$.

$\mathrm{D}$. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1:4$.

Which statements are correct about degrees of freedom?

A. A molecule with $n$ degrees of freedom has $n^2$ different ways of storing energy.

B. Each degree of freedom is associated with $\frac{1}{2}RT$ average energy per mole.

C. A monoatomic gas molecule has $1$ rotational degree of freedom where as diatomic molecule has $2$ rotational degrees of freedom

D. ${\mathrm{CH}}_4$ has a total to $6$ degrees of freedom.

Choose the correct answer from the option given below:

Given below are two statements:

Statement I :The average momentum of a molecule in a sample of an ideal gas depends on temperature.

Statement II : The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2v$.

In the light of the above statements, choose the correct answer from the options given below :

The root mean square speed of smoke particles of mass $5\times {10}^{-17} \mathrm{kg}$ in their Brownian motion in air at NTP is approximately.

[Given $k=1.38\times {10}^{-23}\mathrm{J} {\mathrm{K}}^{-1}$]