At what temperature will the $\text{rms}$ speed of oxygen molecules become just sufficient for escaping from the earth's atmosphere? 
(Given: Mass of oxygen molecule $(m)= 2.76\times 10^{-26}~\text{kg}$, Boltzmann's constant $k_B= 1.38\times10^{-23}~\text{J K}^{-1}$)
1. $2.508\times 10^{4}~\text{K}$
2. $8.360\times 10^{4}~\text{K}$
3. $5.016\times 10^{4}~\text{K}$
4. $1.254\times 10^{4}~\text{K}$

A gas mixture consists of $2$ moles of $\mathrm{O_2}$ and $4$ moles of $\mathrm{Ar}$ at temperature $T.$ Neglecting all the vibrational modes, the total internal energy of the system is:

A gas mixture consist of 2 moles of $O_2$ and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:

(1)4RT 

(2) 15RT

(3)9RT

(4)11RT

One mole of an ideal monatomic gas undergoes a process described by the equation $P{V}^3=$ constant. The heat capacity of the gas during this process is:

1. $\frac{3}{2}R$           

2. $\frac{5}{2}R$

3. $2R$             

4. $R$

The molecules of a given mass of gas have rms velocity of 200 ms^-1 at $27^{\circ}\mathrm{C}$ and 1.0 x 10⁵ Nm^-2 pressure. When the temperature and pressure of the gas are increased to, respectively, $127^{\circ}\mathrm{C}$ and 0.05 X 10⁵ Nm^-2, rms velocity of its molecules in ms^-1 will become:
1. 400/√3
2. 100√2/3
3. 100/3 
4.100√2

A given sample of an ideal gas occupies a volume $V$ at a pressure $P$ and absolute temperature $T$. The mass of each molecule of the gas is $m$. Which of the following gives the density of the gas?
1. $\frac{P}{kT}$
2. $\frac{Pm}{kT}$
3. $\frac{P}{kTV}$
4. $mkT$