At what temperature will the RMS speed of oxygen molecules become just sufficient for escaping from the earth's atmosphere? 
(Given : Mass of oxygen molecule (m) = 2.76 x 10^-26 kg, Boltzmann's constant k_B = 1.38 × 10^-23 J K^-1):

1. $2.508\times {10}^4$ $K$

2. $8.360\times {10}^4$ $K$

3. $5.016\times {10}^4$ $K$

4. $1.254\times {10}^4$ $K$

A gas mixture consists of \(2\) moles of O₂ and \(4\) moles of Ar at temperature \(T.\) Neglecting all the vibrational modes, the total internal energy of the system is:
1. \(15RT\)
2. \(9RT\)
3. \(11RT\)
4. \(4RT\)

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 r.m.s velocity of 200 ms^-1 at 27°C and 1.0 x 10⁵ Nm^-2 pressure. When the temperature and pressure of the gas are increased to ,respectively, 127°C and 0.05 X 10⁵ Nm^-2 , r.m.s 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. \(mkT\)