Kinetic interpretation of temperature, RMS speed

The temperature of a gas is \(-50^\circ\) C. To what temperature the gas should be heated so that the rms speed is increased by 3 times?

Match Column - I and Column - II and choose the correct match from the given choices.

Column - I: (A) Root mean square speed of gas molecules; (B) Pressure exerted by ideal gas; (C) Average kinetic energy of a molecule; (D) Total internal energy of 1 mole of a diatomic gas

Column - II: (P) \(\frac{1}{3}nm\bar{v}^2\); (Q) \(\sqrt{\frac{3RT}{M}}\); (R) \(\frac{5}{2}RT\); (S) \(\frac{3}{2}k_BT\)

The average thermal energy for a mono-atomic gas is : (\(k_B\) is Boltzmann constant and T, absolute temperature)

Increase in temperature of a gas filled in a container would lead to :

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) = \(2.76\times10^{-26}\) kg

Boltzmann's constant \(k_B = 1.38\times10^{-23}\) J K\(^{-1}\))

The molecules of a given mass of a gas have r.m.s. velocity of \(200\ ms^{-1}\) at \(27^\circ C\) and \(1.0\times10^{5}\ Nm^{-2}\) pressure. When the temperature and pressure of the gas are respectively, \(127^\circ C\) and \(0.05\times10^{5}\ Nm^{-2}\), the r.m.s. velocity of its molecules in \(ms^{-1}\) is :

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?