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 $$\mathrm{k}=1.38 \times 10^{-23} \mathrm{JK}^{-1}$$]

A vessel contains $$14 \mathrm{~g}$$ of nitrogen gas at a temperature of $$27^{\circ} \mathrm{C}$$. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be :

Take $$\mathrm{R}=8.32 \mathrm{~J} \mathrm{~mol}^{-1} \,\mathrm{k}^{-1}$$.

A Carnot engine has efficiency of $$50 \%$$. If the temperature of sink is reduced by $$40^{\circ} \mathrm{C}$$, its efficiency increases by $$30 \%$$. The temperature of the source will be:

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 $$2 v$$.

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