Question

At a certain temperature, the R.M.S. velocity for ${{\text{O}}_{\text{2}}}$ is $\text{400 m/sec}$ . At the same temperature, the R.M.S. velocity for ${{\text{H}}_{\text{2}}}$ molecules will be
(A) $\text{100 m/sec}$
(B) $\text{25 m/sec}$
(C) $\text{1600 m/sec}$
(D) $\text{6400 m/sec}$

Answer 1

The R.M.S. velocity or the root mean square velocity of a gas is equal to the square root of the mean of the squares of the velocity of the individual gas molecules. The formula for the same can be given as follows:

Formula used:
$\sqrt{\dfrac{\text{3RT}}{\text{M}}}$ ; where R is universal gas constant, T is the absolute temperature, M is the molecular weight of the gas.

Complete step by step solution:
From the above equations, we can say that, the R.M.S. velocity of ${{\text{O}}_{\text{2}}}$ is,
$\sqrt{\dfrac{\text{3RT}}{\text{32}}}$ = $\text{400 m/sec}$ , where the molecular weight of oxygen is 32.
The R.M.S. velocity of ${{\text{H}}_{\text{2}}}$ molecules is, $\sqrt{\dfrac{\text{3RT}}{2}}$ = ${{\text{V}}_{{{\text{H}}_{\text{2}}}}}$ .
Therefore, the ratio of the R.M.S. velocities is equal to, $\dfrac{\text{400 }}{{{\text{V}}_{{{\text{H}}_{\text{2}}}}}}=\sqrt{\dfrac{\text{3RT}}{\text{32}}}\text{ }\times\text{ }\sqrt{\dfrac{\text{2}}{\text{3RT}}}$
$\Rightarrow \dfrac{\text{400 }}{{{\text{V}}_{{{\text{H}}_{\text{2}}}}}}=\sqrt{\dfrac{1}{16}}$ $\Rightarrow {{\text{V}}_{{{\text{H}}_{\text{2}}}}}=\text{400}\times \text{4}$ = $\text{1600 m/sec}$
Hence, the correct answer is option C.

Note:
The average velocity of the gas is the arithmetic mean of the velocities of different molecules of a gas at a given temperature. The mathematical formula of the average velocity is equal to,
${{\text{V}}_{\text{av}}}\text{=}\sqrt{\dfrac{\text{8RT}}{\text{M}}}$ , where R is universal gas constant, T is the absolute temperature, M is the molecular weight of the gas.
According to the kinetic molecular theory, gaseous particles are in a state of random molecular motion and these particles are always colliding with each other changing directions. Although the velocity of the molecules changes continuously but the distribution remains constant over a range. Hence the average behaviour of the gas molecules is considered which is believed to be constant.