Question

If the root mean square velocity of the molecules of hydrogen at NTP is $1.84\,km{s^{ - 1}}$. Calculate the root mean square velocity of the oxygen molecule at NTP, molecular weight of hydrogen and oxygen are $2$ and $32$ respectively.
A. $1.47\,km{s^{ - 1}}$
B. $0.94\,km{s^{ - 1}}$
C. $1.84\,km{s^{ - 1}}$
D. $0.47\,km{s^{ - 1}}$

Answer 1

Hint: The mean square velocity of the origin is the square root of the velocity average of the square. As such, it has velocity units. The explanation we use for the velocity of rms instead of the average is that the net velocity is zero for a normal gas sample since the particles are travelling in all directions.

Complete step by step answer:
Given,
RMS value of hydrogen at NTP, ${V_1} = 1.84\,km{s^{ - 1}}$
Molar mass of hydrogen, ${M_1} = 0.002kgmo{l^{ - 1}}$
Molar mass of oxygen, ${M_2} = 0.032\,kgmo{l^{ - 1}}$

We know that, the root mean square velocity is given by:
$V = \sqrt {\dfrac{{3RT}}{{{M_1}}}} $
Hence, for ${V_1}$
${V_1} = \sqrt {\dfrac{{3RT}}{{{M_1}}}} $ ....... (i)
For, ${V_2}$
${V_2} = \sqrt {\dfrac{{3RT}}{{{M_2}}}} $ ....... (ii)
Equation (i) divided by equation (ii), we get:
$
 \dfrac{{{V_1}}}
{{{V_2}}} = \sqrt {\dfrac{{{M_2}}}
{{{M_1}}}} \\
 {V_2} = {V_1} \times \sqrt {\dfrac{{{M_1}}}
{{{M_2}}}} \\
 \Rightarrow {V_2} = 1.84 \times \sqrt {\dfrac{2}
{{32}}} \\
 \Rightarrow {V_2} = 0.46\,km{s^{ - 1}} \\
$

Additional Information:
- The product of pressure and volume of one gram molecule of the ideal gas is equal to the product of a number of moles of gas, the constant of the universal gas and the absolute temperature in compliance with the ideal gas law.
- With the assistance of the kinetic theory of gases, the action of gas molecules is explained. At the macroscopic level, it is the analysis of gas molecules.
The five postulates of the kinetic theory of gases follow:
1. Gas is a composition of a vast number of molecules that continuously travel at random.
2. As the distance between the gas molecules is greater than the molecules' size, the volume of the molecules is insignificant.
3. Even negligible are the intermolecular reactions.

Note:
Here we have to be careful while dividing the values of the numerator and the denominator otherwise our answer would be wrong. The root-mean-square velocity, defined as the square root of the mean velocity-square of the molecules in a gas, is the measure of the velocity of particles in a gas. The collision between molecules is still elastic with each other and with the container walls. All the molecules' total kinetic energy depends on the temperature.