According to Maxwell’s velocity distribution law, the number of molecules having velocity in the range \(c\) and \(c+dc\) is given by,

\(\displaystyle{n(c)=4\pi{N}{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}c^2e^{-\frac{mc^2}{2kT}}\ dc}\)

(i) Average Velocity:

The average velocity is

\(\displaystyle{\bar{c}=\frac{1}{N}\int_{0}^{\infty}c\cdot\ n(c)\ dc}\)

where, \(\displaystyle{N=\int_{0}^{\infty}n(c)\ dc}\) is the total number of molecules.

\(\displaystyle{\bar{c}=\frac{1}{N}4\pi{N}{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}\int_{0}^{\infty}c^3\ e^{-\frac{mc^2}{2kT}}\ dc}\)

\(\displaystyle{=4\pi{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}\frac{2kT}{m}\frac{kT}{m}\int_{0}^{\infty}x\ e^{-x}\ dx}\)

by using, \(\displaystyle{\frac{mc^2}{2kT}=x\\ or,\ \frac{m}{2kT}2c\ dc=dx\\ or,\ c\ dc=\frac{kT}{m}\ dx}\)

\(\displaystyle{\bar{c}=8\pi{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}{\left(\frac{kT}{m}\right)}^{2}\Gamma{(2)}}\)

\(\displaystyle{=8\pi{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}{\left(\frac{kT}{m}\right)}^{2}}\)

where, \(\Gamma{(2)}=1\Gamma{(1)}=1\)

\(or,\ \displaystyle{\bar{c}=8\pi\frac{{(kT)}^{1/2}}{{(2\pi)}^{1/2}{m}^{1/2}}}\)

\(or,\ \displaystyle{\bar{c}=\sqrt{\frac{8kT}{m\pi}}}\)

(ii) r.m.s. velocity (\(c_{rms}\)):

The mean square velocity is,

\(\displaystyle{\bar{c^2}=\frac{1}{N}\int_{0}^{\infty}c^2\ n(c)\ dc}\)

\(\displaystyle{=\frac{1}{N}4\pi{N}{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}\int_{0}^{\infty}c^4e^{-\frac{mc^2}{2kT}}\ dc}\)

\(\displaystyle{=4\pi{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}{\left(\frac{2kT}{m}\right)}^{2}\frac{kT}{m}\frac{1}{\sqrt{2kT/m}}\int_{0}^{\infty}\frac{x^2e^{-x}}{\sqrt{x}}\ dx}\)

by using, \(\displaystyle{\frac{mc^2}{2kT}=x\\or,\ \frac{m}{2kT}2c\ dc=dx\\or,\ dc=\frac{kT}{cm}dx}\)

\(\displaystyle{{\bar{c}}^2=4\pi{\left(\frac{m}{2\pi{kT}}\right)}^{3/2}{\left(\frac{2kT}{m}\right)}^{3/2}\frac{kT}{m}\int_{0}^{\infty}x^{3/2}e^x\ dx}\)

\(\displaystyle{={4\pi}\frac{1}{{\pi}^{3/2}}\frac{kT}{m}\Gamma{(5/2)}}\)

\(\displaystyle{={4\pi}\frac{1}{\sqrt{\pi}}\frac{3}{2}\frac{1}{2}\sqrt{\pi}=\frac{3kT}{m}}\)

Therefore,

\(\displaystyle{c_{r.m.s.}=\sqrt{\bar{c^2}}=\sqrt{\frac{3kT}{m}}}\)

(iii) Most Probable Velocity (\(c_{mp}\)):

The most probable velocity is the value of \(c\) for which the number of molecules having that velocity is maximum.

According to the Maxwell’s velocity distribution law, the number of molecules having velocity \(c\) and \(c+dc\) is given by,

\(\displaystyle{n(c)\ dc=4\pi{N}{\frac{m}{2\pi{kT}}}^{3/2}\cdot{c}^2\cdot{e}^{-\frac{mc^2}{2kT}}\ dc}\)

The probability that a molecule may posses a velocity \(c\) and \(c+dc\) is given by,

\(\displaystyle{P(c)\ dc=\frac{n(c)\ dc}{N}=4\pi{\frac{m}{2\pi{kT}}}^{3/2}\cdot{c}^2\cdot{e}^{-\frac{mc^2}{2kT}}\ dc}\)

The probability will be maximum if,

\(\displaystyle{\frac{d}{dc}[4\pi{\frac{m}{2\pi{kT}}}^{3/2}\cdot{c}^2\cdot{e}^{-\frac{mc^2}{2kT}}]=0}\)

\(or,\ \displaystyle{2c\cdot{e}^{-\frac{mc^2}{2kT}}-c^2\frac{m}{2kT}\cdot{2c}\cdot{e}^{-\frac{mc^2}{2kT}}=0}\)

\(or,\ \displaystyle{c^2=\frac{2kT}{m}}\)

\(or,\ \displaystyle{c=\sqrt{\frac{2kT}{m}}}\)

Hence, the most probable velocity is

\(\displaystyle{c_{mp}=\sqrt{\frac{2kT}{m}}}\)