**Various
Speeds of Gas Molecules**

The motion of molecules in a gas is characterized by any of the following three speeds.

**1) Root mean square speed: **It is defined as the square root of mean of
squares of the speed of different molecules.
i.e.

\({{v}_{rms}}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+v_{4}^{2}+……..}{N}}=\sqrt{\overline{{{v}^{2}}}}\),

(i) From the expression of pressure \(\left( P \right)=\frac{1}{3}\rho v_{rms}^{2}\),

\(\Rightarrow {{v}_{rms}}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3PV}{Mass\,of\,gas}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3kT}{m}}\),

Where, \(\rho =\frac{Mass\,of\,gas}{V}=Density\,of\,the\,gas\), \(M=\mu \times \left( Mass\,of\,gas \right)\); \(pV=\mu RT\); \(R=k{{N}_{A}}\); \(k=Boltzmann’s\,constant\); \(m=\frac{M}{{{N}_{A}}}=\) Mass of each molecule.

(ii) With rise in temperature rms speed of gas molecules increases as \({{v}_{rms}}\propto \sqrt{T}\)

(iii) With increase in molecular weight rms speed of gas molecule decreases as \({{v}_{rms}}\propto \frac{1}{\sqrt{M}}\), e.g. rms speed of hydrogen molecules is four times that of oxygen molecules at the same temperature.

(iv) rms speed of gas molecules is of the order of km/sec e.g. at NTP for hydrogen gas:

\(\left( {{v}_{rms}} \right)=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.31\times 273}{2\times {{10}^{3}}}}=1840m/\sec \),

(v) rms speed of gas molecules is \(\sqrt{\frac{3}{\gamma }}\) times that of speed of sound in gas, as \({{v}_{rms}}=\sqrt{\frac{3RT}{M}}\) and \({{v}_{s}}=\sqrt{\frac{\gamma RT}{M}}\)\(\Rightarrow {{v}_{rms}}=\sqrt{\frac{3}{\gamma }}{{v}_{s}}\),

(vi) rms speed of gas molecules does not depends on the pressure of gas because \(P\propto \rho \), if pressure is increased n times then density will also increases by n times but \({{v}_{rms}}\) remains constant.

(vii) Moon has no atmosphere because \({{v}_{rms}}\) of a gas molecule is more than escape velocity\(\left( {{v}_{e}} \right)\).

A planet or satellite will have atmosphere only if \({{v}_{rms}}<{{v}_{e}}\)

(viii) At \(T=0\) ; \({{v}_{rms}}=0\) i.e. the rms speed of molecules of a gas is zero at \(0K\). This temperature is called absolute zero.

**2) Most probable speed: **The particles of a gas have range of speeds.
This is defined as the speed which is possessed by maximum fraction of total number
of molecules of the gas.

Most probable speed \(\left( {{v}_{mp}} \right)=\sqrt{\frac{2P}{\rho }}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2kT}{m}}\),

**3) Average Speed: **It is the arithmetic mean of the speeds of
molecules in a gas temperature.

\({{v}_{avg}}=\frac{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+{{v}_{4}}+……..}{N}\) And according to kinetic theory of gases

Average speed \(\left( {{v}_{avg}} \right)=\sqrt{\frac{8P}{\pi \rho }}=\sqrt{\frac{8}{\pi }\times \frac{RT}{M}}=\sqrt{\frac{8}{\pi }\times \frac{kT}{m}}\).