JEE Main & Advanced Physics Kinetic Theory of Gases Various Speeds of Gas Molecules

The motion of molecules in a gas is characterised 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 \[P=\frac{1}{3}\rho \,v_{rms}^{2}\]

\[\Rightarrow \] \[{{v}_{rms}}=\sqrt{\frac{3P}{\rho }}=\sqrt{\frac{3PV}{\text{Mass of gas}}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3kT}{m}}\]

\[\text{where }\rho =\frac{\text{Mass of gas}}{V}=\text{Density of the gas}\], \[M=\mu \times \](mass of gas), \[pV=\mu RT\], \[R=k{{N}_{A}},\] \[k=\] Boltzmannís 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/s  e.g., at NTP for hydrogen gas

\[({{v}_{rms}})=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.31\times 273}{2\times {{10}^{3}}}}=1840\,m/s\].

(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 (if temperature remains constant) because \[P\propto \rho \](Boyle?s law) if pressure is increased n times then density will also increases by n times but vrms remains constant.

(vii) Moon has no atmosphere because \[{{v}_{rms}}\] of gas molecules is more than escape velocity \[({{v}_{g}})\].

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 0 K. This temperature is called absolute zero.

(2) Most probable speed : The particles of a gas have a range of speeds. This is defined as the speed which is possessed by maximum fraction of total number of molecules of the gas. e.g., if speeds of 10 molecules of a gas are 1, 2, 2, 3, 3, 3, 4, 5, 6, 6 km/s, then the most probable speed is 3 km/s, as maximum fraction of total molecules possess this speed.

Most probable speed \[{{v}_{mp}}=\sqrt{\frac{2P}{\rho }}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2kT}{m}}\]

(3) Average speed : It is the arithmetic mean of the speeds of molecules in a gas at given temperature.

\[{{v}_{av}}=\frac{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+{{v}_{4}}+.....}{N}\] and according to kinetic theory of gases

Average speed \[{{v}_{av}}=\sqrt{\frac{8P}{\pi \rho }}=\sqrt{\frac{8}{\pi }\frac{RT}{M}}=\sqrt{\frac{8}{\pi }\frac{kT}{m}}\]