# Various Speeds of Gas Molecules

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