Explain in detail the Maxwell Boltzmann distribution function.

Explanation

Maxwell-Boltmann speed distribution function:
(i) In any gas, the molecules are moving in random directions.
(ii) The speed of each molecule is not the same even though macroscopic parameters like temperature and pressure are fixed.
(iii) Each molecule collides with every other molecule and they exchange their speed.
(iv) We calculated the rms speed of each molecule and not the speed of each molecule which is rather difficult.
(v) In this scenario we can find the number of gas molecules that move with the speed of `5ms^(-1)` to `10ms^(-1)` or `10ms^(-1)` to `15ms^(-1)` etc.
(vi) In general our interest is to find how many gas molecules have the range of speed from v to v + dv. This is given by Maxwell's derive the distribution of speeds among the molecules of a gas
`N_(v)=4piN((m)/(2pikT))^((3)/(2))v^(2)e^((mv^(2))/(2kT))`
(vii) The above expression is graphically shown as follows

(viii) For a given temperature, the number of molecules having lower speed increases parabolically but decreases exponentially after reaching most probable speed. The rms speed, average speed and most probable speed are indicated in the figure. It can be seen that the rms speed is the greatest among the three.
(ix) The area under the graph will give the total number of gas molecules in the system.
(x) Figure shows the speed distribution graph for two different temperature. As temperature increases, the peak of the curve is shifted to the right. It implies that the average speed of each molecule will increase. But the area under each graph is same since it represents the total number of gas molecules.