Question

The ratio of most probable velocity to that of average velocity is:
(a) $\dfrac{\pi }{2}$
(b) $\dfrac{2}{\pi }$
(c) $\dfrac{{\sqrt \pi }}{2}$
(d) $\dfrac{2}{{\sqrt \pi }}$

Answer 1

Hint: We know that, in a sample of a gas, molecules collide with each other as well as with the walls of the container, as a result of which the speed and direction of the molecules keep on changing. Thus, the speed possessed by all molecules of the gas sample Is not the same.

Complete Step by step answer: So, we understood that molecules are never in a stationary state but they are believed to be in a chaotic or random motion. They travel in all possible directions but with a different velocity. Also, we should note that all collisions between molecules are perfectly elastic, that is there is no change in the energies of the molecules after collision.

Most probable velocity: When a large number of molecules possess the same velocity that velocity is known as the most probable velocity. In Maxwell speed distribution plot and typical speed, the velocity at top of the curve is known as the most probable velocity.
It is given as
${v_{mp}} = \sqrt {\dfrac{{2RT}}{M}} $ --(1)
Where T is temperature, M is molar mass and R is the ideal gas constant.
Average Velocity: It is the average or arithmetic mean of different velocities of all the molecules of gas at a particular temperature (T).
$\bar v = \dfrac{{{v_1} + {v_2} + {v_3} + .......{v_n}}}{n}$
Where ${v_1},{v_2},{v_3}$.. etc. are velocities of gas molecules and n is the total number of molecules.
$\bar v = \sqrt {\dfrac{{8RT}}{{\pi M}}} $ --(2)
Now dividing (1) and (2) we get the ratios,
$\dfrac{{{v_{mp}}}}{{\bar v}} = \dfrac{{\sqrt {\dfrac{{2RT}}{M}} }}{{\sqrt {\dfrac{{8RT}}{{\pi M}}} }}$
We can cancel R,T and M since they are common, remaining are
$\dfrac{{{v_{mp}}}}{{\bar v}} = \sqrt {\dfrac{{2\pi }}{8}} $
$ \Rightarrow \dfrac{{{V_{mp}}}}{{\bar v}} = \sqrt {\dfrac{\pi }{4}} = \dfrac{{\sqrt \pi }}{2}$
Hence, we can say that the ratio is $\dfrac{{\sqrt \pi }}{2}$

Hence, option (c) is correct.

Note: Apart from most probable velocity and average velocity there is also root mean square velocity. It is the square root of the mean of the squares of the speeds of molecules. It is given as
$u = \sqrt {\dfrac{{3RT}}{M}} \;or\;u = \sqrt {\dfrac{{3PV}}{M}} $