Question

The ratio of most probable velocity ($\alpha$), average velocity ($\upsilon$), root mean square velocity ($\mu$) is:
(A) $\sqrt { 2 } :\sqrt { \dfrac { 8 }{ \pi } } :\sqrt { 3 }$
(B) $1\quad :\sqrt { 2 } :\sqrt { 3 }$
(C) $\sqrt { 2 } :\sqrt { 3 } :\sqrt { 8 }$
(D) $1\quad :\sqrt { 8\pi } :\sqrt { 3 }$

Answer 1

Hint: All three of them are related to the kinetic theory of gases, where we determine the different kinds of velocities of gas molecules depending on the temperature and pressure.

Complete step by step answer:
The most probable velocity of a gas at a particular temperature is the velocity possessed by the moment possessed by the maximum fraction of the total number of the molecules.
$\alpha \quad =\quad \sqrt { \dfrac { 2RT }{ M } }$
The average velocity of an object is its total displacement divided by the total time taken. In other words, it is the rate at which an object changes its position from one place to another.
$\upsilon \quad =\quad \sqrt { \dfrac { 8RT }{ \pi M } }$
The root mean square velocity is the square root of the average of the square of the velocity. As such, it has units of velocity. The reason we use the RMS velocity instead of the average is that for a typical gas sample the net velocity is zero since the particles are moving in all directions.
$\mu \quad =\quad \sqrt { \dfrac { 3RT }{ M } }$
Now take the ratio of all of them. We will get the final result as -
$\alpha :\upsilon :\mu \quad =\quad \sqrt { 2 } :\sqrt { \dfrac { 8 }{ \pi } } :\sqrt { 3 }$
Hence the correct option is A.

Note: Maxwell-Boltzmann statistics curve:The static curve used for the representation of velocity and the number of gas molecules in thermal equilibrium. The area of the static curve represents the total number of molecules of the gas in thermal equilibrium. From this distribution function, the most probable speed, the average speed, and the root-mean-square speed can be derived.