Question

What is the ratio of most probable velocity to the average velocity?

Answer 1

Hint: Hint. We must remember that the most probable velocity is the velocity conducted by the maximum number of molecules at the same temperature. And an average velocity is the velocity at which an object changes its position from one place to another.

Complete step by step answer:
The solution In a gas there are a number of particles. The speed travelled by the number of gas particles at the same temperature is known as Most Probable Speed. Most Probable Velocity thus, can be defined as the velocity with which the maximum number of the particles in a gas move at constant temperature.
Mathematically, formula of most probable velocity can be written as follows:
Most probable velocity =\[\;{v_p}\; = \sqrt {\dfrac{{{\text{2RT}}}}{{\text{M}}}} \;m/s\]
Now, average Velocity can be defined as the sum of the speed of all the number of particles divided by the total number of particles. The average velocity of an object can also be defined as total displacement of an object divided by the total time consumed.
Mathematically, formula of the average velocity can be written as follows:
Average velocity =\[{v_{avg}}\; = \sqrt {\dfrac{{{\text{8RT}}}}{{\pi {\text{M}}}}} m/s\].
Where R is the universal gas constant
T is the temperature in kelvin
M is the molar mass of the gas
Now, we will determine the ratio of most probable velocity to the average velocity from their formulas as follows
The most probable velocity \[\left( {{V_p}} \right)\] : average velocity\[\;\left( {{V_{avg}}} \right)\] = \[\dfrac{{\sqrt {\dfrac{{{\text{2RT}}}}{{\text{M}}}} }}{{\sqrt {\dfrac{{{\text{8RT}}}}{{\pi {\text{M}}}}} }}\]
By solving this, we get
\[{V_p}:{V_{avg}} = \;\sqrt {\dfrac{{2\pi }}{8}} \]

\[{V_p}:{V_{avg}} = \dfrac{{\sqrt \pi }}{2}\]

Hence, the ratio of most probable velocity to the average velocity is \[\sqrt \pi :2\]

Note: We can understand that the Value of π is $pi$ and so if we see the ratio of most probable velocity to the average velocity, then we can say that value of average velocity is always greater than that of most probable velocity.