Question

Ratio of average to most probable velocity is:
A. \[1.128\]
B. \[1.124\]
C. \[1.0\]
D. \[1.112\]

Answer 1

Hint: Recall the formula of average velocity for ideal gas and the formula for most probable velocity. The average velocity of gas is the average velocity of each gas particle in the particle and the most probable velocity of the gas is the velocity with most of the gas particle moves.

Formula used:
The average velocity of gas particle is given by,
\[{V_{avg}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} \]
where \[{V_{avg}}\] is the average, velocity \[R\] is the universal gas constant, \[T\] is the absolute temperature of the gas and \[M\] is the molar mass of the gas.
The most probable velocity of ideal gas is given by,
\[{V_m} = \sqrt {\dfrac{{2RT}}{{\pi M}}} \]
where\[{V_m}\] is the most probable velocity, \[R\] is the universal gas constant, \[T\] is the absolute temperature of the gas and \[M\] is the molar mass of the gas.

Complete step by step answer:
We know that the average velocity is the average velocity of each particle of gas in a container with each moving and the most probable velocity of the gas is the velocity with which the maximum number of particles moves.Now, we have to find the ratio of them. We know that the expression for average velocity is given by,
\[{V_{avg}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} \]
The expression for most probable velocity is given by,
\[{V_m} = \sqrt {\dfrac{{2RT}}{{\pi M}}} \]
So, the ratio of these two is,
\[{V_{avg}}:{V_m} = \sqrt {\dfrac{{8RT}}{{\pi M}}} :\sqrt {\dfrac{{2RT}}{{\pi M}}} \]
\[\Rightarrow {V_{avg}}:{V_m} = \sqrt {\dfrac{8}{\pi }} :\sqrt 2 \]
\[\Rightarrow{V_{avg}}:{V_m} = \sqrt {\dfrac{4}{\pi }} :1\]
Which will be equal to,
\[\therefore {V_{avg}}:{V_m} = 1.128:1\]
So, ratio of average to most probable velocity \[1.128\].

Hence, option A is the correct answer.

Note:Apart from average velocity and most probable velocity there is root mean square velocity which is calculated as taking the average of the square of the velocities of each particle and taking square root of that. The expression for root mean square velocity is given by, \[{V_{rms}} = \sqrt {\dfrac{{3RT}}{{\pi M}}} \].