Question

The r.m.s velocity of a gas depends upon which of the following factor(s)?
A. Temperature only
B. Molecular mass only
C. Temperature and molecular mass of gas
D. None of these

Answer 1

Hint: Root mean square velocity or r.m.s velocity is defined as the square root of the mean of the squares of the different velocities.
It is denoted by \[{{\rm{V}}_{{\rm{rms}}}}\].

Formula Used:
\[{{\rm{V}}_{{\rm{rms}}}}{\rm{ = }}\sqrt {\frac{{{\rm{3RT}}}}{{\rm{M}}}} \]
where
R = Universal gas constant
T = Temperature of the gas
M = Gram molecular mass of the gas

Complete Step by Step Solution:
Matter exists in three states i.e., gas, liquid and solid. A gas consists of molecules separated by wide distances.
The molecules are unrestricted to change positions in the area in which it is comprised.
According to the kinetic theory of gases, every gas is made up of a large number of extremely small particles called molecules.
These molecules move continuously in different directions at different velocities. They keep on colliding with each other and with the walls of the container which is called a molecular collision.
The average kinetic energy of the molecules of a gas and the absolute temperature of the gas have a direct relationship.
We know that the expression for the Kinetic gas equation is:-
\[{\rm{pV = }}\frac{1}{3}{\rm{mN}}{{\rm{V}}_{{\rm{rms}}}}^{\rm{2}}{\rm{\;}}\]
where
m=mass of the gas molecule
N=total number of gas molecules
\[{{\rm{V}}_{{\rm{rms}}}}\]=root mean square velocity
\[{\rm{pV = }}\frac{1}{3}{\rm{M}}{{\rm{V}}_{{\rm{rms}}}}^{\rm{2}}{\rm{\;}}\]
where M=mN=gram molecular mass
\[ \Rightarrow {{\rm{V}}_{{\rm{rms}}}}^{\rm{2}}{\rm{ = }}\frac{{{\rm{3pV}}}}{{\rm{M}}}\]
\[ \Rightarrow {{\rm{V}}_{{\rm{rms}}}}{\rm{ = }}\sqrt {\frac{{{\rm{3pV}}}}{{\rm{M}}}} \]
We know that pV=RT for 1 mole of an ideal gas
\[{{\rm{V}}_{{\rm{rms}}}}{\rm{ = }}\sqrt {\frac{{{\rm{3RT}}}}{{\rm{M}}}} \]
So, root mean square velocity is directly proportional to the square root of temperature and inversely proportional to the square root of molecular mass.
So, option C is correct.

Additional Information: Root mean square velocity is expressed in m/s. Temperature and molecular mass are taken in kelvin and kilograms respectively.

Note: r.m.s velocity is directly proportional to the square root of temperature and inversely proportional to the square root of molecular mass. Hence, at a given temperature lightweight molecules like hydrogen, and helium have more rapid motion than molecules with higher molecular mass.