Question

What is the relationship between the average velocity$(v)$, root mean square velocity$(u)$, and most probable velocity $(\alpha )$ [AFMC$1994$]
A.$\alpha :v:u::1:1.128:1.224$
B.$\alpha :v:u::1.128:1:1.224$
C.$\alpha :v:u::1.128:1.224:1$
D.$\alpha :v:u::1.124:1.228:1$

Answer 1

Hint: According to kinetic theory, a gas contains many microscopic particles always in random motion. As a result, they rapidly collide with each other and with the walls of the container. Thus these moving particles have different velocities at a particular temperature and three kinds of velocities are average velocity, root means square velocity and most probable velocity.

Complete step-by-step solution:Average velocity $(v)$is the arithmetic mean of the velocities of different molecules of a gas at a given temperature and it can be expressed in terms of the molecular mass of a gas,$M$ and Universal gas constant,$R$at a particular temperature,$T$.
Average velocity,$v=\sqrt{\dfrac{8RT}{\pi M}}$
Root mean square velocity $(u)$ is the square root of the mean of the velocity of an individual of a gas at a particular temperature$T$ and it can be expressed as
Root mean square velocity,$u=\sqrt{\dfrac{3RT}{M}}$
And finally, the Most probable velocity $(\alpha )$is the velocity at which the maximum number of the particles in a gas move at a particular temperature and it can also be expressed as
Most probable velocity,$\alpha =\sqrt{\dfrac{2RT}{M}}$
Now taking the ratio of most probable velocity$(\alpha )$, average velocity$(v)$and root mean square velocity$(u)$we get,
$\alpha :v:u=\sqrt{\dfrac{2RT}{M}}:\sqrt{\dfrac{8RT}{\pi M}}:\sqrt{\dfrac{3RT}{M}}$
Or,$\alpha :v:u=\sqrt{2}:\sqrt{\dfrac{8}{\pi }}:\sqrt{3}$
Or,$\alpha :v:u=1.414:1.596:1.732$
Or,$\alpha :v:u=\dfrac{1.414}{1.414}:\dfrac{1.596}{1.414}:\dfrac{1.732}{1.414}$ [Dividing each by $1.414$]
Or,$\alpha :v:u=1:1.128:1.224$
Or we can write the above ratio in the following way:
$\alpha :v:u::1:1.128:1.224$

Thus, option (A) is correct.

Note: According to Kinetic theory, the collisions between gaseous particles are perfectly elastic which means the gaseous particles only change their directions and kinetic energies but the total kinetic energy is always conserved.