Question

With the increase in temperature, the difference between r.m.s. velocity and average velocity will
a. Increase
b. Decrease
c. Remain same
d. Decrease becoming almost zero at a high temperature.

Answer 1

Hint: To find the difference with the increase in temperature, first we need to obtain the formulas for root mean square (r.m.s.) velocity and average velocity or specifically the proportionality of velocity with respect to temperature. As they are both velocities, the variables or the components required to find the velocity will be the same and will only differ by some constant factor. Hence, from the constant factor and relation with temperature, we can find the change in difference.

Complete step by step answer:
Before finding the change indifference, let us understand the velocity
Average velocity: As the molecules of gas have different speeds in different directions, to express a single value for velocity, we use average velocity which is defined as the rate of change on the position of the object. Average velocity can be expressed as
${{v}_{av}}=\sqrt{\dfrac{8RT}{\pi M}}$
R.M.S. velocity: The root means square velocity as the name suggests is the velocity of the root of the mean of the square of the velocities of the individual molecules. R.M.S. velocity is expressed as
${{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}$
Now, from the above formulas, we can deduce that velocity is proportional to the temperature of the gas. Hence, with the increase in temperature, the velocity will increase.
For simplicity of explanation, let us suppose $\dfrac{R}{M}=1$
Hence, the velocities are expressed as
${{v}_{rms}}=\sqrt{3T}$ and ${{v}_{av}}=\sqrt{\dfrac{8T}{\pi }}$
$\therefore {{v}_{rms}}=1.73\sqrt{T}$
$\therefore {{v}_{av}}=1.59\sqrt{T}$
Now, if the temperature is increased by $1\;K$ , then the r.m.s. velocity increases by $\;1.73$ and the average velocity increases by $\;1.59$ .
Thus, r.m.s. velocity increases at a faster rate compared to average velocity.
Hence, with the increase in temperature, the difference between the r.m.s. velocity and the average velocity increases.

Hence, the correct answer is Option $(A)$ .

Note:
Other than the above-mentioned two velocities, the third type of velocity namely most probable velocity is also considered which is defined as the velocity possessed by the maximum fraction of molecules at an instant of time. It is expressed as ${{v}_{mp}}=\sqrt{\dfrac{2RT}{M}}$
Hence, from the above-obtained formulas of r.m.s. velocity, average velocity, and most probable velocity, the proportion can be obtained as
${{v}_{rms}}:{{v}_{av}}:{{v}_{mp}}=\sqrt{3}:\sqrt{\dfrac{8}{\pi }}:\sqrt{2}$
This relation can be useful to obtain the value of one velocity with the value of the other velocity given and find the change in difference when any one or more conditions change.