Question

The average velocity of a gas molecule is 400 m/s. The rms velocity at the same temperature will be:
A. 550 m/s
B. 434 m/s
C. 750 m/s
D. 350 m/s

Answer 1

Hint: Under kinetic energy and molecular speed of gases, there are three types of speeds which are related to each other. They are :
1. Average velocity
2. Root mean square velocity
3. Most probable velocity
The ratio of ${{v}_{rms}}:{{v}_{mp}}:{{v}_{avg}}=\sqrt{3}:\sqrt{2}:\sqrt{\dfrac{8}{\Pi }}$

Complete step by step answer:
-According to the kinetic theory of gases, the speed distribution for a gas at a certain temperature can be described by the Maxwell-Boltzmann equation.
-From the distribution function, we can find three types of speeds: average speed, most probable speed and the root mean square speed.
- It is impossible to measure the velocity of each gas molecule in 1 mole under normal conditions. So Maxwell distribution curve is used to find the velocity of gas molecules.

- The relationship between different velocities can be given as
  \[{{v}_{rms}}\] = \[\sqrt{\dfrac{3kBT}{m}}\]
  \[{{v}_{avg}}=\sqrt{\dfrac{8kBT}{m\pi }}\]
  \[{{v}_{mp}}=\sqrt{\dfrac{2kBT}{m}}\]

-Now, looking at the question, we can see that the average velocity is mentioned in the question and we are asked to find the value of the rms velocity.
-Putting the values in the formula, we get
400 = \[\sqrt{\dfrac{8kBT}{m\pi }}\]
Squaring both sides, we get
400x400 =\[\dfrac{8kBT}{m\pi }\]
\[\Rightarrow \]\[\dfrac{kBT}{m}\] = 200 x 100 x π (on rearranging)

Now, from the formula of rms velocity,
\[{{v}_{rms}}\]= \[\sqrt{\dfrac{3kBT}{m}}\]
\[\Rightarrow {{v}_{rms}}=\sqrt{3}x\sqrt{\dfrac{kBT}{m}}\]
Putting the value of $kBT/m$ from above in our equation, we get
\[{{v}_{rms}}\]= 434.16
So, the correct answer is “Option B”.

Note: Always check if temperature is the same or different. In case of different temperatures, we need to put the values in the formula. Also, keep in mind the units of all the same-type variables should be the same.
Eg, the temperatures should be of the same unit. The different velocities should be of the same unit.