Question

There is no atmosphere on moon because
(A). It is closer to earth
(B). It revolves around the earth
(C). It gets light for the sun
(D). The escape velocity of the gas molecules is lesser than their root mean square velocity

Answer 1

- Hint: If a mass should settle or be on the surface of a planet or any of their celestial bodies, its speed must be less than the escape velocity.

Complete step-by-step solution -

We can answer this question by analysing each option one by one and then figure out which is the most appropriate.
First, option A: - It says that there is no atmosphere on the moon because it is closer to earth. However, the moon being closer to earth does not give justification of no atmosphere on the moon.
Option B: - It says that there is no atmosphere on the moon because it revolves around the earth. If this is true, then earth will also not have atmosphere since it also revolves around a celestial body (the sun). Again, this reason also does not justify the given question.
Option C: - It says that the reason for no atmosphere on the moon is that it gets light from the sun. However, even earth gets light from the sun. This option is also incorrect.
Option D: - It says that because the escape velocity of the gas molecules on the moon is less than the root mean square velocity, there is no atmosphere on the moon. If we think about this reason carefully, it might be correct.

Let us first understand what escape velocity is. It is the minimum required velocity (speed) for a body to escape the gravitational pull experienced by it from the celestial body by which it is being attracted. Which means if a body on the surface (or within the gravity) of a celestial body somehow gains a velocity greater than or equal to the escape velocity for that body, it will come out of the gravity of that celestial body and will no longer experience any gravitational pull.
Therefore, if a gas molecule on the moon has a velocity greater than or equal to its escape velocity, the molecule will no longer be in the gravity of the moon. However, a gas has millions of molecules and each molecule will have a different velocity. That is why we consider the root mean square velocity as the speed of the gas. Root mean square velocity of a gas is the square root of the mean of the squares of the velocities of all the molecules of the gas. It gives us the most probable speed of a molecule of the gas.
From the above statements, we can understand that if the root mean square velocity of the gas must be greater than or equal to the escape velocity of the gas. Therefore, option D is the correct option.

Note: Some may think why the root mean square value of the velocity is taken and why not the normal mean of velocities of all the molecules. In a gas, all the molecules move randomly in any direction. Hence, if the mean of the velocities of all the molecules is calculated, it will be equal to zero. Therefore, the normal mean or average is not considered.