Question

The most probable velocity of a molecule is one kmps. find the rms velocity of the molecule in kmps.
$\begin{aligned} & a)1.128 \\\ & b)1.224 \\\ & c)1.5 \\\ & d)1.086 \\\ \end{aligned}$

Answer 1

We know the value of rms velocity. First, find out the ratio between the rms velocity and the most probable velocity of a molecule. As we know the ratio and the value of most probable velocity, we can easily find out the value of rms velocity.
Formula used:
$\begin{aligned} & {{V}_{rms}}=\sqrt{\dfrac{3RT}{M}} \\\ & \Rightarrow {{V}_{mp}}=\sqrt{\dfrac{2RT}{M}} \\\ \end{aligned}$

Complete answer:
We know the value of most probable velocity. Let’s first find out the ratio between the rms velocity and the most probable velocity.
$\begin{aligned} & {{V}_{rms}}=\sqrt{\dfrac{3RT}{M}} \\\ & \Rightarrow {{V}_{mp}}=\sqrt{\dfrac{2RT}{M}} \\\ & \dfrac{{{V}_{rms}}}{{{V}_{mp}}}=\sqrt{\dfrac{3}{2}} \\\ \end{aligned}$
As we have obtained the ratio between the root mean square velocity and the most probable velocity,
Let’s substitute the value of most probable velocity in the above equation we get,
$\begin{aligned} & \dfrac{{{V}_{rms}}}{1}=\sqrt{\dfrac{3}{2}} \\\ & {{V}_{rms}}=1.22kmps \\\ \end{aligned}$

So, the correct answer is “Option B”.

Additional Information:
Most probable velocity is the velocity obtained by most of the atoms in a molecule at the same temperature. As the atoms possess different velocities at different temperatures, it is important to consider their velocities at the same temperature. Rms velocity is the square root of the average of the square of the velocities possessed by the atoms in a molecule at the same temperature. Both the rms and the most probable velocities depend on the temperature, molar gas constant and the molar mass of the given gas. The root mean square velocity is the greatest of both average velocity and most probable velocity.

Note:
While calculating the most probable velocity, the speed of all atoms in the molecule must be taken at the same temperature. As the electrons in the atoms vibrate more and their energy increases, their velocities also increase. Therefore, the velocities of the atoms will increase as temperature increases. Thus, the noted values must be taken at the same temperature.