Question

Define most probable speed, average speed and root mean square speed. How are they related to each other?

Answer 1

Hint: The Maxwell-Boltzmann equation is based on the theory of kinetic gases. It basically describes the speeds of a gas at a certain temperature in an easy manner. We can say that it gives us a graph between speed i.e. fraction of molecules w.r.t temperature.

Complete answer:
Root mean square speed which is abbreviated as RMS value is the square root of the mean of squares of speed of individual gas molecules whereas RMS of gas molecules can be given by the formula $\sqrt{{\displaystyle \frac{3RT}{M}}}$ where R = molar gas constant, T = temperature in kelvin, M = Molar mass of the gas. So root mean square speed is shown by the formula
$v=\sqrt{{\displaystyle \frac{3RT}{M}}}$, i.e. square root of speed of gas molecules.
Average speed is defined as the arithmetic mean of the mean of the speeds of different molecules of a gas at a given temperature which can be represented by the formula as follows:
$u=\sqrt{{\displaystyle \frac{8RT}{\pi M}}}$
Whereas the most probable speed is given by $a=\sqrt{{\displaystyle \frac{2RT}{M}}}$ and defined as the speed which corresponds to the peak of the curve.
Hence in this way we can define most probable speed, average speed and root mean square speed and relation can be shown as:
a : v : u = 1 : 1.128 : 1.224

Note:
From Maxwell-Boltzmann distribution curve we can also find the values for most probable speed, average speed and root mean square speed. Speed of an object defines the rate of change of position with respect to time and speed is a scalar quantity. The SI unit of speed is metre per second.