# 5.7 Kinetic Energy And Molecular Speeds

Molecules of gases remain in continuous motion. While moving they collide with each other and with the walls of the container. This results in change of their speed and redistribution of energy.

So the speed and energy of all the molecules of the gas at any instant are not the same. Thus, we can obtain only average value of speed of molecules.

If there are n number of molecules in a sample and their individual speeds are u1, u2,…….un, then average speed of molecules uav can be calculated as follows:

uav = (u 1 + u 2 + .........u n) / n

Maxwell and Boltzmann have shown that actual distribution of molecular speeds depends on temperature and molecular mass of a gas. Maxwell derived a formula for calculating the number of molecules possessing a particular speed.

Fig. shows schematic plot of number of molecules vs. molecular speed at two different temperatures T1 and T 2 (T 2 is higher than T 1 ) . The distribution of speeds shown in the plot is called Maxwell-Boltzmann distribution of speeds. The graph shows that number of molecules possessing very high and very low speed is very small. The maximum in the curve represents speed possessed by maximum number of molecules .

This speed is called most probable speed, ump. This is very close to the average speed of the molecules . On increasing the temperature most probable speed increases. Also, speed distribution curve broadens at higher temperature.

Broadening of the curve shows that number of molecules moving at higher speed increases. Speed distribution also depends upon mass of molecules. At the same temperature, gas molecules with heavier mass have slower speed than lighter gas molecules .

For example, at the same temperature lighter nitrogen molecules move faster than heavier chlorine molecules. Hence, at any given temperature, nitrogen molecules have higher value of most probable speed than the chlorine molecules.

We know that kinetic energy of a particle is given by the expression:

Kinetic Energy = (1/2)mu2

Therefore,  if  we  want  to  know  average translational kinetic energy, for the movement of a gas particle in a straight line, we require the value of mean of square of speeds, of all molecules.

The mean square speed is the direct measure of the average kinetic energy of gas molecules. If we take the square root of the mean of the square of speeds then we get a value of speed which is different from most probable speed and average speed. This speed is called root mean square speed and is given by the expression as follows: 