Maxwell’s Distribution of Velocities and Energies
Maxwell’s Distribution of Molecular Velocities
Actually, as a result of collisions, a redistribution of both velocity and energy takes place.
• But for convenience, it was supposed that all molecules in a given gas at a given temperature were moving with a constant root-mean-square velocity u.
By utilizing probability considerations, Maxwell and Boltzmann, have, in fact, shown that
• the actual distribution of molecular velocities depends on the temperature and molecular weight of a gas, and is given by the equation :
• This equation is known as the Maxwell-Boltzmann distribution law for molecular velocities.
where,
• dn = the number of molecules out of a total molecules n having velocities between c and c+dc,
• M = molecular weight of the gas &
• T = temperature of the gas.
where p denotes the fraction of molecules
Fig.1: Distribution of Molecular Velocities
It is clear from the above plots of p vs c that
• the probability of a molecule with zero velocity is very small.
• Further, the probability of molecules with velocities greater than zero increases, passes through a maximum, and then falls rapidly towards zero.
The important features are:
1. The fraction of molecules with too low or too high velocities is very small,
2. Most probable velocity➠There is a certain velocity for which the fraction of molecules is maximum viz., called most probable velocity.
• Thus it is the velocity which is possessed by maximum number of molecules of the gas at a given temperature.
• The most probable velocity is in any gas not a constant, but shifts towards higher values of c with increase in temperature; i.e., at higher temperatures higher velocities are more probable than at low.
• Mathematical analysis shows that the most probable velocity, 𝛼(alpha), is not equal either to the root- mean-square velocity u or the average velocity of all the molecules v.
Effect of Temperature on Maxwell’s distribution of velocities :
It is clear from the curves of Fig.1 that
• On increasing the temperature ,the peak of the distribution curve shifts downwards and the curve is flattened.
• On increasing the temperature, the entire curve shifts more and more towards the right.
• On increasing the temperature, the most probable velocity (𝛼)increases.
• On increasing the temperature, the fraction of molecules possessing the most probable velocity (𝛼) decreases.
• At higher temperatures higher velocities are more probable than at low temperature.
Maxwell’s distribution of energies
According to Maxwell’s distribution of velocities:
As we know, for any specific velocity c the kinetic energy of a gas per mole is E = M c² / 2
substituting this relation in the above equation, the distribution of kinetic energies in a gas is obtained as:
where,
Now dividing above equation by dE, we get:
• where p’ =probability for incidence of kinetic energies of translation of magnitude E.
• Note- The plots of p’ vs E obtained from this equation are very similar to those of p vs c as shown in Fig.1
