Hi!!
Plz explain the topic- "MAXWELL DISTRIBUTION OF MOLECULAR SPEEDS".
In the context of the Kinetic Molecular Theory of Gases, a gas contains a large number of particles in rapid motions. Each particle has a different speed, and each collision between particles changes the speeds of the particles. An understanding of the properties of the gas requires an understanding of the distribution of particle speeds.
The Maxwell distribution describes the distribution of particle speeds in an ideal gas. The distribution may be characterized in a variety of ways.
Average Speed
The average speed is the sum of the speeds of all of the particles divided by the number of particles.
Most Probable Speed
The most probable speed is the speed associated with the highest point in the Maxwell distribution. Only a small fraction of particles might have this speed, but it is more likely than any other speed.
Now max wellian function is a probability function given as:
- i is the microstate .
- Ei is the energy level of microstate i.
- T is the equilibrium temperature of the system.
- gi is the degeneracy factor, or number of degenerate microstates which have the same energy level
- k is the Boltzmann constant.
- Ni is the number of molecules at equilibrium temperature T, in a state i which has energy Ei and degeneracy gi.
- N is the total number of molecules in the system.
we can solve for the velocity function too:
where the distribution for a single direction is:
Usually, we are more interested in the speeds of molecules rather than their component velocities. The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is
and the increment of volume is
where θ and ϕ are the "course" (azimuth of the velocity vector) and "path angle" (elevation angle of the velocity vector). Integration of the normal probability density function of the velocity, above, over the course (from 0 to 2π) and path angle (from 0 to π), with substitution of the speed for the sum of the squares of the vector components, yields the probability density function
for the speed. This equation is simply the Maxwell distribution with distribution parameter :
