Maxwell-Boltzmann Distribution
The distribution of speeds in an ideal gas
obeys the Maxwell-Boltzmann distribution.
The probability of finding a molecule with a particular speed v is given
by:
| P(v) |
= 4π |
( |
| m
|  |
| 2πkT
|
|
) |
3/2 |
v2 e-mv2/2kT
|
where m is the mass of a single molecule of the gas.
Basic features of the MB distribution:
- Low temperature → narrow distribution;
high temperature →
wide distribution.
- Low temperature → peak close to v=0;
high temperature →
peak at large v.
- It is very, very unlikely to find very fast molecules.
- It is also unlikely to find very slow molecules.
- Most molecules have speed near the peak of the distribution.
The distribution is characterized by three speeds. These are:
| The most probable speed:
vmp |
= |
( |
| 2kT
| |
| m
|
|
) |
½ |
| The average speed:
vav |
= |
( |
| 8kT
| |
| πm
|
|
) |
½ |
| The root-mean-square speed:
vrms |
= |
( |
| 3kT
| |
| m
|
|
) |
½ |
All these characteristic speeds scale as (kT/m)½
≈ 500 m/s.