MaxwellBoltzmann Distribution
The distribution of speeds in an ideal gas
obeys the MaxwellBoltzmann distribution.
The probability of finding a molecule with a particular speed v is given
by:
P(v) 
= 4π 
( 
m
 
2πkT


) 
^{3/2} 
v^{2} 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:
v_{mp} 
= 
( 
2kT
 
m


) 
^{½} 
The average speed:
v_{av} 
= 
( 
8kT
 
πm


) 
^{½} 
The rootmeansquare speed:
v_{rms} 
= 
( 
3kT
 
m


) 
^{½} 
All these characteristic speeds scale as (kT/m)^{½}
≈ 500 m/s.