Ludwig Eduard Boltzmann
(influenced by the work of James
Clerk Maxwell) developed the kinetic theory of matter to
predict the distribution of speeds of molecules in a gas to be expected at a
given temperature. 

This work suggested that
if we plot a graph of the number of
molecules, N, which have speeds in narrow ranges, from v to v+Δv, against speed,
v, then we should find the following distribution. 



The results are based on the assumption that
the molecules move independently of each other except for collisions
between molecules which have a duration much less than the time
between collisions. 

v_{o} corresponds to the "peak" of
the curve, called the most probable speed. 

If we take a narrow
range of speeds centered on v_{o} we should find that this
range has the greatest number of molecules. 

v_{m} is the mean (or average) speed
of the molecules (a little greater than v_{o} as the curve
is not symmetrical). 





As the temperature of
the gas increases, both the range of speeds and the mean (average)
speed increases. 

Therefore, for the same quantity of gas, at
a higher temperature, the distribution is as shown by the
red curve. 

These predictions have been verified by
experiment, thus giving support to the kinetic theory. 



Some approximate average speeds (actually "root mean square"
speeds) for molecules of different gases at different temperatures
are given in the table below. 

The speeds are in ms^{1}. 

Gas 
273K 
373K 
H_{2} 
1.81×10^{3} 
2.10×10^{3} 
He 
1.30×10^{3} 
1.52×10^{3} 
N_{2} 
4.90×10^{2} 
5.73×10^{2} 




Notice that, for a given temperature, the
lower mass molecules have much higher average speed 



Notice also that these figures illustrate the
fact that absolute temperature of the gas is proportional
to the average kinetic energy (which, remember, is
proportional to speed squared) of its molecules,
as shown below: 



Compare this figure with the ratios of the
speeds squared for each type of gas 


