Kinetic Molecular Theory
The most immediately useful bit of information you can pull from the definition
of the kinetic molecular theory provided in the
summary is that the average kinetic
energy
of a gas is proportional to the absolute temperature.
âàùT 

@@Equation @@ has a number of very serious implications. First of
all, any two gases at the same temperature will have the same kinetic energy.
Remember that kinetic energy E_{k} = 1/2mv^{2}, and that average kinetic energy
= 1/2m.
Here's where things get complicated. After some mathematical maneuvering we
find a more exact expression for the average velocity $\overline{v}$:
k is the Boltzmann constant. Think of the Boltzmann constant as the gas constant R for individual molecules. Analogously,
m is the mass per
molecule, just as
M is the mass per mole. If you multiply
k by Avogadro's
number, you'll get
R.
Let's take a breather. In order to keep things simple, I have refrained from
including derivations. If you are at all mathematically inclined, however, I
suggest that you take a look in a good physics book (look under statistical
mechanics or ideal gases) at the derivations of and the other
equations I introduce. Alternatively, ask your instructor to show you. The
derivations can be painful, but they will prove to you that these equations have
meaning.
Let's get back into the fray. There are two other characterizations of v that
you should know: the most probable velocity $v_p$ and the root mean
square velocity $v_{\mbox{rms}}$. The most probable velocity is exactly
what it sounds like: the velocity at which the greatest number of molecules in a
gas travel. It can be expressed mathematically:
v_{p} = = 

The root mean square velocity, which measures the typical velocity of molecules in a gas, is slightly tricky. To derive its value, find the square root of the mean of the squares of the average velocity. It is easier to understand
mathematically:
Make sure that you see that
v_{rms} = , NOT
v_{rms} = . The latter equation reduces to
v_{rms} = , which is not the case.
v_{rms}
requires the mean of the squares of the velocities. Square the velocities
first, then take their mean.
When solving for these values of v, be sure to reduce all variables to
SI units. M is particularly insidiousit must be in kg/m^{3} if all the
other units are SI.