Kinetic Molecular Theory

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.

@@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 = 1/2mv², 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:

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 insidious--it must be in kg/m³ if all the other units are SI.