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.
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@@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 $\overline{v}$:
= = |
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} = = |
v_{rms} | = | ||
= | |||
= |
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/m^{3} if all the other units are SI.