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.
You'll often see the range of speeds plotted against the number of molecules on a Maxwell-Boltzmann speed distribution. Plotting the values of , v _{p} , and v _{rms} , we find that:
v _{p} < < v _{rms} |
A Maxwell-Boltzmann speed distribution changes with temperature. As discussed with the kinetic molecular theory, higher temperatures lead to higher velocities. Thus the distribution of a gas at a hotter temperature will be broader than it is at lower temperatures.
The Maxwell-Boltzmann speed distribution also depends on the molecular mass of the gas. Heavier molecules have, on average, less kinetic energy at a given temperature than light molecules. Thus the distribution of lighter molecules like H _{2} is much broader and faster than the distribution of a heavier molecule like O _{2} :
Diffusion is the spread of one substance through another. The fact that molecules collide when they diffuse is the reason why it takes a considerable amount of time for a gas to travel from one place to another. Think, for example, of a smell released at one point in the room. Because gas molecules move at such fast velocities, if there were no collisions the smell would fill the room instantly.
The collision between gas molecules makes the calculation of the rate of diffusion difficult. Instead, we'll focus on the mean free path. The mean free path λ is the mean distance a molecule travels before it impacts another molecule; given the huge number of collisions in a gas, the mean free path is vastly smaller than any typical room or container. The mean free path is calculated with the following formula:
λ = |
rate of collision = |
Effusion is the rate at which a gas passes through a small hole into a vacuum. The rate of effusion of a gas is directly proportional to v _{rms} :
= = |