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.

Maxwell-Boltzmann Speed Distributions

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:

These three measures of v are not equal because the distribution is not symmetrical about its peak. Such is the case because the lowest possible speed is zero, while the highest is classically infinite. For this reason, the peak of the distribution (v_p) will always be to the left of the average speed (). The difference between v_p and v_rms is even more exaggerated because it involves the mean of squares.

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 total area underneath the Maxwell-Boltzmann speed distributions is equal to the total number of molecules. If the area under the two curves is equal, than the total number of molecules in each distribution are equal.

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 H₂ is much broader and faster than the distribution of a heavier molecule like O₂:

Diffusion and Mean Free Path

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:

N is the total number of molecules present. The rate of collisions is simply v_rms divided by the mean free path:

Effusion

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: