Kinetic-Molecular Theory and Effusion
Gases are in constant motion: smells slowly spread through rooms, smoke molecules drift even without wind, and pollutants scatter through the atmosphere. Even if the air is totally still, the continuous and random motion of gas molecules in a mixture will eventually result in an even distribution of gases in any sized container. This phenomenon is called diffusion, and it depends on the speed of the gases involved. The speed of a gas, its kinetic energy, and its temperature are related in a description of the microscopic behavior of gases, called the kinetic-molecular theory. There are four main points in the kinetic-molecular theory:
1. Gas molecules are separated by large distances, relative to the size of the molecules themselves
2. Gas molecules are continuously moving at high speed and in random directions
3. The average kinetic energy of a gas is determined only by the temperature; all molecules have the same average kinetic energy regardless of their mass (speed may differ between molecule types)
4. Collisions between gas molecules or between a molecule and the container occur without loss of energy (they are elastic)
In addition, the "ideal gas" has no intermolecular attraction and its molecules occupy zero volume.
It is the failure of these laws at high pressure and low temperature that led to the formation of the van der Waals equation. However, at normal conditions, these laws are close enough to reality that they can be used with no significant penalty in accuracy.
The average kinetic energy of a gas depends on its temperature; the higher the temperature, the faster the gas molecules are moving, and therefore the more kinetic energy they have. We can say that the kinetic energy of a gas is proportional to its temperature. Using our friend the proportionality constant (C), we have:
KE = CT
If you have taken physics, you know that the kinetic energy of a molecule is also equal to 1/2 times its mass, times its speed squared:
KE = 1/2mu2
However, each molecule will have a slightly different velocity (u); we're interested in the average speed, since we're considering the entire gas. An average is indicated by placing a bar over the averaged factor; since this is hard to represent in HTML, we'll underline the averaged factors; but be aware that when written out as normal, these factors will have bars over them, not under.
KE = 1/2mu2
Finally, we will set these two equations equal to each other:
1/2mu2 = CT
Now, the kinetic energy for all gas molecules must be the same at a given temperature; however, more massive gas molecules must have lower speeds and lighter molecules greater speed in order to have the same kinetic energy. An equation relating molecular mass, temperature, and average speed was derived by James Clerk Maxwell, who also did important work in electromagentic science:
(u2)1/2 = (3RT/M)1/2
where u is the average, or mean, velocity of the gas' molecules in meters per second (the term (u2)1/2 is often called the root-mean-square speed, or rms speed); R is the ideal gas constant (8.3145 J/K * mol); T is the temperature in kelvins, and M is the molar mass in kilograms per mole (kilograms are needed to make the units cancel properly). This equation is called Maxwell's equation in his honor.
Let's take this equation our for a spin... Say we have a nitrogen molecule, N2, at room temperature, 298 kelvin. How fast is the average nitrogen molecule moving? Using Maxwell's equation, simply plug in the temperature and molar mass:
rms speed = ((3 * 8.3145 J/K * mol * 298 K)/0.028 kg/mol)1/2 = 515.238 (J/kg)1/2
Well, our answer's in joules per kilogram, not meters per second! But fear not--one joule is equivalent to 1 kilogram times meters squared per second squared, so when the kilograms cancel and the units are square-rooted, we get the familiar meters per second; in this case, the average nitrogen molecule is moving at:
515.238 (m2/s2)1/2 = 515.238 m/s (or about 1150 miles per hour!)
Another type of gaseous motion is called effusion, in which a gas or mixture of gases pass through a tiny hole into another low-pressure container. A Scottish chemist named Thomas Graham studied the movement of effusing gases in the nineteenth century, and he discovered that the rates of effusion of two compounds were inversely proportional to the square root of molar masses:
(Rate of gas 1 / Rate of gas 2) = (Molar mass of gas 2 / Molar mass of gas 1)1/2
This equation, not surprisingly, is named in Graham's honor; it is called Graham's law. The units of rate are irrelevant, as long as they're the same for both gases, since they cancel in the division.
The important thing to note about this law is that it allows us to find the molar mass of an unknown gas by comparing its rate of effusion to that of a known gas. For example, let's say a sample of helium gas passes through an effusion barrier at the rate of .00067 liters per hour; the unknown gas effuses at .00015 liters per hour. What is its molar mass? We'll set up Graham's law as follows:
(.00067/.00015) = (M / 4)1/2
Some quick division and squaring yields a molar mass of 79.8.
This concludes our discussion of gases. The remainder of the chapter will focus on solutions and the equilibrium between the three states of matter.