As part of our training to become teaching assistants, UI assigns each graduate student a section to teach from Zumdahl & Zumdahl’s general chemistry text. We’re supposed to stand up in front of the other graduate students and give a five-minute presentation on our subject…with the implication that this little exercise will help prepare us for teaching hordes of ravenous, grade-hungry freshmen and sophomores. Many are saying the whole situation is “a lot of hot air,” which brings me to my assigned topic, real-life corrections to the ideal gas law.
Johannes Diderik van der Waals, known affectionately to his friends as “J.D. van der Wizzle,” was the first chemist to bring corrections to the gas law into the mainstream. He ended up winning the Nobel Prize for his efforts, and although his improved gas law doesn’t hold up to intense quantitative scrutiny, the qualitative considerations that led him to his results were spot-on. And his results were certainly better than those provided by the ideal gas law alone.
To progress from the ideal gas law to a more accurate equation, all you really have to do is look at the kinetic molecular theory of gases (characteristics of an ideal gas), say “that’s wrong,” and look at the resulting effect on a “real” gas’s pressure, volume, and temperature. For instance, start with the ideal assumption that gas molecules take up no volume. Throwing that out the window, we must admit that gas molecules take up volume–and that the actual volume of empty space available to each individual molecule is thus less than the entire volume of the container. All the molecules are the same size, and the number of them is proportional to n, so the volume taken up by the actual gas molecules is proportional to n. The effective volume available to each gas molecule is thus V–nb, where V is the volume of the container and b is empirically determined. Thus, we can correct the IGL slightly by putting it in this form:
P(V–nb) = nRT
This says that when pressure and temperature are constant, adding moles of gas to a container increases its volume faster than we would expect according to the kinetic theory because, well, molecules have volume.
The KMTG also states that pressure is the result of gas molecules striking the sides of their container, and that they do not interact with each other at all. Assuming that gas molecules do interact with each other leads to the conclusion that collisions of the molecules with the walls are opposed and slightly lessened by gas-gas interactions. The extent of interactions depends on the concentration of gas molecules, n/V. The dependence is actually square for reasons that would take too long to explain here, and so the pressure predicted by the ideal gas law is essentially P+a(n/V)2, where P is the observed, “real” pressure. Putting it all together,
[P+a(n/V)2](V–nb) = nRT
Voila! The van der Waals equation of gases.