Ideal Gases and the Kinetic Theory

We have seen in previous notes that the gas equation:

PV = nRT

was pretty much in its modern form by 1860 with the inclusion of Avogadro's hypothesis. The remarkable success of this simple equation prompted several scientists to attempt to derive this equation from a set of simple assumptions. The workers primarily responsible for this were James Clerk Maxwell, Ludwig Boltzmann (1844 - 1906), and Rudolph Clausius (1822 - 1888). These scientists began by establishing a set of assumptions or hypotheses concerning the nature of matter. By the mid 1800's the ideas of atoms and molecules seemed less strange than they did in 1800 so the assumptions built upon imagining a gas to consist of particles (atoms or molecules).

The hypotheses can be summarized as:
• A gas is made up of a large number of particles (atoms or molecules) whose size is negligible compared to the size of the container. These particles are also small compared to the distances between particles.
• The molecules are in constant, random motion. The distribution of kinetic energies in the gas are described by the Maxwell-Boltzmann equation.
• The attractions and repulsions between the particles are negligible (i.e., small or nonexistant intermolecular forces)
• Collisions between molecules can transfer energy, but the total average energy of the collection of molecules remains constant. The collisions are "elastic" meaning that particles bounce off of one another rather than sticking.
• The average kinetic energy is proportional to the temperature (in K) of the gas.

Note that the nature or kind of gas does not enter into these equations so the assumptions should be true whether we're talking about oxygen or sulfurhexafluoride.

Using these assumptions, Maxwell, Boltzmann, and Clausius developed mathematical models of the behavior of these gases. We have already discussed the subject of pressure and how pressure arises from the collision of a gas particle with the sides of a container. Pressure arises from the transfer of momentum of the particle as it changes direction at the wall.

Examining the assumptions made above, it should be obvious that the ideal gas law (PV = nRT) should break down at some point. If the gas is squeezed into a very small volume (or alternatively high pressure) where the volume of the molecules themselves is no longer negligible or if the gas is at a sufficiently low temperature so that the intermolecular forces become significant compared to the kinetic energy, we might expect the equation to fail. And it does. To deal with this we must have a more complex set of gas laws called real gas equations. The simpliest of these equations is the one devloped by the Dutch chemist Johannes van der Waals and is discussed in the next notes.