In this section we shall discuss several more formula as well as a few interesting definitions. But first, we shall begin by considering a hypothetical situation. Imagine yourself staring at a plot of the states of a gas, say Carbon Dioxide. Of course, starting with a gas, you appropriately picture a dot on the graph where you begin. If you instantly raise the pressure while holding the temperature constant, what happens? That's right, you soon get a liquid. But don't do that yet! If you did, start over with the dot as a gas. Now raise the temperature while holding the pressure constant. What happens? That's right again, seemingly nothing. There's no where to go. Keep doing that for a while, raising the temperature while holding the pressure constant, until you hit Tc. You remember Tc, the critical temperature. We talked about him early? Good, I'm glad you do. Because he's important now, so pay attention! Once you reach Tc it's safe to raise the pressure, so do so. That's it, take it up nice and high until you reach Pc. Then feel free to drop the temperature slowly until you reach the original one. Now gently lower the pressure until your plot tells you you have a liquid. Now ask yourself what the heck happened because that ain't no liquid in your closed tube.
What you just discovered is called a supercritical fluid. It is neither gas nor liquid. These are widely used as solvents because they can be quite sensitively "fine-tuned;" adjusted to meet exactly what's needed so far as pressure, temperature, polarity (through addition of other substances) and the like. Plus, they're good because you don't just jump straight to a liquid from a gas (which is messy) and this way is a whole lot cooler. There will eventually be a link at the bottom of this page leading to a graphical demonstration of this (so you don't have to use your TV-addled imagination), but for now this is all there is. And that's all there is to say specifically on supercritical fluids for now.
Now let us skip merrily ahead to look at energy. Specifically, let's look at potential energy as a function of intermolecular distance. As one gas molecule approaches another from infinite, its potential energy slowly decreases (like when an eraser approaches earth from above). However, eventually this "bottoms out" to a minimum potential energy and potential energy becomes quite high... because one atom has just intruded on another's space. If you continue getting closer, the atoms continue to push apart leading to infinity along the Y-axis. This graph too will be available shortly from a lower link. For now, though, just rely on this description and the following explanation: Potential Energy as a function of the distance is equal to:
4*epsilon*[ (sigma/r)^12 - (sigma/r)^6 ]
where sigma and epsilon are dependant on the molecule in question. (and yes, this equation too will be scanned in legibly in a below link). This equation, is by the way, called the Lennard-Jones Equation.
This graphical question and the existance of this equation bring up a question... how do we determine the minimum potential energy of a molecule? Well, we take the derivative of the equation and set it to zero. In doing this we discover that potential is at a minimum when
r = the sixth root of two * sigma, or when potential equals negaitve epsilon.
This solution works very well for non-polar gases especially and is called the 12-6 potential (also the Lennard-Jones potential).
Now comes the big fun part in our discussion of Real Gasses: The Kinetic Model of Gases. By making a few assumptions and taking what we know and throwing them all into a blender, we can come up with the following:
A gas is made of molecules
Gas molecules are in continuous random motion
The volume taken up by the molecules is much less than the total free volume
The only energy transfers occuring are in elastic collisions (here we assum that there are no intermolecular forces, and hence no potential energy).
Finally, we assume that everything can be described using Newtonian physics.
Now, let's get busy!