Obviously, the Ideal Gas Law is not followed in all circumstances. If it were it would not be the "Ideal" Gas Law, only the Gas Law. But when then do gases not follow the law? And how may we successfully understand their behaviour in these times? The answer lies upon staunch analysis of real gases (as opposed to the fake kind).
For one mole of gas, (P * V) / (R * T) = 1 if the gas is ideal when n = 1. However, let us introduce a new variable, Z(P) such that Z, called the compressibility factor, is a function of the pressure. If T is set constant and (P * V) / (R * T) = Z(P) then, the graph of (P,Z(P)) is a horizontal line. This means, of course, that an ideal gas has Z equal to one and the graph is a straight line horizontal at unit one. However, not all gases are ideal.
At higher pressures, strong deviations from the ideal gas are seen. These deviations are higher when Z is further from one, or when P is high and T is low (as V is held constant). The deviation of the gases from zero to four hundred atms is due largely to intermolecular forces, which hold the molecules togeather more than in an ideal gas. Above four hundred atms, however, the dominant force causing deviation is the size of the molecule in question. The larger the molecule, the bigger the deviation, just like the stronger the intermolecular forces, the bigger the deviation. Both of these factors are due to molecular freedom of motion. At high pressures the larger molecules are too large to move freely without crashing into a neighbor. And at lower pressures van der Waal's forces prevent purely independent motion by individual molecules.
Now set V(bar) equal to the volume per mole. P * V (bar) = R* T * (1 + BP + CP^2...) where B and C are functions of the temperature and dependant on the gas in question. This is an example of a virial equation, and virial equations, as we shall see, are very good for non-ideal gases.
The next important piece of P-chem conversation involves critical temperature (Tc), critical pressure (Pc), critical volume (Vc), and reduced variables. Imagine putting a liquid in a clear closed tube. Don't ask me how you got the liquid into a closed tube, but imagine that you did so and that you filled it one half the way up or so. Above the liquid, say water, is gas. At a high pressure, it is possible to liquify the gas. But the higher the temperature, the higher is the pressure necessary to liquify the gas. Until the temperature hits Tc, of course (and correspondingly Pc, Vc, etc). Once Tc is exceeded, the phase boundary ceases to exist between the liquid and the gas; in other words no miniscus is visible and it is difficult to tell liquid from gas near the surface. Now Tc is different for every gas, but if you divide the actual temperature of the tube by the Tc of the gas in the tube (known from experimentation and observation), you get what is called a reduced variable, in this case T-reduced (Tr). By doing the same thing with pressure and V(bar), three reduced variables, Tr, Pr, and V(bar)r are arrived upon. When you plot one of these variables opposite the reduced compressibility factor, (Zr, Xr), a non-ideal graph appears. This is precisely what we want, as it turns out that all gases obey this or a similar graph. The reduced variables, then, provide a method of augmenting our knowledge of non-ideal graphs and provide an avenue for non-ideal gas behaviour prediction.
This is a good approximation, but there are better ones. Consider for a moment what happens to a gas at high pressure. If the gas is at a low pressure, the volume of the container is essentially the volume that the gas molecules have to move around in, as they are fairly spread out. However, upon compression the volume of the gas molecules themselves may (and often does) become significant; in otherwords the effective volume (or the area the gas molecules are free to wander in) is lessened due to the sheer bulk of compressed gases. Because of this PV = RT need modified slightly, to P * ( V - Vof factor) = RT. But also at high temperatures the molecules are forced closer togeather, which allos for more and stronger intermolecular forces. These forces hold the molecules loosely togeather. The forces slow the molecules doewn. But pressure is based upon molecules randomly crashing into surfaces. So if the molecules crashing into surfaces are slowed by intermolecular forces, then the pressure created by these molecules is lessened and PV = RT again needs correction, such that (P + ?)*(V-Vfactor) = RT. This looks like a messy equation, and you're right, it is, so a gentleman by the name of van der Waal's worked it out. He discovered:
(P + (a*n^2)/V^2) * (V - n*B)= nRT, where a and b depend on the particular molecule and are determinable only through experimentation. Through use of this equation the size of the molecule in question can be determined (through calculation of V(bar) at high P) and also the behaviour of a non-ideal gas can be more accurately determined.
That is all for this lecture, stay tuned for more. Thank you and goodnight.