You may see some effects as shown in your graph, but I believe the deviations that are shown are greatly exaggerated. I'd be curious to know what the source of the graph is, and if these substances do, in fact, show this behavior. ## \\ ## Editing...Presumably the curve is for constant temperature, and what it is showing is basically a huge drop in the expected volume as the pressure is increased, but then as the pressure is increased further, the volume is actually larger than expected from the ideal gas law. The Van der Waals equation (## (P+an^2/V^2)(V-nb)=nRT ##), as opposed to the ideal gas equation, can serve to account for slight differences from the ideal gas law, and the forces accounting for the variations in the pressure part of the equation are often referred to as van der Waals forces. Perhaps if you solve for ## PV ## in this equation and graph it vs. ## P ##, (for constant ## T ##), if you have a significant ## a ## parameter from the molecule, you might get behavior as shown in your graph, but certainly not to the extent that is shown. ## \\ ## Additional editing: A little algebra on the Van der Waals equation gives ## PV=nRT-an^2/V +Pnb+an^3 b/V^2 ## and since ## PV ## is approximately equal to ## nRT ##, we can call ## 1/V=CP ## for some constant ## C ## (to a good approximation). Let ## nRT=G ## for some constant ##G ##, and we have ## PV=G-an^2CP+nbP+an^3 b C^2 P^2 ## with ## n ## a constant and ## a ## and ## b ## also constant for a substance. If ## a ## is large enough, the ## -an^2CP ## term could be larger than the ## nbP ## term and could dominate the deviation of ## PV ## from ideality (of being equal to ## G ##) before the ## an^3 b C^2 P^2 ## term takes over for large ## P ##. Perhaps this is what the graph you have is attempting to illustrate. For Helium, the attractive forces between molecules (which is what the ## a ## term represents) are very small, (so that ## a ## is very small), and any deviation from ideality is dominated by the ## nbP ## term. The ## b ## term in the Van der Waals equation results from atoms or molecules having a finite size. With a very small ## a ##, the ## an^3 b C^2P^2 ## term most likely is small for Helium for large ## P ##. Alternatively, for ## CH_4 ##, the attractive forces between molecules is apparently quite significant, accounting for a sizable ## a ## parameter, but certainly not to the extent depicted by the graph. ## \\ ## One additional comment: For complete mathematical rigor, because we ultimately looked at the ## P^2 ## term in the analysis, we should really write ## 1/V=CP+DP^2+...## in an iterative solution for ## 1/V ##. In a slightly simplified analysis, we assumed ## D ## was negligibly small, but with complete mathematical rigor, we should have kept the ## DP^2 ## term in the algebra that followed.
