# Deriving a Real Gas equation

I've read about the ideal gas law and the van der Waal's gas law. I know that the ideal gas law is horribly inaccurate for temperatures close to, but above the critical temperatures, so it can only be used for permanent gases like nitrogen, oxygen, etc. The van der Waal's equation is a little better in this regard, but also fails for temperatures below the critical one, notaby by the "van der Waal's loops".

I looked into how the van der Waal's equation and how it was derived. Reading the Wikipedia page about it, I find that van der Waal's assumed that gas molecules are hard incompressible objects that have a certain volume (evident by the constant $$b$$, defined as the volume occupied by one mole of molecules themselves), and these molecules attract each other (I couldn't find how these molecules attracted each other, like perhaps inverse square or cube?). I find the rigid volume assumption a little too generous. I also know that molecules attract each other at long ranges due to London dispersion forces, but these dispersion forces also act as repulsions when the molecules get too close. We can describe this by the potential energy of the two molecules:

Differentiating this curve (a.k.a Morse potential) should give us the force between two molecules. Now what we have is a single expression for both attraction and repulsion, rather than one for attraction and another one for repulsion (as hard spheres, in van der Waal's attraction). This single expression of attraction and repulsion gives the molecules some "squishy" nature which I believe is how real gases behave.

Naturally, I want to derive a gas equation using the Morse potential graph, rather than how van der Waal's has assumed. This is where I hit my roadblock. So far I have compared the derivation of the van der Waal's equation, and I have tried to make modifications as per the Morse potential's force law. Here are some conclusions I came to:

1. The original van der Waal's equation is $$(P+\frac{a}{V^2})(V-b)=RT$$ I believe the new equation that I plan to derive will not have any volume correction parameter, since with our assumption of how molecules are "squishy", there is no correct volume correction.

2. I plan to incorporate repulsions into the pressure correction factor as well. My equation should be of the form $$(P+K_0)V=RT$$ where $$K_0$$ is a factor that corrects pressure due to both attractions and repulsions.

I looked at the derivation of van der Waal's and found this

Next, we introduce a (not necessarily pairwise) attractive force between the particles. van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous; furthermore, he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size.[citation needed] Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles, which is not true, as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density

$$C=\frac{N_A}{V_m}$$

The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by

$$a'C^2 = a'\left(\frac{N_A}{V_m}\right)^2 = \frac{a}{{V_m}^2}$$

So we can see that the pressure correction factor is described by considering the force experienced by the particle is proportional to the number density of the particle. This seems well and good for the van der Waal's equation, since it only considered attractions. In my case, I dont think it's so simple. Molecules closer to the molecule in consideration would repel it, and those further would attract it. I have no idea on how to incorporate this in terms of number density, even though I believe it's possible (denser gas would mean more attractions as well as more repulsions, and depending on the Morse potential is what would decide if actual pressure is more or less). This where I need help. How do I incorporate the Morse potential curve into the pressure correction factor?

• Have you reviewed also other real gas models ? en.m.wikipedia.org/wiki/Real_gas – Poutnik Aug 18 at 17:23
• @Poutnik I have. But I wish to see if what I've written above can help me get an equation. I dont want the "best" gas equation, but hopefully just one that's more accurate than the van der Waal's equation. – Pritt Balagopal Aug 19 at 1:51