# Is this equation of state original?

I created this equation of state as a personal undergraduate summer project (here is the link) seven years ago:

$$\frac{PV}{nRT}-1 \propto \left(\frac{nRT}{PV}\right)^2 \left[\left(\frac{nRT}{PV}\right)^2 - 1\right]$$

I tried to show it to my professors but none of them replied to my emails. I had since forgotten about it until I recently came across it in the cloud. Here is a table comparing its theoretical predictions for the critical temperatures of various gases to the actual experimental values • How do you use it in order to make predictions of $T_c$? – user1420303 May 29 '16 at 12:08
• @user1420303 I'm guessing $$\left(\frac{\partial p}{\partial V}\right)_T = \left(\frac{\partial^2 p}{\partial V^2}\right)_T = 0$$ although the link says $T_c = \varepsilon/(k \ln 2)$ (not sure how the derivation works) – orthocresol May 29 '16 at 12:23
• @orthocresol Sorry, I was not able to open the link before, it took me to the table image with the SE App . I am going to read it from my PC in minutes. – user1420303 May 29 '16 at 13:59
• @user1420303 It was broken just now, I fixed it. ;) – orthocresol May 29 '16 at 14:27
• I think it is original, although I'm not sure about some points in the derivation. – user1420303 May 29 '16 at 17:27

Extending beyond the question of originality, I read your demonstration and what bothers me to some extent is the link you establish between $Z$ (the compression factor) and $U(r)$, the intermolecular potential energy. First, let's state that linking $Z$ to $U$ is indeed a common way to derive equations of state. However, the way you are doing it is lacking a sound basis. The sentence “From these explanations it can be said that $Z-1=U(r)$” (page 1) is the problem:
1. Yes, if $U(r)>0$ then $Z>1$, and the other way around… but that does not mean that the two quantities are proportional.
2. In fact, $Z$ is a scalar (function of $P$ and $T$), while $U(r)$ is a function of distance (and not thermodynamic variables). So they cannot be linked directly.
You intuited between $Z$ and $U$, which exists, but it needs to be formalized a bit more carefully. Standard approaches to do so rely on perturbation theory or mean-field method. You can check for examples the classical derivations of the van der Waals equation of state to see how this can be done. But it requires some fundamentals of statistical physics…