Is this equation of state original?
Yes, this equation of state is original. While I cannot absolutely guarantee that it has never been used or proposed before, I can say that I have not encountered it in any “classical” model — and I teach thermodynamics and statistical physics, so I have seen many textbooks and classical exercices :)

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:


Yes, if $U(r)&gt;0$ then $Z&gt;1$, and the other way around… but that does not mean that the two quantities are proportional.
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…