At 20:19 of this video ,
The professor writes $ Z = \frac{ P \overline{V_{real} } }{RT}$, and he writes,
$$ \frac{ P \overline{V_{real} } }{RT} = \sum_{i=0}^{n} \frac{ a_i} { V_{real} }$$
But I did not know that we could just Taylor expand in inverse monomials, like,
$$ f(x) = \sum_{k=0}^{n} \frac{ a_i}{x^i}$$
Let me consider the family of cubic equations of state (which are used billions of time every single day in the oil and gas industry.
They write $$P=\frac {R\,T}{V-b} - \frac {a(T)}{V^2+ub V+w b^2}$$ Changing the values of $u$ and $w$, you find Van der Waals, Redlich-Kwong, Peng-Robinson (and so on) equations of state.
$b$, the so-called covolume, is very small; it corresponds more or less to the minimum voluma occupied by a molecule.
Multiplying both sides by $\frac V{R\,T}$ and simplifying, we have $$\frac {P\,V} {R\,T}=\frac {V}{V-b}-\frac {a(T)}{R\,T}\frac {V }{V^2+ub V+w b^2}$$ For commodity, let $\alpha(T)=\frac {a(T)}{R\,T}$ and write $$\frac {P\,V} {R\,T}=\frac {V}{V-b}-\alpha(T)\frac {V }{V^2+ub V+w b^2}$$ Now, for gases, $V$ is quite large. So, using long division or Taylor series, we have $$\frac {P\,V} {R\,T}=\left(1+\frac{b}{V}+\frac{b^2}{V^2}+\frac{b^3}{V^3}+\cdots \right)-\alpha(T) \left(\frac{1}{V}-\frac{b u}{V^2}+\frac{b^2 u^2-b^2 w}{V^3}+\cdots\right)$$ that is to say $$\frac {P\,V} {R\,T}=1+\frac{b-\alpha (T)}{V}+\frac{b (b+u \alpha (T))}{V^2}+\frac{b^2 \left(b+\alpha (T) \left(w-u^2\right)\right)}{V^3}+\cdots$$ or, in short, $$\frac {P\,V} {R\,T}=1+\sum_{n=1}^p \frac{c_n(T)}{V^n}$$ This is the virial form of the equation of state.
The virial equation is important because it can be derived directly from statistical mechanics. The first term $1$ corresponds to the ideal gas, the second to corresponds to interactions between pairs of molecules, the third to interactions between triplets and so on.
The most sophisticated equations of state (often called "quasi-experimental" equations of state) are virial based. A simple one is the so-called Benedict–Webb–Rubin equation of state $$ p = \rho RT + \left(B_0 RT - A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 + \left(bRT - a - \frac{d}{T}\right) \rho^3 +$$ $$ \alpha\left(a + \frac{d}{T}\right) \rho^6 + \frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)$$ where $\rho=\frac 1 V$.