# Virial Expansions of Hard Spheres

[Equation of State for Nonattracting Rigid Spheres (1969)](https://aip.scitation.org/doi/pdf/10.1063/1.1672048)

Equations for discribing states of fluids have been studied around the 1960's.

* Percus-Yevick equations
\begin{eqnarray}
Z = \frac{PV}{Nk_BT} = \frac{1+y+y^2}{(1-y)^3}
\end{eqnarray}
where
\begin{eqnarray}
y = \frac{b}{4V}, ~~ b = \frac{2}{3}(N\Pi\sigma^3)
\end{eqnarray}
This relations were also found by Reiss, Frisch, and Lebowitz.

* Thiele
\begin{eqnarray}
Z = \frac{PV}{Nk_BT} = \frac{1+2y + 3y^2}{(1-y)^2}
\end{eqnarray}

* Emprical form by Guggenheim
\begin{eqnarray}
Z = \frac{PV}{Nk_BT} = \frac{1}{(1-y)^4}
\end{eqnarray}

* Ree and Hoover (Deriviation from already known viral coefficients)
\begin{eqnarray}
&& \frac{1+1.75399y+2.31704y^2+1.108928y^3}{1 - 2.246004y+ 1.301056y^2}\\
=&& (1+1.75399y+2.31704y^2+1.108928y^3) \{1 + (2.246004y- 1.301056y^2) + (2.246004y- 1.301056y^2)^2 + (2.246004y- 1.301056y^2)^3 \cdots \}\\
=&& 1 + (1.75399 + 2.246004)y + (−
1.301056 + 2.246004
^2 + 1.75399 \cdot 2.246004 + 2.31704
) + \cdots\\
=&& 1 + 4y + 10y + 18.36 + 28.2y^4+39.5y^5\cdots
\end{eqnarray}

* Carnahan and Stariling
Derived formula by assumptions that all virial coefficients are integers.
\begin{eqnarray}
Z =&& 1 + 4y + 10y^2 + 18y^3 + \cdots \\
=&& 1 + \sum_{n=1}^{} n(n+3)y^n
\end{eqnarray}
Thus the close form will be:
\begin{eqnarray}
Z = \frac{1+y+y^2-y^3}{(1-y)^3}
\end{eqnarray}
and differs from the Percus-Yevick only by $y^3$ term at the numerator.

> Proof
\begin{eqnarray}
(1-y)^3 Z =&& (1-3y+3y^2-y^3)Z\\
=&& (1-3y+3y^2-y^3) \left( 1 + \sum_{n=1}^{} n(n+3)y^n\right)\\
=&& (1-3y+3y^2-y^3) + \left( \sum_{n=1}^{} n(n+3)y^n + \sum_{n=2}^{} -3(n-1)(n+2)y^{n} + \sum_{n=3}^{} 3(n-2)(n+1)y^{n} + \sum_{n=4}^{} -(n-3)ny^{n}\right)\\
=&& (1-3y+3y^2-y^3) + (4y + 10y^2 + 18y^3 -12y^2 -30y^3 + 12y^3)\\
=&& 1+y+y^2-y^3
\end{eqnarray}

## Comparsions with calculations
[Alder and Wainwright](https://aip.scitation.org/doi/pdf/10.1063/1.1731425) calculated $Z$ by [method](https://aip.scitation.org/doi/pdf/10.1063/1.1730376) develped in 1959.
The form by Carnahan and Stariling seems to be best analytical form, which fits the best with Alder and Wainwright's.

## Memo
What is Alder and Wainwright's method and why is it thought to be the most precise?

What is $\sigma,b$ when definig $y$?

Are there any limitations of formula by Carnahan and Stariling? I thought the formula is phenomological.
Still in nucleation theory, the formula is thought to be precise when using DFT approach.

The value of these phenomological formula might be the memorability. Also by closed form, the calculation cost is low.