# Carnahan-Starling equation of state

The **Carnahan-Starling equation of state** is an approximate (but quite good) equation of state for the fluid phase of the hard sphere model in three dimensions. It is given by (Ref ^{[1]} Eqn. 10).

\[ Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }. \]

where:

- \( Z \) is the compressibility factor
- \( p \) is the pressure
- \( V \) is the volume
- \( N \) is the number of particles
- \( k_B \) is the Boltzmann constant
- \( T \) is the absolute temperature
- \( \eta \) is the packing fraction:

\[ \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} \]

- \( \sigma \) is the hard sphere diameter.

## Contents

## Virial expansion[edit]

It is interesting to compare the virial coefficients of the Carnahan-Starling equation of state (Eq. 7 of ^{[1]}) with the hard sphere virial coefficients in three dimensions (exact up to \(B_4\), and those of Clisby and McCoy ^{[2]}):

\(n\) | Clisby and McCoy | \(B_n=n^2+n-2\) |

2 | 4 | 4 |

3 | 10 | 10 |

4 | 18.3647684 | 18 |

5 | 28.224512 | 28 |

6 | 39.8151475 | 40 |

7 | 53.3444198 | 54 |

8 | 68.5375488 | 70 |

9 | 85.8128384 | 88 |

10 | 105.775104 | 108 |

## Thermodynamic expressions[edit]

From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref ^{[3]} Eqs. 2.6, 2.7 and 2.8)

Pressure (compressibility):

\[\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}\]

Configurational chemical potential:

\[\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}\]

Isothermal compressibility:

\[\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}\]

where \(\eta\) is the packing fraction.

Configurational Helmholtz energy function:

\[ \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}\]

## The 'Percus-Yevick' derivation[edit]

It is interesting to note (Ref ^{[4]} Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution of the Percus Yevick integral equation for hard spheres via the compressibility route, to one third via the pressure route, i.e.

\[Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }\]

The reason for this seems to be a slight mystery (see discussion in Ref. ^{[5]} ).

## Kolafa correction[edit]

Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy ^{[6]}:

\[ Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }. \]

## See also[edit]

## References[edit]

- ↑
^{1.0}^{1.1}N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics**51**pp. 635-636 (1969) - ↑ Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics
**122**pp. 15-57 (2006) - ↑ Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics
**103**pp. 9388-9396 (1995) - ↑ G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics
**54**pp. 1523-1525 (1971) - ↑ Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry
**93**pp. 6916-6919 (1989) - ↑ Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics
**140**136101 (2014)