Talk:Pressure

2019-11-21T11:12:53Z

129.69.120.91:

The formula for different dimensions looks, well well, interesting, if the system is 2d or 1d, what would be the volume? Can you please specify this?

Carnahan-Starling equation of state

2018-06-18T08:46:16Z

129.69.120.91:

[[Image:CS_EoS_plot.png|thumb|350px|right]]
The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> Z </math> is the [[compressibility factor]]
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The CS eos is not applicable for packing fractions greater than 0.55 <ref>https://arxiv.org/pdf/cond-mat/0605392.pdf</ref>.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 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)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 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==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 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)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\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}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 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)] </ref> 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.

:<math>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 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 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)]</ref> ).
== Kolafa correction ==
Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy <ref>[http://dx.doi.org/10.1063/1.4870524 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)]</ref>:

: <math>
Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
</math>

== See also ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

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Carnahan-Starling equation of state