Carnahan-Starling equation of state - Revision history

132.230.196.72: Added the errors given by Clisby and McCoy to the table of virial coefficients.

2023-03-21T13:25:51Z

Added the errors given by Clisby and McCoy to the table of virial coefficients.

← Older revision		Revision as of 15:25, 21 March 2023
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	\| 4 \|\| 18.3647684 \|\| 18		\| 4 \|\| 18.3647684 \|\| 18
	\|-		\|-
	\| 5 \|\| 28.~~224512~~ \|\| 28		\| 5 \|\| 28.22451(26) \|\| 28
	\|-		\|-
	\| 6 \|\| 39.~~8151475~~ \|\| 40		\| 6 \|\| 39.81515(93) \|\| 40
	\|-		\|-
	\| 7 \|\| 53.~~3444198~~ \|\| 54		\| 7 \|\| 53.3444(37) \|\| 54
	\|-		\|-
	\| 8 \|\| 68.~~5375488~~ \|\| 70		\| 8 \|\| 68.538(18) \|\| 70
	\|-		\|-
	\| 9 \|\| 85.~~8128384~~ \|\| 88		\| 9 \|\| 85.813(85) \|\| 88
	\|-		\|-
	\| 10 \|\| 105.~~775104~~ \|\| 108		\| 10 \|\| 105.78(39) \|\| 108
	\|}		\|}

Carl McBride: Added reference details

2020-11-10T10:15:34Z

Added reference details

← Older revision		Revision as of 12:15, 10 November 2020
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	</math>		</math>
	== Liu correction ==		== Liu correction ==
	Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:		Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357 Hongqin Liu "Carnahan Starling type equations of state for stable hard disk and hard sphere fluids", arXiv:2010.14357]</ref>:

	: <math>		: <math>

Hqliu1018: /* Liu correction */

2020-10-28T13:46:05Z

Liu correction

← Older revision		Revision as of 15:46, 28 October 2020
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	: <math>		: <math>
	Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\eta^5 }{(1-\eta)^3 }.		Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
	</math>		</math>

Hqliu1018: /* Liu correction */

2020-10-28T13:45:43Z

Liu correction

← Older revision		Revision as of 15:45, 28 October 2020
Line 84:		Line 84:

	: <math>		: <math>
	Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\~~ata~~^5 }{(1-\eta)^3 }.		Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\eta^5 }{(1-\eta)^3 }.
	</math>		</math>


	== See also ==		== See also ==

Hqliu1018 at 13:44, 28 October 2020

2020-10-28T13:44:54Z

← Older revision		Revision as of 15:44, 28 October 2020
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	Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.		Z = \frac{ 1 + \eta + \eta^2 - \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
	</math>		</math>
			== Liu correction ==
			Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357]</ref>:

			: <math>
			Z = \frac{ 1 + \eta + \eta^2 - \frac{8}{13}\eta^3 - \eta^4 + frac{1}{2}\ata^5 }{(1-\eta)^3 }.
			</math>


	== See also ==		== See also ==

Carl McBride: Slight tidy

2018-06-18T12:23:28Z

Slight tidy

← Older revision		Revision as of 14:23, 18 June 2018
Line 18:		Line 18:

	*<math> \sigma </math> is the [[hard sphere model \| hard sphere]] diameter.		*<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>.		The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 <ref>[https://arxiv.org/abs/cond-mat/0605392 Hongqin Liu "A very accurate hard sphere equation of state over the entire stable and metstable region", arXiv:cond-mat/0605392 (2006)]</ref>.
	==Virial expansion==		==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>):		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"		{\| style="width:40%; height:100px" border="1"
	\|-		\|-

129.69.120.91 at 08:46, 18 June 2018

2018-06-18T08:46:16Z

← Older revision		Revision as of 10:46, 18 June 2018
Line 18:		Line 18:

	*<math> \sigma </math> is the [[hard sphere model \| hard sphere]] diameter.		*<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==		==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>):		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>):

Carl McBride: Added a Kolafa correction section

2014-04-08T11:16:26Z

Added a Kolafa correction section

← Older revision		Revision as of 13:16, 8 April 2014
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	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> ).		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 ==		== See also ==

128.131.48.66: /* Thermodynamic expressions */

2013-09-30T09:45:44Z

Thermodynamic expressions

← Older revision		Revision as of 11:45, 30 September 2013
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	Isothermal [[compressibility]]:		Isothermal [[compressibility]]:

	:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>		:<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]].		where <math>\eta</math> is the [[packing fraction]].

Carl McBride: Enlarged image

2010-12-09T16:27:28Z

Enlarged image

← Older revision		Revision as of 18:27, 9 December 2010
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	[[Image:CS_EoS_plot.png\|thumb\|right]]		[[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).		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).