SIMPLE algorithm - Revision history

210.212.249.130 at 19:12, 2 December 2014

2014-12-02T19:12:24Z

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	An '''osmotic coefficient''' ''φ'' is a quantity which characterises the deviation of a [[solvent]] from [[ideal solution\|ideal behaviour]], referenced to [[Raoult's law]]. The osmotic coefficient based on [[molality]] ''b'' is defined by:
	~~: <math>\varphi=\frac{\mu_A^*-\mu_A}{RTM_A\sum_i b_i}\,</math>~~

	~~and on an [[amount fraction]] basis by:~~

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	~~where <math>\mu_A^*</math> is the [[chemical potential]] of the pure solvent and <math>\mu_A</math> is the [[chemical potential]]~~ of the ~~solvent in a solution~~, ~~''M''<sub>A</sub> is its [[molar mass]]~~, ''~~x''~~<~~sub~~>A<~~/sub~~> ~~its [[amount fraction]]~~, ~~''R'' the [[gas constant]] and ''T'' the [[temperature]] in~~ [~~[kelvin]]s.<ref>{{cite web \|url=~~http://~~goldbook~~.~~iupac~~.~~org~~/~~O04342.html \|title=Gold book definition}}<~~/~~ref> The latter osmotic~~
	~~coefficient is sometimes called the '''rational osmotic coefficient'''. The values for the two definitions are different, but since~~

	~~:<math>\ln x_A =~~ - ~~\ln(1 + M_A \sum_i b_i) \approx~~ - ~~M_A \sum_i b_i,</math>~~

	~~the two definitions are similar~~, and ~~in fact both approach 1 as the concentration goes~~ to ~~zero~~.

	~~In a single solute solution~~, ~~the (molality based) osmotic coefficient and the solute activity coefficient~~ are ~~related to the [[excess chemical potential\|excess Gibbs free energy]]~~ <~~math~~>~~G_{ex}~~<~~/math~~> by the ~~relations:~~

	~~:<math>RTb(1~~-~~\varphi) = G_{ex} - b \frac{dG_{ex}}{db}</math>~~
	~~:<math>RT\ln\gamma = \frac{dG_{ex}}{db}</math>~~

	~~and there is thus~~ a ~~differential relationship between them (temperature and pressure held constant):~~

	~~<!--A few other equations~~ for ~~reference, not worth putting in final article:~~
	~~:<math>d \mu_A + M_A b RT d \ln (\gamma b) = 0</math>~~
	~~:<math>d (\varphi b) = b d \ln(\gamma b)</math>~~
	~~:<math>d((\varphi~~ -~~1)b + b) = b d \ln\gamma + b d\ln b</math>~~
	~~-->~~
	~~:<math>d((\varphi -1)b) = b d \ln\gamma</math>~~

	~~In [[ionic solution]]s~~, ~~[[Debye~~-~~Hückel theory]] implies~~ that <~~math~~>~~(\varphi - 1)\sum_i b_i~~<~~/math~~> is ~~[[asymptotic]]~~ to <~~math~~>~~-\frac 2 3 A I^{3/2}~~<~~/math~~>~~, where ''I'' is [[ionic strength]] and ''A'' is~~ the ~~Debye-Hückel constant (equal to about 1.17 for water at 25 °C)~~. This ~~means that, at least at low concentrations,~~ the ~~vapor pressure~~ of ~~the solvent~~ will ~~be greater than that predicted by Raoult's law~~. ~~For instance, for solutions~~ of ~~[[magnesium chloride]], the vapor pressure is slightly greater than that predicted by Raoult's law up to~~ a ~~concentration~~ of 0.~~7 mol/kg~~, ~~after which~~ the ~~vapor pressure is lower than Raoult's law predicts~~.

	~~For aqueous solutions, the osmotic coefficients can~~ be ~~calculated theoretically by [[Pitzer equations]]<ref name="davies">I. Grenthe and H. Wanner, ''Guidelines for the extrapolation~~ to ~~zero ionic strength'',~~ http://~~www~~.~~nea~~.fr/~~html~~/~~dbtdb~~/~~guidelines/tdb2~~.~~pdf~~<~~/ref~~> ~~or TCPC model.~~<~~ref~~>X. Ge, ~~X. Wang~~, ~~M. Zhang~~, ~~S. Seetharaman. Correlation~~ and ~~Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model~~. ~~J. Chem. Eng. data. 52 (2007) 538-547~~.http://~~pubs~~.~~acs~~.~~org~~/~~doi~~/~~abs~~/10.~~1021/je060451k~~<~~/ref~~><~~ref~~>X. Ge, M. ~~Zhang~~, M. ~~Guo~~, X. ~~Wang~~, ~~Correlation and Prediction~~ of ~~Thermodynamic Properties~~ of ~~Non~~-~~aqueous Electrolytes by~~ the ~~Modified TCPC Model~~. J. ~~Chem~~. ~~Eng~~. ~~data~~. ~~53 (2008)149~~-~~159~~.~~http://pubs.acs.org/doi/abs/10~~.~~1021/je700446q~~<~~/ref~~><~~ref~~>X. Ge, M. ~~Zhang~~, M. ~~Guo~~, X. ~~Wang. Correlation and Prediction of thermodynamic properties~~ of ~~Some Complex Aqueous Electrolytes by the Modified Three~~-~~Characteristic-Parameter Correlation Model~~. J. ~~Chem~~. ~~Eng~~. ~~Data~~. ~~53(2008)950-958~~.~~http~~://~~pubs~~.~~acs~~.~~org~~/~~doi/abs/10~~.~~1021/je7006499~~<~~/ref~~><~~ref~~>~~X. Ge~~, X. ~~Wang. A Simple Two-Parameter Correlation Model for Aqueous Electrolyte across a wide range of temperature. J. Chem. Eng. Data. 54(2009)179-186~~.http://~~pubs.acs~~.~~org~~/~~doi~~/~~abs~~/10.~~1021/je800483q</ref>~~

	~~==See also==~~
	* [[Bromley equation]]
	* [[Pitzer equation]]
	* [[Davies equation]]
	* [[van'~~t Hoff factor]]~~

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Physical chemistry]]~~

en>Susilehtola: /* Algorithm */ - use proper font (text) for textual abbreviations in equations

2013-12-10T13:59:57Z

Algorithm: - use proper font (text) for textual abbreviations in equations

← Older revision		Revision as of 15:59, 10 December 2013
Line 1:		Line 1:
	~~Surely the second option would be more beneficial for any website. It~~ is ~~very easy to customize plugins according to~~ the ~~needs~~ of a ~~particular business~~. The ~~Word~~ - ~~Press Dashboard :~~ an ~~administrative management tool that supports FTP content upload 2. For more information about~~ [~~http~~://~~www.anjomanma.cloudsite.ir~~/~~S80P wordpress backup~~] ~~review our own web-page. If you are using videos on your site then this~~ is the ~~plugin to use. It~~'s ~~as simple as hiring a Wordpress plugin developer or learning how to create what is needed~~. <br><br>~~Right starting from social media support to search engine optimization, such plugins are easily available within~~ the ~~Word - Press open source platform~~. ~~After all~~, ~~Word~~ - ~~Press is free~~, ~~many of~~ the ~~enhancements for Word - Press like themes and plugins~~ are ~~also free~~, and ~~there is plenty of free information online about how~~ to ~~use Word - Press~~. ~~There are number of web services that offer Word press development across~~ the ~~world. These four plugins will make this effort easier~~ and the ~~sites run effectively as well as make other widgets added~~ to a ~~site easier to configure. W3C compliant HTML~~ and ~~a good open source powered by Word~~ - ~~Press CMS site is regarded as the prime minister.~~ <br><br>~~Your Word~~ - ~~Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content~~ is ~~meant~~ to ~~make your choice easier~~ and ~~reassure you that~~ the ~~decision~~ to ~~go ahead with this conversion is requited with rich benefits:~~. ~~Whether or not it~~'s ~~an viewers on your web page~~, ~~your social media pages~~, ~~or your web page, those who have~~ a ~~present and effective viewers~~ of "~~fans~~" ~~are best best~~ for ~~provide provides~~, ~~reductions~~, and ~~deals to help re-invigorate their viewers~~ and ~~add to their main point here~~. ~~Nonetheless, with stylish Facebook themes obtainable on~~ the ~~Globe Broad Internet, half of your enterprise is done previously~~. ~~Article Source~~: ~~Stevens works in Internet and Network Marketing~~. <br><br>~~You can add keywords but it is best to leave this alone~~. ~~High Quality Services: These companies help you in creating high quality Word - Press websites~~. ~~One~~ of the ~~great features of Wordpress is its ability to integrate SEO into your site~~. ~~Contact Infertility Clinic Providing One stop Fertility Solutions at~~:. ~~Fortunately~~, ~~Word - Press Customization Service is available these days~~, ~~right from custom theme design~~, ~~to plugin customization~~ and ~~modifying your website, you can take any bespoke service for your Word~~ - ~~Press development project~~. <br><br>A ~~sitemap is useful~~ for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a ~~site isn't competent enough in grabbing the attention~~ of ~~potential consumers~~. ~~You can select color of your choice, graphics of your favorite, skins, photos, pages, etc~~. ~~Word~~ - ~~Press is an open source content management system which is easy to use and offers many user friendly features~~. ~~Article Source~~: ~~Hostgator discount coupons for your Wordpress site here~~.		An '''osmotic coefficient''' ''φ'' is a quantity which characterises the deviation of a [[solvent]] from [[ideal solution\|ideal behaviour]], referenced to [[Raoult's law]]. The osmotic coefficient based on [[molality]] ''b'' is defined by:
			: <math>\varphi=\frac{\mu_A^*-\mu_A}{RTM_A\sum_i b_i}\,</math>

			and on an [[amount fraction]] basis by:

			:<math>\varphi=-\frac{\mu_A^*-\mu_A}{RT \ln x_A}\,</math>

			where <math>\mu_A^*</math> is the [[chemical potential]] of the pure solvent and <math>\mu_A</math> is the [[chemical potential]] of the solvent in a solution, ''M''<sub>A</sub> is its [[molar mass]], ''x''<sub>A</sub> its [[amount fraction]], ''R'' the [[gas constant]] and ''T'' the [[temperature]] in [[kelvin]]s.<ref>{{cite web \|url=http://goldbook.iupac.org/O04342.html \|title=Gold book definition}}</ref> The latter osmotic
			coefficient is sometimes called the '''rational osmotic coefficient'''. The values for the two definitions are different, but since

			:<math>\ln x_A = - \ln(1 + M_A \sum_i b_i) \approx - M_A \sum_i b_i,</math>

			the two definitions are similar, and in fact both approach 1 as the concentration goes to zero.

			In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient are related to the [[excess chemical potential\|excess Gibbs free energy]] <math>G_{ex}</math> by the relations:

			:<math>RTb(1-\varphi) = G_{ex} - b \frac{dG_{ex}}{db}</math>
			:<math>RT\ln\gamma = \frac{dG_{ex}}{db}</math>

			and there is thus a differential relationship between them (temperature and pressure held constant):

			<!--A few other equations for reference, not worth putting in final article:
			:<math>d \mu_A + M_A b RT d \ln (\gamma b) = 0</math>
			:<math>d (\varphi b) = b d \ln(\gamma b)</math>
			:<math>d((\varphi -1)b + b) = b d \ln\gamma + b d\ln b</math>
			-->
			:<math>d((\varphi -1)b) = b d \ln\gamma</math>

			In [[ionic solution]]s, [[Debye-Hückel theory]] implies that <math>(\varphi - 1)\sum_i b_i</math> is [[asymptotic]] to <math>-\frac 2 3 A I^{3/2}</math>, where ''I'' is [[ionic strength]] and ''A'' is the Debye-Hückel constant (equal to about 1.17 for water at 25 °C). This means that, at least at low concentrations, the vapor pressure of the solvent will be greater than that predicted by Raoult's law. For instance, for solutions of [[magnesium chloride]], the vapor pressure is slightly greater than that predicted by Raoult's law up to a concentration of 0.7 mol/kg, after which the vapor pressure is lower than Raoult's law predicts.

			For aqueous solutions, the osmotic coefficients can be calculated theoretically by [[Pitzer equations]]<ref name="davies">I. Grenthe and H. Wanner, ''Guidelines for the extrapolation to zero ionic strength'', http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf</ref> or TCPC model.<ref>X. Ge, X. Wang, M. Zhang, S. Seetharaman. Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model. J. Chem. Eng. data. 52 (2007) 538-547.http://pubs.acs.org/doi/abs/10.1021/je060451k</ref><ref>X. Ge, M. Zhang, M. Guo, X. Wang, Correlation and Prediction of Thermodynamic Properties of Non-aqueous Electrolytes by the Modified TCPC Model. J. Chem. Eng. data. 53 (2008)149-159.http://pubs.acs.org/doi/abs/10.1021/je700446q</ref><ref>X. Ge, M. Zhang, M. Guo, X. Wang. Correlation and Prediction of thermodynamic properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model. J. Chem. Eng. Data. 53(2008)950-958.http://pubs.acs.org/doi/abs/10.1021/je7006499</ref><ref>X. Ge, X. Wang. A Simple Two-Parameter Correlation Model for Aqueous Electrolyte across a wide range of temperature. J. Chem. Eng. Data. 54(2009)179-186.http://pubs.acs.org/doi/abs/10.1021/je800483q</ref>

			==See also==
			* [[Bromley equation]]
			* [[Pitzer equation]]
			* [[Davies equation]]
			* [[van't Hoff factor]]

			==References==
			{{reflist}}

			[[Category:Physical chemistry]]

en>Kri: Italicalized instead

2012-08-16T21:05:27Z

Italicalized instead

New page

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