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← Older revision		Revision as of 07:45, 4 February 2014
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	~~They contact me Emilia~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota~~. For many ~~years he~~'s ~~been working~~ as a ~~receptionist~~. ~~To do aerobics~~ is a ~~factor~~ that I'~~m totally addicted~~ to.<br><br>~~Here is my webpage~~ :: ~~healthy food delivery~~ ([http://~~Tinyurl~~.~~com~~/~~k7cuceb company website~~])		An '''activity coefficient''' is a factor used in [[thermodynamics]] to account for deviations from ideal behaviour in a [[mixture]] of [[chemical substance]]s.<ref>{{GoldBookRef\|title=Activity coefficient\|file=A00116}}</ref> In an [[ideal mixture]], the microscopic interactions between each pair of [[chemical species]] are the same (or macroscopically equivalent, the [[enthalpy change of solution]] and volume variation in mixing is zero) and, as a result, properties of the mixtures can be expressed directly in terms
			of simple [[concentration]]s or [[partial pressure]]s of the substances present e.g. [[Raoult's law]]. Deviations from ideality are accommodated by modifying the concentration by an ''activity coefficient''. Analogously, expressions involving gases can be adjusted for non-ideality by scaling [[partial pressure]]s by a [[fugacity]] coefficient.

			The concept of activity coefficient is closely linked to that of [[activity (chemistry)\|activity in chemistry]].

			== Thermodynamic definition ==
			The [[chemical potential]], <math> \mu_B</math>, of a substance B in an [[ideal mixture]] is given by
			:<math> \mu_B = \mu_{B}^{\ominus} + RT \ln x_B \,</math>
			where <math>\mu_{B}^{\ominus}</math> is the chemical potential in the [[standard state]] and x<sub>B</sub> is the [[mole fraction]] of the substance in the mixture.

			This is generalised to include non-ideal behavior by writing
			:<math> \mu_B = \mu_{B}^{\ominus} + RT \ln a_B \,</math>
			when <math>a_B</math> is the activity of the substance in the mixture with
			:<math> a_B = x_B \gamma_B</math>
			where <math>\gamma_B</math> is the activity coefficient. As <math>\gamma_B</math> approaches 1, the substance behaves as if it were ideal. For instance, if <math>\gamma_B \approx 1</math>, then [[Raoult's Law]] is accurate. For <math>\gamma_B > 1 </math> and <math>\gamma_B < 1 </math>, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

			In many cases, as <math>x_B</math> goes to zero, the activity coefficient of substance B approaches a constant; this relationship is Henry's Law for the solvent. These relationships are related to each other through the [[Gibbs-Duhem]] equation.<ref>R. DeHoff, ''Thermodynamic in Materials Science'', Taylor and Francis, 2006. pp230-1</ref>
			Note that in general activity coefficients are dimensionless.

			Modifying mole fractions or concentrations by activity coefficients gives the ''effective activities'' of the components, and hence allows expressions such as [[Raoult's law]] and [[equilibrium constant]]s constants to be applied to both ideal and non-ideal mixtures.

			Knowledge of activity coefficients is particularly important in the context of [[electrochemistry]] since the behaviour of [[electrolyte]] solutions is often far from ideal, due to the effects of the [[ionic atmosphere]]. Additionally, they are particularly important in the context of [[soil chemistry]] due to the low volumes of solvent and, consequently, the high concentration of [[electrolytes]].<ref>{{cite book \| author= Jorge G. Ibanez\| coauthors= Margarita Hernandez-Esparza, Carmen Doria-Serrano, Mono Mohan Singh\| title= Environmental Chemistry: Fundamentals\| year= 2007\| publisher= Springer\| isbn= 978-0-387-26061-7}}</ref>
			==Dependence on state parameters==
			The derivative of the activity coefficient to temperature and respectively pressure are connected to the excess molar properties.
			: <math>\bar{V^E}_i= RT \frac{\partial (\ln(\gamma_i))}{\partial P}</math>

			: <math>\bar{H^E}_i= -RT^2 \frac{\partial (\ln(\gamma_i))}{\partial T}</math>

			== Application to chemical equilibrium ==
			At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The [[Gibbs free energy]] change for the reactions, <math>\Delta_r G</math>, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

			:<math> \alpha A + \beta B \rightleftharpoons \sigma S + \tau T</math>
			:<math> \Delta_r G = \sigma \mu_S + \tau \mu_T - (\alpha \mu_A + \beta \mu_B) = 0\,</math>
			Substitute in the expressions for the chemical potential of each reactant:

			:<math> \Delta_r G = \sigma \mu_S^\ominus + \sigma RT \ln a_S + \tau \mu_T^\ominus + \tau RT \ln a_T -(\alpha \mu_A^\ominus + \alpha RT \ln a_A + \beta \mu_B^\ominus + \beta RT \ln a_B)=0</math>
			Upon rearrangement this expression becomes

			:<math> \Delta_r G =\left(\sigma \mu_S^\ominus+\tau \mu_T^\ominus -\alpha \mu_A^\ominus- \beta \mu_B^\ominus \right) + RT \ln \frac{a_S^\sigma a_T^\tau} {a_A^\alpha a_B^\beta} =0</math>

			The sum <math>\left(\sigma \mu_S^\ominus+\tau \mu_T^\ominus -\alpha \mu_A^\ominus- \beta \mu_B^\ominus \right)</math> is the standard free energy change for the reaction, <math>\Delta_r G^\ominus</math>. Therefore

			:<math> \Delta_r G^\ominus = -RT \ln K </math>

			''K'' is the [[equilibrium constant]]. Note that activities and equilibrium constants are dimensionless numbers.

			This derivation serves two purposes. It shows the relationship between standard free energy change and equilibrium constant. It also shows that an equilibrium constant is defined as a quotient of activities. In practical terms this is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

			:<math>K= \frac{[S]^\sigma[T]^\tau}{[A]^\alpha[B]^\beta} \times \frac{\gamma_S^\sigma \gamma_T^\tau}{\gamma_A^\alpha \gamma_B^\beta}</math>
			where [S] denotes the [[concentration]] of S, etc. In practice equilibrium constants are [[Determination of equilibrium constants\|determined]] in a medium such that the quotient of activity coefficient is constant and can be ignored, leading to the usual expression
			:<math>K= \frac{[S]^\sigma[T]^\tau}{[A]^\alpha[B]^\beta}</math>
			which applies under the conditions that the activity quotient has a particular (constant) value.

			== Measurement and prediction of activity coefficients ==
			[[File:UNIQUACRegressionChloroformMethanol.png\|thumb\|UNIQUAC [[Regression analysis\|Regression]] of Activity Coefficients ([[Chloroform]]/[[Methanol]] Mixture)]]
			Activity coefficients may be measured experimentally or calculated theoretically, using the [[Debye-Hückel equation]] or extensions such as [[Davies equation]],<ref>C.W. Davies, ''Ion Association'',Butterworths, 1962</ref> [[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> [[Specific ion interaction theory]] (SIT)<ref>{{cite web\|url=http://www.iupac.org/web/ins/2000-003-1-500 \|title=Project: Ionic Strength Corrections for Stability Constants \|accessdate=2008-11-15 \|publisher=IUPAC \| archiveurl= http://web.archive.org/web/20081029193538/http://www.iupac.org/web/ins/2000-003-1-500\| archivedate= 29 October 2008 <!--DASHBot-->\| deadurl= no}}</ref> may also be used. Alternatively correlative methods such as [[UNIQUAC]], [[Non-Random Two Liquid model\|NRTL]], [[MOSCED]] or [[UNIFAC]] may be employed, provided fitted component-specific or model parameters are available.

			A new alternative for activity coefficients prediction, which is less dependent on model parameters, is the COSMO-RS method. In this methods the required information comes from quantum mechanics calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.<ref name = Klamt>Andreas Klamt, "COSMO-RS: From Quantum Chemistry to Fluid Phase Thermodynamics and Drug Design", Elsevier, 2005.</ref>

			For uncharged species, the activity coefficient γ<sub>0</sub> mostly follows a [[salting-out]] model:<ref name = Butler>J.N. Butler, "Ionic Equilibrium", John Wiley and Sons, Inc., 1998.</ref>

			:<math> \log_{10}(\gamma_{0}) = b I</math>

			This simple model predicts activities of many species (dissolved undissociated gases such as CO<sub>2</sub>, H<sub>2</sub>S, NH<sub>3</sub>, undissociated acids and bases) to high ionic strengths (up to 5 mol/kg). The value of the constant b for CO<sub>2</sub> is 0.11 at 10 °C and 0.20 at 330 °C.<ref>A.J. Elis and R.M. Golding, Am. J. Sci, 162, p 47-60, 1963.</ref><ref>S.D.Malinin, Geokhimiya, 3, p. 235-245, 1959.</ref>

			For water (solvent), the activity a<sub>w</sub> can be calculated using:<ref name = "Butler"/>

			:<math> \ln(a_{w}) = \frac{-\nu m}{55.51} </math>φ

			where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, ''m'' is the molality of the salt dissolved in water, φ is the [[osmotic coefficient]] of water, and the constant 55.51 represents the molality of water. In the above equation, the activity of a solvent (here water) is represented as inversely proportional to the number of particles of salt versus that of the solvent.

			==References==
			{{reflist}}

			{{DEFAULTSORT:Activity Coefficient}}
			[[Category:Equilibrium chemistry]]

en>Kri: /* Description of the method / x_ should be in bold font

2012-09-01T14:30:04Z

Description of the method: x_* should be in bold font

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They contact me Emilia. Years in the past we moved to North Dakota. For many years he's been working as a receptionist. To do aerobics is a factor that I'm totally addicted to.<br><br>Here is my webpage :: healthy food delivery ([http://Tinyurl.com/k7cuceb company website])

Conjugate gradient method - Revision history

en>Gknor: Matlab->MATLAB

117.211.79.6 at 09:08, 20 February 2014

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

en>Kri: /* Description of the method */ x_* should be in bold font

en>Kri: /* Description of the method / x_ should be in bold font