View source for Physics:Osmotic coefficient

{{Short description|Quantity characterizing the deviation of a solvent from ideal behavior}}
An '''osmotic coefficient''' <math>\phi</math> is a quantity which characterises the deviation of a [[Physics:Solvent|solvent]] from [[Physics:Ideal solution|ideal behaviour]], referenced to [[Physics:Raoult's law|Raoult's law]]. It can be also applied to solutes. Its definition depends on the ways of expressing [[Chemistry:Chemical composition|chemical composition]] of mixtures.

The osmotic coefficient based on [[Chemistry:Molality|molality]] ''m'' is defined by:
<math display="block">\phi = \frac{\mu_A^* - \mu_A}{RTM_A \sum_i m_i}</math>

and on a [[Chemistry:Mole fraction|mole fraction]] basis by:

<math display="block">\phi = -\frac{\mu_A^* - \mu_A}{RT \ln x_A}</math>

where <math>\mu_A^*</math> is the [[Chemistry:Chemical potential|chemical potential]] of the pure solvent and <math>\mu_A</math> is the [[Chemistry:Chemical potential|chemical potential]] of the solvent in a solution, ''M''<sub>A</sub> is its [[Physics:Molar mass|molar mass]], ''x''<sub>A</sub> its [[Chemistry:Mole fraction|mole fraction]], ''R'' the [[Physics:Gas constant|gas constant]] and ''T'' the [[Physics:Temperature|temperature]] in [[Kelvin]].<ref>{{GoldBookRef|title=osmotic coefficient| file = O04342}}</ref> The latter osmotic
coefficient is sometimes called the '''rational osmotic coefficient'''. The values for the two definitions are different, but since

<math display="block">\ln x_A = - \ln \left(1 + M_A \sum_i m_i \right) \approx - M_A \sum_i m_i,</math>

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

== Applications ==
For liquid solutions, the osmotic coefficient is often used to calculate the salt [[Chemistry:Activity coefficient|activity coefficient]] from the solvent activity, or vice versa. For example, freezing point depression measurements, or measurements of deviations from ideality for other [[Chemistry:Colligative properties|colligative properties]], allows calculation of the salt activity coefficient through the osmotic coefficient.

==Relation to other quantities==
In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient <math>\gamma </math> are related to the [[Chemistry:Excess chemical potential|excess Gibbs free energy]] <math>G^E</math> by the relations:

:<math>RTm(1-\phi) = G^E - m \frac{dG^E}{dm}</math>
:<math>RT\ln\gamma = \frac{dG^E}{dm}</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((\phi -1)m) = m d (\ln\gamma)</math>

=== Liquid electrolyte solutions ===
For a single salt solute with molal activity (<math>\gamma_\pm m</math>), the osmotic coefficient can be written as <math>\phi=\frac{-\ln(a_A)}{\nu m M_A}</math>where <math>\nu</math> is the stochiometric number of salt and <math>a_A</math> the activity of the solvent. <math>\phi</math> can be calculated from the salt activity coefficient via:<ref>{{Cite book|last=Pitzer|first=Kenneth S.|title=Activity Coefficients in Electrolyte Solutions|publisher=CRC Press|year=2018}}</ref>

:<math>\phi = 1 + \frac{1}{m}\int_0^m md \left( \ln (\gamma_{\pm}) \right)</math>

Moreover, the activity coefficient of the salt <math>\gamma_{\pm}</math> can be calculated from:<ref>{{Cite book|last=Pitzer|first=Kenneth|title=Activity Coefficients in Electrolyte Solutions|last2=|first2=|publisher=CRC Press|year=1991|isbn=978-1-315-89037-1|pages=13}}</ref>

: <math>\ln (\gamma_{\pm}) = \phi-1+\int^m_0 \frac{\phi-1}{m} dm</math>

According to [[Chemistry:Debye–Hückel theory|Debye–Hückel theory]], which is accurate only at low concentrations, <math display="inline"> (\phi - 1) \sum_i m_i</math> is asymptotic to <math display="inline"> -\frac 2 3 A  I^{3/2}</math>, where ''I'' is [[Chemistry:Ionic strength|ionic strength]] and ''A'' is the Debye–Hückel constant (equal to about 1.17 for water at 25&nbsp;°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 [[Chemistry:Magnesium chloride|magnesium chloride]], the [[Physics:Vapor pressure|vapor pressure]] is slightly greater than that predicted by Raoult's law up to a concentration of 0.7&nbsp;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 [[Physics:Pitzer equations|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>{{cite journal|last1=Ge|first1=Xinlei|last2=Wang|first2=Xidong|last3=Zhang|first3=Mei|last4=Seetharaman|first4=Seshadri|title=Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model|journal=Journal of Chemical & Engineering Data|volume=52|issue=2|year=2007|pages=538–547|issn=0021-9568|doi=10.1021/je060451k}}</ref><ref>{{cite journal|last1=Ge|first1=Xinlei|last2=Zhang|first2=Mei|last3=Guo|first3=Min|last4=Wang|first4=Xidong|title=Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model|journal=Journal of Chemical & Engineering Data|volume=53|issue=1|year=2008|pages=149–159|issn=0021-9568|doi=10.1021/je700446q}}</ref>
<ref>{{cite journal|last1=Ge|first1=Xinlei|last2=Zhang|first2=Mei|last3=Guo|first3=Min|last4=Wang|first4=Xidong|title=Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model|journal=Journal of Chemical & Engineering Data|volume=53|issue=4|year=2008|pages=950–958|issn=0021-9568|doi=10.1021/je7006499}}</ref><ref name="GeWang2009">{{cite journal|last1=Ge|first1=Xinlei|last2=Wang|first2=Xidong|title=A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures†|journal=Journal of Chemical & Engineering Data|volume=54|issue=2|year=2009|pages=179–186|issn=0021-9568|doi=10.1021/je800483q}}</ref>

==See also==
* [[Physics:Bromley equation|Bromley equation]]
* Pitzer equation
* [[Physics:Davies equation|Davies equation]]
* [[Van 't Hoff factor|van 't Hoff factor]]
* [[Physics:Law of dilution|Law of dilution]]
* [[Chemistry:Thermodynamic activity|Thermodynamic activity]]
* [[Chemistry:Ion transport number|Ion transport number]]

==References==
{{reflist}}

[[Category:Physical chemistry]]

{{Sourceattribution|Osmotic coefficient}}