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An osmotic coefficient $\phi$ is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of mixtures.

The osmotic coefficient based on molality b is defined by:

\phi ={\frac {\mu _{A}^{*}-\mu _{A}}{RTM_{A}\sum _{i}b_{i}}}\,

and on a mole fraction basis by:

\phi =-{\frac {\mu _{A}^{*}-\mu _{A}}{RT\ln x_{A}}}\,

where $\mu _{A}^{*}$ is the chemical potential of the pure solvent and $\mu _{A}$ is the chemical potential of the solvent in a solution, M_A is its molar mass, x_A its mole fraction, R the gas constant and T the temperature in kelvins.^[1] The latter osmotic coefficient is sometimes called the rational osmotic coefficient. The values for the two definitions are different, but since

\ln x_{A}=-\ln(1+M_{A}\sum _{i}b_{i})\approx -M_{A}\sum _{i}b_{i},

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

Relation to other quantities

In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient are related to the excess Gibbs free energy $G^{E}$ by the relations:

RTb(1-\phi )=G^{E}-b{\frac {dG^{E}}{db}}

RT\ln \gamma ={\frac {dG^{E}}{db}}

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

d((\phi -1)b)=bd(\ln \gamma )

In ionic solutions, Debye–Hückel theory implies that $(\varphi -1)\sum _{i}b_{i}$ is asymptotic to $-{\frac {2}{3}}AI^{3/2}$ , 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^[2] or TCPC model.^[3]^[4] ^[5]^[6]

References

^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "osmotic coefficient". doi:10.1351/goldbook.O04342
^ I. Grenthe and H. Wanner, Guidelines for the extrapolation to zero ionic strength, http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf
^ Ge, Xinlei; Wang, Xidong; Zhang, Mei; Seetharaman, Seshadri (2007). "Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model". Journal of Chemical & Engineering Data. 52 (2): 538–547. doi:10.1021/je060451k. ISSN 0021-9568.
^ Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model". Journal of Chemical & Engineering Data. 53 (1): 149–159. doi:10.1021/je700446q. ISSN 0021-9568.
^ Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model". Journal of Chemical & Engineering Data. 53 (4): 950–958. doi:10.1021/je7006499. ISSN 0021-9568.
^ Ge, Xinlei; Wang, Xidong (2009). "A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures†". Journal of Chemical & Engineering Data. 54 (2): 179–186. doi:10.1021/je800483q. ISSN 0021-9568.

[1] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "osmotic coefficient". doi:10.1351/goldbook.O04342

[davies-2] I. Grenthe and H. Wanner, Guidelines for the extrapolation to zero ionic strength, http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf

[3] Ge, Xinlei; Wang, Xidong; Zhang, Mei; Seetharaman, Seshadri (2007). "Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model". Journal of Chemical & Engineering Data. 52 (2): 538–547. doi:10.1021/je060451k. ISSN 0021-9568.

[4] Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model". Journal of Chemical & Engineering Data. 53 (1): 149–159. doi:10.1021/je700446q. ISSN 0021-9568.

[5] Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (2008). "Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model". Journal of Chemical & Engineering Data. 53 (4): 950–958. doi:10.1021/je7006499. ISSN 0021-9568.

[GeWang2009-6] Ge, Xinlei; Wang, Xidong (2009). "A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures†". Journal of Chemical & Engineering Data. 54 (2): 179–186. doi:10.1021/je800483q. ISSN 0021-9568.

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@@ Line 1: / Line 1: @@
-An '''osmotic coefficient''' ''φ'' is a quantity which characterises the deviation of a [[solvent]] from [[ideal solution|ideal behaviour]], referenced to [[Raoult's law]]. It can be also applied to solutes. Its definition depends on the ways of expressing [[chemical composition]] of mixtures.
+An '''osmotic coefficient''' <math>\phi</math> is a quantity which characterises the deviation of a [[solvent]] from [[ideal solution|ideal behaviour]], referenced to [[Raoult's law]]. It can be also applied to solutes. Its definition depends on the ways of expressing [[chemical composition]] of mixtures.
 The osmotic coefficient based on [[molality]] ''b'' is defined by:
-: <math>\varphi=\frac{\mu_A^*-\mu_A}{RTM_A\sum_i b_i}\,</math>
+: <math>\phi=\frac{\mu_A^*-\mu_A}{RTM_A\sum_i b_i}\,</math>
 and on a [[mole fraction]] basis by:
-:<math>\varphi=-\frac{\mu_A^*-\mu_A}{RT \ln x_A}\,</math>
+:<math>\phi=-\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 [[mole fraction]], ''R'' the [[gas constant]] and ''T'' the [[temperature]] in [[kelvin]]s.<ref>{{GoldBookRef|title=osmotic coefficient| file = O04342}}</ref> The latter osmotic
@@ Line 17: / Line 17: @@
 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^E</math> by the relations:
-:<math>RTb(1-\varphi) = G^E - b \frac{dG^E}{db}</math>
+:<math>RTb(1-\phi) = G^E - b \frac{dG^E}{db}</math>
 :<math>RT\ln\gamma = \frac{dG^E}{db}</math>
@@ Line 27: / Line 27: @@
 :<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>
+:<math>d((\phi -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&nbsp;mol/kg, after which the vapor pressure is lower than Raoult's law predicts.

Revision as of 00:15, 28 February 2021

Relation to other quantities

See also

References