An activity coefficient [ [http://www.iupac.org/goldbook/A00116.pdf Gold Book definition] ] is a factor used in
thermodynamicsto account for deviations from ideal behaviour in a mixtureof chemical substances. In an ideal mixturethe interactions between each pair of chemical speciesare the same (or more formally, the enthalpyof mixing is zero) and, as a result, properties of the mixtures can be expressed directly in termsof simple concentrations or partial pressures of the substances present e.g. Raoult's law. Deviations from ideality are accommodated by modifying the concentration by an "activity coefficient". Analogously, expressions involved gases can be adjusted for non-ideality by scaling partial pressures by a fugacitycoefficient.
The chemical potential, , of a substance B in an
ideal mixtureis given by:where is the chemical potential in the standard stateand xB is the mole fractionof the substance in the mixture.
This is generalised to include non-ideal behaviour by writing:when is the activity of the substance in the mixture with:where is the activity coefficient. As mole fraction or concentration of B tends to zero, the behaviour of the mixture more closely approximates to ideal, and so the activity coefficients (of both solute and solvent) tend to unity in very dilute solutions.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 lawand equilibrium constants constants to be applied to both ideal and non-ideal mixtures.
Knowledge of activity coefficients is particularly important in the context of
electrochemistrysince the behaviour of electrolytesolutions is often far from ideal, due the effects of the ionic atmosphere.
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 energychange for the reactions, , is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as
::Substitute in the expressions for the chemical potential of each reactant:
:Upon rearrangement this expression becomes
The sum is the standard free energy change for the reaction, ΔG
"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 than 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
:where [S] denotes the
concentrationof S, etc. In practice equilibrium constants are determined in a medium such that the quotient of activity coefficient is constant and can be ignored, leading to the usual expression:which applies under the conditions that the activity quotient has a particular (constant) value.
Measurement and prediction of activity coefficients
Activity coefficients may be measured experimentally or calculated theoretically, using the
Debye-Hückel equationor extensions such as Davies equation [ C.W. Davies, "Ion Association",Butterworths, 1962] or Pitzer equationsI. Grenthe and H. Wanner, "Guidelines for the extrapolation to zero ionic strength", http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf] . Specific Ion Theory (SIT) [ [http://public.kubsu.ru/aquasolsoft SIT theory] ] may also be used. Alternatively correlative methods such as UNIFACmay be employed, provided fitted model parameters are available
For uncharged species, the activity coefficient γ0 mostly follows a "salting-out" modelJ.N. Butler, "Ionic Equilibrium", John Wiley and Sons, Inc., 1998.] :
This simple model predicts activities of many species (dissolved undissociated gases such as CO2, H2S, NH3, undissociated acids and bases) to high ionic strengths (up to 5 mol/kg). The value of the constant b for CO2 is 0.11 at 10 °C and 0.20 at 330 °C [A.J. Elis and R.M. Golding, Am. J. Sci, 162, p 47-60, 1963.] [S.D.Malinin, Geokhimiya, 3, p. 235-245, 1959.] .
where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, "m" is the molal concentration of the salt dissolved in water, Φ is the
osmotic coefficientof water, and the constant 55.51 represents the molal concentration 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.
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