Activity coefficient

An activity coefficient [ [http://www.iupac.org/goldbook/A00116.pdf Gold Book definition] ] is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture the interactions between each pair of chemical species are the same (or more formally, the enthalpy of 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 fugacity coefficient.

Thermodynamics

The chemical potential, $mu\_B$, of a substance B in an ideal mixture is given by:$mu\_B\; =\; mu\_\{B\}^\{ominus\}\; +\; RT\; ln\; x\_B\; ,$where $mu\_\{B\}^\{ominus\}$ is the chemical potential in the standard state and x_B is the mole fraction of the substance in the mixture.

This is generalised to include non-ideal behaviour by writing:$mu\_B\; =\; mu\_\{B\}^\{ominus\}\; +\; RT\; ln\; a\_B\; ,$when $a\_B$ is the activity of the substance in the mixture with:$a\_B\; =\; x\_B\; gamma\_B$where $gamma\_B$ 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 law and equilibrium constants 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 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 energy change for the reactions, $Delta\; G$, 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^O. Therefore

:$Delta\; G^ominus\; =\; -RT\; ln\; K$

"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 concentration of 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 equation or 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 UNIFAC may 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.] :

:$log\_\{10\}(gamma\_\{0\})\; =\; b\; I$

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.] .

For water (solvent), the activity a_w can be calculated using:

:$ln(a\_\{w\})\; =\; frac\{-\; u\; m\}\{55.51\}\; phi$

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 coefficient of 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.

References