# Activity coefficient

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An activity coefficient [1] 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 terms of 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.

These examples can be understood by considering the chemical potential of each substance in the mixture. The chemical potential, ${\displaystyle \mu _{B}}$, of a substance B in an ideal mixture is given by

${\displaystyle \mu _{B}=\mu _{B}^{\ominus }+RT\ln x_{B}\,}$

where ${\displaystyle \mu _{B}^{\ominus }}$ is the chemical potential in the standard state and xB is the mole fraction of the substance in the mixture.

This is generalised to include non-ideal behaviour by writing

${\displaystyle \mu _{B}=\mu _{B}^{\ominus }+RT\ln a_{B}\,}$

when ${\displaystyle a_{B}}$ is the activity of the substance in the mixture with

${\displaystyle a_{B}=x_{B}\gamma _{B}}$

where ${\displaystyle \gamma _{B}}$ is the activity coefficient. As mole fraction or concentration tends 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.

## 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[2] or Pitzer equations[3]. Specific Ion Theory (SIT) [4] 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" model[5]:

${\displaystyle \log _{10}(\gamma _{0})=bI}$

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[6][7].

For water (solvent), the activity aw can be calculated using[5]:

${\displaystyle \ln(a_{w})={\frac {-\nu m}{55.51}}\phi }$

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

1. Gold Book definition
2. C.W. Davies, Ion Association,Butterworths, 1962
3. I. Grenthe and H. Wanner, Guidelines for the extrapolation to zero ionic strength, http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf
4. SIT theory
5. J.N. Butler, "Ionic Equilibrium", John Wiley and Sons, Inc., 1998.
6. A.J. Elis and R.M. Golding, Am. J. Sci, 162, p 47-60, 1963.
7. S.D.Malinin, Geokhimiya, 3, p. 235-245, 1959.