Activity coefficient

Experimental determination of activity coefficients

Activity coefficients may be determined experimentally by making measurements on non-ideal mixtures. Use may be made of Raoult's law or Henry's law to provide a value for an ideal mixture against which the experimental value may be compared to obtain the activity coefficient. Other colligative properties, such as osmotic pressure may also be used.

Radiochemical methods

Activity coefficients can be determined by radiochemical methods.

At infinite dilution

Activity coefficients for binary mixtures are often reported at the infinite dilution of each component. Because activity coefficient models simplify at infinite dilution, such empirical values can be used to estimate interaction energies. Examples are given for water:

Theoretical calculation of activity coefficients

Activity coefficients of electrolyte solutions may be calculated theoretically, using the Debye–Hückel equation or extensions such as the Davies equation, Pitzer equations or TCPC model. Specific ion interaction theory (SIT) may also be used.

For non-electrolyte solutions correlative methods such as UNIQUAC, NRTL, MOSCED or UNIFAC may be employed, provided fitted component-specific or model parameters are available. COSMO-RS is a theoretical method which is less dependent on model parameters as required information is obtained from quantum mechanics calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.

For uncharged species, the activity coefficient γ₀ mostly follows a salting-out model:

${\displaystyle {\log }_{10}({\gamma }_0)=bI}$

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.

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

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

where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, b is the molality of the salt dissolved in water, φ is the osmotic coefficient of water, and the constant 55.51 represents the molality 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.

Link to ionic diameter

The ionic activity coefficient is connected to the ionic diameter by the formula obtained from Debye–Hückel theory of electrolytes:

${\displaystyle \log ({\gamma }_i)=-\frac{A{z}_i^2\sqrt{I}}{1+Ba\sqrt{I}}}$

where A and B are constants, z_i is the valence number of the ion, and I is ionic strength.

Concentrated ionic solutions

Ionic activity coefficients can be calculated theoretically, for example by using the Debye–Hückel equation. The theoretical equation can be tested by combining the calculated single-ion activity coefficients to give mean values which can be compared to experimental values but only if the two ionic functions are strictly independent of one other (e.g. not determined by regression of experimental data).

Stokes–Robinson model

For concentrated ionic solutions the hydration of ions must be taken into consideration, as done by Stokes and Robinson in their hydration model from 1948. The activity coefficient of the electrolyte is split into electric and statistical components by E. Glueckauf who modifies the Robinson–Stokes model.

The statistical part includes hydration index number h, the number of ions from the dissociation and the ratio r between the apparent molar volume of the electrolyte and the molar volume of water and molality b.

Concentrated solution statistical part of the activity coefficient is:

${\displaystyle \ln {\gamma }_s=\frac{h-\nu }{\nu }\ln \left(1+\frac{br}{55.5}\right)-\frac{h}{\nu }\ln \left(1-\frac{br}{55.5}\right)+\frac{br(r+h-\nu )}{55.5\left(1+\frac{br}{55.5}\right)}}$

The Stokes–Robinson model has been analyzed and improved by other investigators. The problem with this widely accepted idea that electrolyte activity coefficients are driven at higher concentrations by changes in hydration is that water activities are completely dependent on the concentration of the ions themselves, as imposed by a thermodynamic relationship called the Gibbs-Duhem equation. This means that the activity coefficients and the corresponding water activities are linked together fundamentally, regardless of molecular-level hypotheses. Due to this high correlation, such hypotheses are not independent enough to be satisfactorily tested.

Ion trios

The rise in activity coefficients found with most aqueous strong electrolyte systems can also be explained by increasing electrostatic repulsions between ions of the same charge which are forced together as the available space between them decreases. In this way, the initial attractions between cations and anions at the low concentrations described by Debye and Hueckel are progressively overcome. It has been proposed that these electrostatic repulsions take place predominantly through the development of so-called ion trios in which two ions of like charge interact, on average and at distance, with the same counterion as well as with each other. This model accurately reproduces the experimental patterns of activity and osmotic coefficients exhibited by numerous 3-ion aqueous electrolyte mixtures. This is problematic at trace concentrations for models based on ionic hydration or molecular attractions.

Dependence on state parameters

The derivative of an activity coefficient with respect to temperature is related to excess molar enthalpy by

${\displaystyle {\overline{H}}_i^{\mathsf{E}}=-R{T}^2\frac{\mathrm{\partial }}{\mathrm{\partial }T}\ln ({\gamma }_i)}$

Similarly, the derivative of an activity coefficient with respect to pressure can be related to excess molar volume.

${\displaystyle {\overline{V}}_i^{\mathsf{E}}=RT\frac{\mathrm{\partial }}{\mathrm{\partial }P}\ln ({\gamma }_i)}$

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, Δ_rG, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

${\displaystyle {\alpha }_{\mathrm{A}}+{\beta }_{\mathrm{B}}={\sigma }_{\mathrm{S}}+{\tau }_{\mathrm{T}},}$

${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G=\sigma {\mu }_{\mathrm{S}}+\tau {\mu }_{\mathrm{T}}-(\alpha {\mu }_{\mathrm{A}}+\beta {\mu }_{\mathrm{B}})=0}$

Substitute in the expressions for the chemical potential of each reactant:

${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G=\sigma {\mu }_S^{\ominus }+\sigma RT\ln a_{\mathrm{S}}+\tau {\mu }_{\mathrm{T}}^{\ominus }+\tau RT\ln a_{\mathrm{T}}-(\alpha {\mu }_{\mathrm{A}}^{\ominus }+\alpha RT\ln a_{\mathrm{A}}+\beta {\mu }_{\mathrm{B}}^{\ominus }+\beta RT\ln a_{\mathrm{B}})=0}$

Upon rearrangement this expression becomes

${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G=\left(\sigma {\mu }_{\mathrm{S}}^{\ominus }+\tau {\mu }_{\mathrm{T}}^{\ominus }-\alpha {\mu }_{\mathrm{A}}^{\ominus }-\beta {\mu }_{\mathrm{B}}^{\ominus }\right)+RT\ln \frac{a_{\mathrm{S}}^{\sigma }{a}_{\mathrm{T}}^{\tau }}{a_{\mathrm{A}}^{\alpha }{a}_{\mathrm{B}}^{\beta }}=0}$

The sum σμ^o
_S + τμ^o
_T − αμ^o
_A − βμ^o
_B is the standard free energy change for the reaction, ${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}{G}^{\ominus }}$.

Therefore,

${\displaystyle {\mathrm{\Delta }}_r{G}^{\ominus }=-RT\ln K}$

where 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 that 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

${\displaystyle K=\frac{[\mathrm{S}{]}^{\sigma }[\mathrm{T}{]}^{\tau }}{[\mathrm{A}{]}^{\alpha }[\mathrm{B}{]}^{\beta }}\times \frac{{\gamma }_{\mathrm{S}}^{\sigma }{\gamma }_{\mathrm{T}}^{\tau }}{{\gamma }_{\mathrm{A}}^{\alpha }{\gamma }_{\mathrm{B}}^{\beta }}}$

where [S] denotes the concentration of S, etc. In practice equilibrium constants are determined in a medium such that the quotient of activity coefficients is constant and can be ignored, leading to the usual expression

${\displaystyle K=\frac{[\mathrm{S}{]}^{\sigma }[\mathrm{T}{]}^{\tau }}{[\mathrm{A}{]}^{\alpha }[\mathrm{B}{]}^{\beta }}}$

which applies under the conditions that the activity quotient has a particular (constant) value.