An activity coefficient is a dimensionless quantity used in thermodynamics to describe departures from ideal behavior in mixtures. It relates a component’s thermodynamic activity to a specified measure of composition, such as mole fraction or normalized concentration. Activity coefficients allow equations for ideal systems to be extended to real solutions, whose components interact differently from those in the chosen ideal reference. Their numerical values depend on the composition scale and reference convention. (old.iupac.org)
Definition and standard states
The activity of component (i) is connected to its chemical potential by
[ \mu_i=\mu_i^\circ+RT\ln a_i, ]
where (\mu_i^\circ) refers to the selected standard state, (R) is the gas constant, and (T) is absolute temperature. On a molality scale,
[ a_i=\gamma_i^{(m)}\frac{m_i}{m^\circ}, ]
with (m^\circ=1\ \mathrm{mol,kg^{-1}}). Analogous definitions use molar concentration, (a_i=\gamma_i^{(c)}c_i/c^\circ), or mole fraction, (a_i=\gamma_i^{(x)}x_i). Both activity and its coefficient are dimensionless. (goldbook.iupac.org)
The standard-state convention determines the limiting behavior. In the pure-component, or Raoult’s-law, convention, (\gamma_i^{(x)}) approaches unity as (x_i) approaches one. For a solute under an infinite-dilution convention associated with Henry’s law, the coefficient instead approaches unity as the solute concentration approaches zero. Consequently, dilution does not make every possible activity coefficient approach one: the reference convention must be specified. (skripte.tf.uni-kiel.de)
Changing the standard state changes the numerical activity and activity coefficient, while leaving the physical chemical potential unchanged when (\mu_i^\circ) is adjusted consistently. Coefficients quoted on different composition scales therefore cannot generally be substituted for one another. (old.iupac.org)
Thermodynamic interpretation
For a mole-fraction description referenced to an ideal solution, the partial molar excess Gibbs free energy is
[ \overline G_i^{,E}=RT\ln\gamma_i. ]
Thus, (\gamma_i>1) signifies a chemical potential higher than the corresponding ideal-reference value, whereas (\gamma_i<1) signifies a lower value. The coefficient is not a fraction of molecules that are “active”; it expresses a free-energy difference. A value of one for a single component at one composition does not establish ideal behavior throughout the mixture. (nvlpubs.nist.gov)
Activity coefficients of different components are thermodynamically coupled. At fixed temperature and pressure, the Gibbs–Duhem equation gives, for a mixture described on the mole-fraction scale,
[ \sum_i x_i,d\ln\gamma_i=0. ]
This constraint prevents independent, arbitrary composition dependences from being assigned to all components. It also provides a basis for testing the consistency of experimental phase-equilibrium data and fitted models. (arxiv.org)
Electrolyte solutions
Activity coefficients are especially important for electrolytes, where interactions between charged species produce nonideality even in relatively dilute solutions. The molality-based ionic strength is
[ I_m=\frac12\sum_i m_i z_i^2, ]
where (z_i) is the charge number of each ion. The squared charge makes multivalent ions contribute more strongly than monovalent ions at equal molality. (mooseframework.inl.gov)
Individual-ion activity coefficients cannot be determined independently by ordinary thermodynamic measurements without adopting an additional convention. Electrically neutral combinations, however, permit determination of a mean ionic activity coefficient. For an electrolyte producing (\nu_+) cations and (\nu_-) anions per formula unit,
[ \gamma_\pm= \left(\gamma_+^{\nu_+}\gamma_-^{\nu_-}\right)^{ 1/(\nu_++\nu_-)}. ]
For a 1:1 electrolyte, this reduces to (\gamma_\pm=(\gamma_+\gamma_-)^{1/2}). The mean coefficient describes the neutral electrolyte combination rather than a separately measurable property of either ion. (goldbook.iupac.org)
The Debye–Hückel theory describes the low-ionic-strength limit. Its limiting law can be written
[ \log_{10}\gamma_i= -Az_i^2\sqrt{I_m/m^\circ}, ]
where the dimensionless coefficient (A) depends on temperature and solvent properties. Extended Debye–Hückel expressions and the Davies equation provide approximations beyond this limit. More concentrated solutions require treatment of additional interactions; Pitzer equations, for example, are used for high-salinity waters beyond the applicability of basic Debye–Hückel models. (mooseframework.inl.gov)
Models and experimental determination
For molecular liquid mixtures, activity coefficients are commonly obtained from excess-Gibbs-energy models. Established examples include Wilson, the non-random two-liquid model (NRTL), and UNIQUAC. Their parameters are fitted to mixture-property data. Group-contribution methods such as UNIFAC instead estimate mixture behavior using molecular functional groups and previously determined interaction parameters. (trc.nist.gov)
Experimental phase equilibria constrain activity coefficients. Under an ideal-vapor approximation and with negligible liquid-pressure correction, modified Raoult’s law is
[ y_iP=x_i\gamma_iP_i^{\mathrm{sat}}, ]
where (y_i) is vapor mole fraction and (P_i^{\mathrm{sat}}) is the pure component’s saturation vapor pressure. Measurements of vapor and liquid compositions and pressure therefore permit coefficients to be inferred. At higher pressures, vapor nonideality and liquid-pressure corrections must also be considered. Data consistency and the temperature and composition ranges of fitted parameters remain important limitations. (arxiv.org)
Chemical applications
In chemical equilibrium, the thermodynamic equilibrium constant is constructed from activities rather than uncorrected concentrations. Activity coefficients consequently affect calculated aqueous speciation, mineral solubility, and interactions between dissolved species and surfaces. Geochemical programs use activity models together with mass balances and equilibrium equations to represent these processes. (wwwbrr.cr.usgs.gov)
The distinction also appears in pH, defined by (\mathrm{pH}=-\log_{10}a_{\mathrm{H^+}}), not simply by hydrogen-ion concentration. Because this definition involves a single-ion activity, practical pH scales require conventions and calibrated standards; treating its activity coefficient as unity is an approximation, not part of the definition. (goldbook.iupac.org)