Activity coefficient

The chemical potential, ${\displaystyle \mu _{\mathrm {B} }}$, of a substance B in an ideal mixture of liquids or an ideal solution is given by

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

where μ^o
_B is the chemical potential of a pure substance ${\displaystyle \mathrm {B} }$, and ${\displaystyle x_{\mathrm {B} }}$ is the mole fraction of the substance in the mixture.

This is generalised to include non-ideal behavior by writing

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

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

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

where ${\displaystyle \gamma _{\mathrm {B} }}$ is the activity coefficient, which may itself depend on ${\displaystyle x_{\mathrm {B} }}$. As ${\displaystyle \gamma _{\mathrm {B} }}$ approaches 1, the substance behaves as if it were ideal. For instance, if ${\displaystyle \gamma _{\mathrm {B} }}$ ≈ 1, then Raoult's law is accurate. For ${\displaystyle \gamma _{\mathrm {B} }}$ > 1 and ${\displaystyle \gamma _{\mathrm {B} }}$ < 1, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

In many cases, as ${\displaystyle x_{\mathrm {B} }}$ goes to zero, the activity coefficient of substance B approaches a constant; this relationship is Henry's law for the solvent. These relationships are related to each other through the Gibbs–Duhem equation.[2] Note that in general activity coefficients are dimensionless.

In detail: Raoult's law states that the partial pressure of component B is related to its vapor pressure (saturation pressure) and its mole fraction ${\displaystyle x_{\mathrm {B} }}$ in the liquid phase,

${\displaystyle p_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }p_{\mathrm {B} }^{\sigma }\;,}$

with the convention ${\displaystyle \lim _{x_{\mathrm {B} }\to 1}\gamma _{\mathrm {B} }=1\;.}$ In other words: Pure liquids represent the ideal case.

At infinite dilution, the activity coefficient approaches its limiting value, ${\displaystyle \gamma _{\mathrm {B} }}$^∞. Comparison with Henry's law,

${\displaystyle p_{\mathrm {B} }=K_{\mathrm {H,B} }x_{\mathrm {B} }\quad {\text{for}}\quad x_{\mathrm {B} }\to 0\;,}$

immediately gives

${\displaystyle K_{\mathrm {H,B} }=p_{\mathrm {B} }^{\sigma }\gamma _{\mathrm {B} }^{\infty }\;.}$

In other words: The compound shows nonideal behavior in the dilute case.

The above definition of the activity coefficient is impractical if the compound does not exist as a pure liquid. This is often the case for electrolytes or biochemical compounds. In such cases, a different definition is used that considers infinite dilution as the ideal state:

${\displaystyle \gamma _{\mathrm {B} }^{\dagger }\equiv \gamma _{\mathrm {B} }/\gamma _{\mathrm {B} }^{\infty }}$

with ${\displaystyle \lim _{x_{\mathrm {B} }\to 0}\gamma _{\mathrm {B} }^{\dagger }=1\;,}$ and

${\displaystyle \mu _{\mathrm {B} }=\underbrace {\mu _{\mathrm {B} }^{\ominus }+RT\ln \gamma _{\mathrm {B} }^{\infty }} _{\mu _{\mathrm {B} }^{\ominus \dagger }}+RT\ln \left(x_{\mathrm {B} }\gamma _{\mathrm {B} }^{\dagger }\right)}$

The ${\displaystyle ^{\dagger }}$ symbol has been used here to distinguish between the two kinds of activity coefficients. Usually it is omitted, as it is clear from the context which kind is meant. But there are cases where both kinds of activity coefficients are needed and may even appear in the same equation, e.g., for solutions of salts in (water + alcohol) mixtures. This is sometimes a source of errors.

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 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 to the effects of the ionic atmosphere. Additionally, they are particularly important in the context of soil chemistry due to the low volumes of solvent and, consequently, the high concentration of electrolytes.[3]

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.

Activity coefficients of electrolyte solutions may be calculated theoretically, using the Debye–Hückel equation or extensions such as the Davies equation,[16] Pitzer equations[17] or TCPC model.[18][19][20][21] Specific ion interaction theory (SIT)[22] 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.[23]

For uncharged species, the activity coefficient γ₀ mostly follows a salting-out model:[24]

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

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.[25]

For water as solvent, the activity a_w can be calculated using:[24]

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

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.

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

${\displaystyle \log(\gamma _{i})=-{\frac {Az_{i}^{2}{\sqrt {I}}}{1+Ba{\sqrt {I}}}}}$

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

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

${\displaystyle {\bar {H}}_{i}^{\mathsf {E}}=-RT^{2}{\frac {\partial }{\partial T}}\ln(\gamma _{i})}$

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

${\displaystyle {\bar {V}}_{i}^{\mathsf {E}}=RT{\frac {\partial }{\partial P}}\ln(\gamma _{i})}$

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

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

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

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

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

Upon rearrangement this expression becomes

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

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

Therefore,

${\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K}$

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 coefficient 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.

• AIOMFAC online-model An interactive group-contribution model for the calculation of activity coefficients in organic–inorganic mixtures.
• Electrochimica Acta Single-ion activity coefficients