In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour.^[1] In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (or macroscopically equivalent, the enthalpy change of solution and volume variation in 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 involving gases can be adjusted for non-ideality by scaling partial pressures by a fugacity coefficient.

The concept of activity coefficient is closely linked to that of activity in chemistry.

Thermodynamic definition

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]

Ionic solutions

For solution of substances which ionize in solution the activity coefficients of the cation and anion cannot be experimentally determined independently of each other because solution properties depend on both ions. Single ion activity coefficients must be linked to the activity coefficient of the dissolved electrolyte as if undissociated. In this case a mean stoichiometric activity coefficient of the dissolved electrolyte, γ_±, is used. It is called stoichiometric because it expresses both the deviation from the ideality of the solution and the incomplete ionic dissociation of the ionic compound which occurs especially with the increase of its concentration.

For a 1:1 electrolyte, such as NaCl it is given by the following:

${\displaystyle \gamma _{\pm }={\sqrt {\gamma _{+}\gamma _{-}}}}$

where ${\displaystyle {\mathrm{\gamma <!--\; \gamma \; -->}}_{+}}$Zdroj:https://en.wikipedia.org?pojem=Activity_coefficients
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