In
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, an activity coefficient is a factor used to account for deviation of a
mixture
In chemistry, a mixture is a material made up of two or more different chemical substances which are not chemically bonded. A mixture is the physical combination of two or more substances in which the identities are retained and are mixed in the ...
of
chemical substance
A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wi ...
s from ideal behaviour. In an
ideal mixture
In chemistry, an ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing by definition; the closer to zer ...
, the microscopic interactions between each pair of
chemical species
A chemical species is a chemical substance or ensemble composed of chemically identical molecular entity, molecular entities that can explore the same set of molecular energy levels on a characteristic or delineated time scale. These energy levels ...
are the same (or macroscopically equivalent, the
enthalpy change of solution
In thermochemistry, the enthalpy of solution ( heat of solution or enthalpy of solvation) is the enthalpy change associated with the dissolution of a substance in a solvent at constant pressure resulting in infinite dilution.
The enthalpy of sol ...
and volume variation in mixing is zero) and, as a result, properties of the mixtures can be expressed directly in terms of simple
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
s or
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
s 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
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
coefficient.
The concept of activity coefficient is closely linked to that of
activity in chemistry.
Thermodynamic definition
The
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
,
, of a substance B in an
ideal mixture
In chemistry, an ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing by definition; the closer to zer ...
of liquids or an
ideal solution is given by
:
,
where ''μ'' is the chemical potential of a pure substance
, and
is the
mole fraction
In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
of the substance in the mixture.
This is generalised to include non-ideal behavior by writing
:
when
is the activity of the substance in the mixture,
:
,
where
is the activity coefficient, which may itself depend on
. As
approaches 1, the substance behaves as if it were ideal. For instance, if
≈ 1, then
Raoult's law is accurate. For
> 1 and
< 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
goes to zero, the activity coefficient of substance B approaches a constant; this relationship is
Henry's law
In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulat ...
for the solvent. These relationships are related to each other through the
Gibbs–Duhem equation In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:
:\sum_^I N_i \mathrm\mu_i = - S \mathrmT + V \mathrmp
where N_i is the number of moles of com ...
.
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
in the liquid phase,
:
with the convention
In other words: Pure liquids represent the ideal case.
At infinite dilution, the activity coefficient approaches its limiting value,
∞. Comparison with
Henry's law
In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulat ...
,
:
immediately gives
:
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:
:
with
and
:
The
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 constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
s to be applied to both ideal and non-ideal mixtures.
Knowledge of activity coefficients is particularly important in the context of
electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...
since the behaviour of
electrolyte
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon dis ...
solutions is often far from ideal, due to the effects of the
ionic atmosphere Ionic Atmosphere is a concept employed in Debye-Hückel theory which explains the electrolytic conductivity behaviour of solutions. It can be generally defined as the area at which a charged entity is capable of attracting an entity of the opposit ...
. Additionally, they are particularly important in the context of
soil chemistry
Soil chemistry is the study of the chemical characteristics of soil. Soil chemistry is affected by mineral composition, organic matter and environmental factors. In the early 1850s a consulting chemist to the Royal Agricultural Society in England, ...
due to the low volumes of solvent and, consequently, the high concentration of
electrolytes
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon di ...
.
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
Sodium chloride , commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. With molar masses of 22.99 and 35.45 g/ ...
it is given by the following:
:
where
and
are the activity coefficients of the cation and anion respectively.
More generally, the mean activity coefficient of a compound of formula
is given by
: