Editing Equilibrium constant (section)

== Dimensionality ==
An equilibrium constant is related to the standard [[Gibbs free energy]] of reaction change, <math>\Delta_R G^\ominus </math>,<ref>[https://iupac.org/wp-content/uploads/2019/05/IUPAC-GB3-2012-2ndPrinting-PDFsearchable.pdf. Green Book (IUPAC), Quantities, Units and Symbols in Physical Chemistry, page 61, édition 2007.]</ref> for the reaction by the expression
:<math>\Delta_R G^\ominus = \left( \frac{\partial G}{\partial \xi } \right)_{P,T} = -RT \ln K  .</math>
Therefore, ''K'', must be a [[Dimensionless quantity|dimensionless number]] from which a logarithm can be derived. In the case of a simple equilibrium
:<chem>A + B <=> AB, </chem>
the thermodynamic equilibrium constant is defined in terms of the [[thermodynamic activity|activities]], {AB}, {A} and {B}, of the species in equilibrium with each other:
:<math>K = \frac{\{AB\}}{\{A\}\{B\}}  .</math>
Now, each activity term can be expressed as a product of a concentration <math>[X]</math> and a corresponding [[activity coefficient]], <math>\gamma(X)</math>. Therefore,
:<math>K = \frac{[AB]}{[A][B]}\times\frac{\gamma(AB)}{\gamma(A)\gamma(B)} = \frac{[AB]}{[A][B]}\times \Gamma .</math>
When <math>\Gamma</math>, the quotient of activity coefficients, is set equal to 1, we get 
:<math>K = \frac{[AB]}{[A][B]}   .</math>
''K'' then appears to have the dimension of 1/concentration. This is what usually happens in practice when an equilibrium constant is calculated as a quotient of concentration values. This can be avoided by dividing each concentration by its standard-state value (usually mol/L or bar), which is standard practice in chemistry.<ref name="Atkins7th" />

The assumption underlying this practice is that the quotient of activities is constant under the conditions in which the equilibrium constant value is determined. These conditions are usually achieved by keeping the reaction temperature constant and by using a medium of relatively high [[ionic strength]] as the solvent. It is not unusual, particularly in texts relating to biochemical equilibria, to see an equilibrium constant value quoted with a dimension. The justification for this practice is that the concentration scale used may be either mol dm<sup>−3</sup> or mmol dm<sup>−3</sup>, so that the concentration unit has to be stated in order to avoid there being any ambiguity.

''Note''. When the concentration values are measured on the [[mole fraction]] scale all concentrations and activity coefficients are dimensionless quantities.

In general equilibria between two reagents can be expressed as
:<chem>{\mathit{p}A} + \mathit{q}B <=> A_\mathit{p}B_\mathit{q}  , </chem>
in which case the equilibrium constant is defined, in terms of numerical concentration values, as
:<math chem="">K = \frac{[\ce{A}_p\ce{B}_q]}{[\ce A]^p[\ce B]^q} .</math>
The apparent dimension of this ''K'' value is concentration<sup>1−p−q</sup>; this may be written as M<sup>(1−p−q)</sup> or mM<sup>(1−p−q)</sup>, where the symbol M signifies a [[molar concentration]] ({{math|1=1M = 1 mol dm<sup>−3</sup>}}). The apparent dimension of a [[dissociation constant]] is the reciprocal of the apparent dimension of the corresponding [[association constant]], and ''vice versa''.

When discussing the [[chemical thermodynamics|thermodynamics]] of chemical equilibria it is necessary to take dimensionality into account. There are two possible approaches.
# Set the dimension of {{math|&Gamma;}} to be the reciprocal of the dimension of the concentration quotient. This is almost universal practice in the field of stability constant determinations. The "equilibrium constant" <math>\frac{K}{\Gamma}</math>, is dimensionless. It will be a function of the ionic strength of the medium used for the determination. Setting the numerical value of {{math|&Gamma;}} to be 1 is equivalent to re-defining the [[standard state]]s. 
# Replace each concentration term <math>[X]</math> by the dimensionless quotient <math>\frac{[X]}{[X^0]}</math>, where <math>[X^0]</math> is the concentration of reagent {{mvar|X}} in its standard state (usually 1&nbsp;mol/L or 1 bar).<ref name="Atkins7th" /> By definition the numerical value of <math>\gamma(X^0)</math> is 1, so {{math|&Gamma;}} also has a numerical value of 1. 
In both approaches the numerical value of the stability constant is unchanged. The first is more useful for practical purposes; in fact, the unit of the concentration quotient is often attached to a published stability constant value in the biochemical literature. The second approach is consistent with the standard exposition of [[Debye–Hückel theory]], where <math>\gamma(AB)</math>, ''etc''. are taken to be pure numbers.