Editing Acid dissociation constant (section)

== Equilibrium constant ==
An acid dissociation constant is a particular example of an [[equilibrium constant]]. The dissociation of a [[monoprotic acid]], HA, in dilute solution can be written as
: <chem>HA <=> A- + H+</chem>

The thermodynamic equilibrium constant {{tmath|K^\ominus}} can be defined by<ref name="rr">{{cite book |last=Rossotti |first=F.J.C. |title=The Determination of Stability Constants |author2=Rossotti, H. |publisher=McGraw–Hill |year=1961}} Chapter 2: Activity and Concentration Quotients pp 5-10</ref>
:<math chem>K^\ominus = \frac{\{\ce{A^-}\} \{\ce{H+}\}}{\ce{\{HA\} }}</math>

where <math>\{X\}</math> represents the [[activity (chemistry)|activity]], at equilibrium, of the chemical species X. <math>K^\ominus</math> is [[dimensionless]] since activity is dimensionless. Activities of the products of dissociation are placed in the numerator, activities of the reactants are placed in the denominator. See [[activity coefficient]] for a derivation of this expression.

[[File:PK acetic acid.png|thumb|200px|alt=Illustration of the effect of ionic strength on the p K A of an acid. In this figure, the p K A of acetic acid decreases with increasing ionic strength, dropping from 4.8 in pure water (zero ionic strength) and becoming roughly constant at 4.45 for ionic strengths above 1 molar sodium nitrate, N A N O 3.|Variation of p''K''<sub>a</sub> of acetic acid with ionic strength.]]

Since activity is the product of [[concentration]] and [[activity coefficient]] (''γ'') the definition could also be written as
:<math chem>K^\ominus = {\frac{[\ce{A^-}] [\ce{H+}]}\ce{[HA] }\Gamma}, \quad \Gamma=\frac{\gamma_\ce{A^-} \ \gamma_\ce{H+}}{\gamma_\ce{HA} \ } </math>
where <math>[\text{HA}]</math> represents the concentration of HA and {{tmath|\Gamma}} is a quotient of activity coefficients.

To avoid the complications involved in using activities, dissociation constants are [[Determination of equilibrium constants|determined]], where possible, in a medium of high [[ionic strength]], that is, under conditions in which {{tmath|\Gamma}} can be assumed to be always constant.<ref name=rr /> For example, the medium might be a solution of 0.1&nbsp;[[molar (unit)|molar]] (M) [[sodium nitrate]] or 3&nbsp;M [[potassium perchlorate]]. With this assumption,
:<math>K_\text{a} = \frac{K^\ominus}{\Gamma} = \mathrm{\frac{[A^-] [H^+]}{[HA]}}</math>
:<math chem>\mathrm{p}K_\ce{a} = -\log_{10}\frac{[\ce{A^-}][\ce{H^+}]}{[\ce{HA}]} = \log_{10}\frac{\ce{[HA]}}{[\ce{A^-}][\ce{H+}]}</math>
is obtained.<!-- <ref name=IUPAC-NMR>{{cite journal | last = Popov| first = K.| author2 = Ronkkomaki, H.| author3 = Lajunen, L.H.J.| year = 2006| title = Guidelines for NMR measurements for Determination of High and Low p''K''<sub>a</sub> Values| url = http://media.iupac.org/publications/pac/2006/pdf/7803x0663.pdf | format = PDF | journal = Pure Appl. Chem. | volume = 78 | issue = 3 | pages = 663–675 | doi = 10.1351/pac200678030663}}</ref> Irrelevant in this context--> Note, however, that all published dissociation constant values refer to the specific ionic medium used in their determination and that different values are obtained with different conditions, as shown for [[acetic acid]] in the illustration above. When published constants refer to an ionic strength other than the one required for a particular application, they may be adjusted by means of [[Specific ion interaction theory|specific ion theory]] (SIT) and other theories.<ref>{{cite web | url= https://iupac.org/projects/project-details/?project_nr=2000-003-1-500 | title= Project: Ionic Strength Corrections for Stability Constants | publisher = International Union of Pure and Applied Chemistry | access-date = 2019-03-28}}</ref>

===Cumulative and stepwise constants===
A cumulative equilibrium constant, denoted by {{tmath|\mathrm \beta ,}} is related to the product of stepwise constants, denoted by {{tmath|\mathrm K .}} For a dibasic acid the relationship between stepwise and overall constants is as follows
:<chem>H2A <=> A^2- + 2H+</chem>
:<math chem>\beta_2 = \frac{\ce{[H2A]}}{[\ce{A^2-}][\ce{H+}]^2}</math>
:<math chem>\log \beta_2 = \mathrm{p}K_\ce{a1} + \mathrm{p}K_\ce{a2}</math>

Note that in the context of metal-ligand complex formation, the equilibrium constants for the formation of metal complexes are usually defined as ''association'' constants. In that case, the equilibrium constants for ligand protonation are also defined as association constants. The numbering of association constants is the reverse of the numbering of dissociation constants; in this example <math chem>\log \beta_1 = \mathrm{p}K_\ce{a2}</math>

=== Association and dissociation constants ===
When discussing the properties of acids it is usual to specify equilibrium constants as acid dissociation constants, denoted by ''K''<sub>a</sub>, with numerical values given the symbol p''K''<sub>a</sub>.
:<math chem>K_\text{dissoc} = \frac{ \ce{[A- ][H+]}}{\ce{[HA]}}: \mathrm{p}K_\text{a} = -\log K_\text{dissoc} </math>

On the other hand, association constants are used for bases. 
:<math chem="">K_\text{assoc} = \frac{\ce{[HA]}}{\ce{[A- ][H+]}} </math>

However, [[Determination of equilibrium constants#Implementations|general purpose computer programs]] that are used to derive equilibrium constant values from experimental data use association constants for both acids and bases. Because stability constants for a [[Coordination complex|metal-ligand complex]] are always specified as association constants, ligand protonation must also be specified as an association reaction.<ref>{{Cite book |last1=Rossotti |first1=Francis J. C |url=https://books.google.com/books?id=zY8zAAAAIAAJ&q=The%20determination%20of%20stability%20constants%20:%20and%20other%20equilibrium%20constants%20in%20solution |title=The determination of stability constants : and other equilibrium constants in solution |last2=Rozotti |first2=Hazel |publisher=McGraw-Hill |year=1961 |isbn=9781013909146 |location=New York |pages=5{{--}}10 |language=en |archive-url=https://archive.org/details/determinationofs0000ross/page/10/mode/2up |archive-date=7 February 2020}}</ref> The definitions show that the value of an acid dissociation constant is the reciprocal of the value of the corresponding association constant:
: <math>K_\text{dissoc} = \frac{1}{K_\text{assoc}}</math>
: <math>\log K_\text{dissoc} = - \log K_\text{assoc}</math>
: <math>\mathrm{p}K_\text{dissoc} = - \mathrm{p}K_\text{assoc}</math>

Notes
# For a given acid or base in water, {{nowrap|1=p''K''<sub>a</sub> + p''K''<sub>b</sub> = p''K''<sub>w</sub>}}, the [[self-ionization of water|self-ionization constant of water]].
# The association constant for the formation of a [[Supramolecular chemistry|supramolecular]] complex may be denoted as K<sub>a</sub>; in such cases "a" stands for "association", not "acid".
# For polyprotic acids, the numbering of stepwise association constants is the reverse of the numbering of the dissociation constants. For example, for [[phosphoric acid]] (details in the [[#polyprotic acids|polyprotic acids]] section below):
::<math>\begin{align}
\log K_{\text{assoc},1} &= \mathrm{p}K_{\text{dissoc},3} \\
\log K_{\text{assoc},2} &= \mathrm{p}K_{\text{dissoc},2} \\
\log K_{\text{assoc},3} &= \mathrm{p}K_{\text{dissoc},1}
\end{align}</math>

=== Temperature dependence ===
All equilibrium constants vary with [[temperature]] according to the [[van 't Hoff equation]]<ref>
{{cite book
| title = Physical Chemistry
| last1 = Atkins
| first1 = P.W.
| last2 = de Paula
| first2 = J.
| year = 2006
| publisher = Oxford University Press
| isbn = 0-19-870072-5
}} Section 7.4: The Response of Equilibria to Temperature</ref>
:<math alt="The derivative of the natural logarithm of any equilibrium constant K with respect to the [[absolute temperature]] T equals the standard enthalpy change for the reaction divided by the product R times T squared. Here R represents the gas constant, which equals the thermal energy per mole per kelvin. The standard enthalpy is written as Delta H with a superscript plimsoll mark represented by the image strikeO. This equation follows from the definition of the Gibbs energy Delta G equals R times T times the natural logarithm of K.">
 \frac{\mathrm{d} \ln\left(K\right)}{\mathrm{d}T} = \frac{\Delta H^\ominus}{RT^2}
</math>

{{tmath|R}} is the [[gas constant]] and {{tmath|T}} is the [[kelvin|absolute temperature]]. Thus, for [[exothermic]] reactions, the standard [[enthalpy change]], {{tmath|\Delta H^\ominus}}, is negative and ''K'' decreases with temperature. For [[endothermic]] reactions, {{tmath|\Delta H^\ominus}} is positive and ''K'' increases with temperature.

The standard enthalpy change for a reaction is itself a function of temperature, according to [[Gustav Kirchhoff#Kirchhoff's law of thermochemistry|Kirchhoff's law of thermochemistry]]:
:<math>\left(\frac{\partial\Delta H}{\partial T}\right)_p = \Delta C_p</math>
where {{tmath|\Delta C_p}} is the [[Specific heat capacity|heat capacity]] change at constant pressure. In practice {{tmath|\Delta H^\ominus}} may be taken to be constant over a small temperature range.

===Dimensionality===
In the equation
:<math>K_\mathrm{a} = \mathrm{\frac{[A^-] [H^+]}{[HA]}},</math>

''K''<sub>a</sub> appears to have [[dimensional analysis|dimensions]] of concentration. However, since <math>\Delta G = -RT\ln K</math>, the equilibrium constant, {{tmath|K}}, ''cannot'' have a physical dimension. This apparent paradox can be resolved in various ways. 
# Assume that the quotient of activity coefficients has a numerical value of 1, so that {{tmath|K}} has the same numerical value as the thermodynamic equilibrium constant <math>K^\ominus</math>.
# Express each concentration value as the ratio c/c<sup>0</sup>, where c<sup>0</sup> is the concentration in a [hypothetical] standard state, with a numerical value of 1, by definition.<ref>{{cite book |last1 = Petrucci |first1 = Ralph H. |last2 = Harwood |first2 = William S. |last3 = Herring |first3 = F. Geoffrey |date=2002 |title = General chemistry: principles and modern applications |url = https://archive.org/details/generalchemistry00hill |url-access = registration |edition=8th |page=[https://archive.org/details/generalchemistry00hill/page/633 633] |quote=Are you wondering... How using activities makes the equilibrium constant dimensionless? |publisher=Prentice Hall |isbn = 0-13-014329-4}}</ref> 
# Express the concentrations on the [[mole fraction]] scale. Since mole fraction has no dimension, the quotient of concentrations will, by definition, be a pure number.

The procedures, (1) and (2), give identical numerical values for an equilibrium constant. Furthermore, since a concentration {{tmath|c_i}} is simply proportional to mole fraction {{tmath|x_i}} and density {{tmath|\rho}}:
:<math>c_i = \frac{x_i\rho}{M} </math>
and since the molar mass {{tmath|M}} is a constant in dilute solutions, an equilibrium constant value determined using (3) will be simply proportional to the values obtained with (1) and (2).

It is common practice in [[biochemistry]] to quote a value with a dimension as, for example, "''K''<sub>a</sub> = 30&nbsp;mM" in order to indicate the scale, millimolar (mM) or micromolar (μM) of the [[concentration]] values used for its calculation.