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==Overview==

An '''acid dissociation constant''', denoted by K<sub>a</sub>, is an [[equilibrium constant]] for the dissociation of a '''weak acid'''. According to the [[Brønsted-Lowry]] theory of [[acid]]s and [[Base (chemistry)|base]]s, an acid is a [[proton]] donor (HA, where H represents an acidic [[hydrogen atom]]), and a base is a proton acceptor.  In [[aqueous solution]], water can function as a base, as in the following general example.
:HA + H<sub>2</sub>O <math>\rightleftharpoons</math> A<sup>-</sup> + H<sub>3</sub>O<sup>+</sup>
Acid dissociation constants are also known as the '''acidity constant''' or the '''acid-ionization constant'''. The term is also used for pK<sub>a</sub>, which is equal to minus the decimal [[logarithm]] of K<sub>a</sub> ([[cologarithm]] of K<sub>a</sub>).

{{Chemical equilibria}}

==Definitions==

=== Monoprotic acids ===
When an acid, HA, dissolves in water, some molecules of the acid 'dissociate' to form [[hydronium ions]] and the [[Brønsted-Lowry|conjugate base]], (A<sup>-</sup>), of the acid.

: <math>H A \rightleftharpoons H^{+} + A^{-}</math>

It is understood that <math>H^{+}</math> stands for the hydronium ion and that each species in this equilibrium may solvate to a greater or lesser extent. The acid dissociation constant is [[equilibrium constant|defined]] as

:<math>K_a = \frac{[\mbox{H}^+][\mbox{A}^- ]} {[\mbox{HA}]}</math>

where the square brackets are usually taken to signify [[concentration]]. <math>H_2 O</math> is omitted from these expressions because in dilute solution the concentration of water may be assumed to be constant. Values of <math>K_a</math> vary over many orders of magnitude, so it is common to take the [[logarithm]] to base ten of the value. 

:<math>\operatorname{p}K_a = -\log_{10} K_a</math>

It is easier to compare the strengths of different acids by comparing <math>pK_a</math> values as they vary over a much smaller range.

=== Polyprotic acids ===
A [[polyprotic acid]] is one that has more than one proton to dissociate. Typical examples are [[malonic acid]], which has two ionizable protons, and [[phosphoric acid]], which has three. The constant for dissociation of the first proton may be denoted as <math>pK_{a1}</math> and the constants for dissociation of successive protons as <math>pK_{a2}</math>, etc.

It is generally true that successive pK values increase (Pauling's first rule).<ref name="pauling"> {{Greenwood&Earnshaw}} p. 50</ref> For example, for a diprotic acid, <math>H_2 A</math>, the two equilibria are

:<math>H_2 A \rightleftharpoons HA^{-} + H^{+}</math>
:<math>HA^{-} \rightleftharpoons A^{2-} + H^{+}</math>

it can be seen that the second proton is removed from a negatively charged species. Since the proton carries a positive charge extra work is needed to remove it. Therefore <math>pK_{a2} > pK_{a1}</math>. There are a few exceptions to this rule which occur when there is a major structural change such as in the sequence

:{| class="wikitable"
|-
| <math>VO_2^{+}(aq) \rightleftharpoons H_3 VO_4 + H^+</math>
| <math>pK_{a1} = 4.2</math>
|-
| <math>H_3 VO_4 \rightleftharpoons H_2 VO_4^{-} + H^{+}</math>
| <math>pK_{a2} = 2.60</math>
|-
| <math>H_2 VO_4^{-} \rightleftharpoons HVO_4^{2-} + H^{+}</math>
| <math>pK_{a3} = 7.92</math>
|-
| <math>HVO_4^{2-} \rightleftharpoons VO_4^{3-} + H^{+}</math>
| <math>pK_{a4} = 13.27</math>
|}

The pKs of vanadic acid, <math>H_3 VO_4</math>, follow Pauling's rule just like [[phosphoric acid]] (values below). All species in this series are tetrahedral, but <math>VO_2 \left( H_2 O \right)_4 ^{+}</math> is octahedral and <math>pK_{a2} < pK_{a1}</math>. <ref>{{Greenwood&Earnshaw}} Chapter 22</ref>

===Bases===
Historically the equilibrium constant <math>K_b</math> for a base was defined as the dissociation constant of <math>BH</math>, the acid conjugate to the base, <math>B</math>. Ionic charges on <math>B</math> and/or <math>BH</math> are omitted here because the base may be a neutral species such as [[ammonia]] or a charged species such as the [[ethanoate]] ion (acetate).

:<math>B + H_2 O \rightleftharpoons BH + OH^{-}</math>

Using similar reasoning to that used before

:<math>K_b = \frac{[\mbox{BH}][\mbox{OH}^-]} {[\mbox{B}]}</math>        
:<math>\operatorname{p}K_b = -\log_{10} K_b\,</math>

Now, the concentration of the [[hydroxide]] ion is related to the concentration of the hydronium by <math>K_w = \left[ H^{+} \right]\left[ OH^{-} \right]</math>, therefore

:<math>[\mbox{OH}^-] = \frac{K_w}{[\mbox{H}^+]}</math>; 

<math>K_w</math> is the constant for the [[self-ionization of water]]. Substituting the expression for <math>\left[ OH^{-} \right]</math> into the expression for <math>K_b</math>

:<math>K_b = K_w \frac{ \left[ \mbox{BH} \right] }{ \left[ \mbox{B} \right] \left[ \mbox{H}^{+} \right] } = \frac{K_w}{K_a}</math>        

It follows that

:<math>pK_b = pK_w - pK_a</math>

In water at 25 °C <math>pK_w</math> is 14 so then <math>pK_b = 14 - pK_a</math>.

In effect there is no need to define <math>pK_b</math> separately from <math>pK_a</math>, but it is done because <math>pK_b</math> values can be found in literature.

=== Protonation constants ===
The protonation constant for a substance <math>S</math>, is given by

:<math>S + H^{+} \rightleftharpoons SH</math>
:<math>K_{assoc} = \frac{[\mbox{SH}]} {[\mbox{S}][\mbox{H}^+]}</math>

Again, ionic charges are omitted from <math>S</math> and <math>SH</math>.
<math>K_{assoc}</math> is an [[equilibrium constant|association constant]].

* For any substance <math>\log_{10} \left( K_{assoc} \right) = pK_a</math>

=== Temperature dependence ===
All equilibrium constants vary with temperature according the [[van 't Hoff equation]]
:<math>\frac {d\ln K} {dT} = \frac{{\Delta H_m}^{\Theta}} {RT^2}.</math> 
Thus, for [[exothermic]] reactions, (ΔH is negative) ''K'' decreases with temperature, but for [[endothermic]] reactions (ΔH is positive) ''K'' increases with temperature.

==Usage==
[[Image:weak acid speciation.png|left]]
The [[pH]] of a solution of '''weak acid''' can be expressed in terms of the extent of dissociation. After rearranging the expression defining the dissociation constant, and putting pH = -log<sub>10</sub>[H<sup>+</sup>], one obtains  
:<math>\mbox{pH} = \mbox{pK}_a - \log \frac{[HA]}{[A^-]}</math>
This is a form of the [[Henderson-Hasselbalch equation]]. It can be deduced from this expression that
* when the acid is 1% dissociated, that is, when <math>\frac{[HA]}{[A^-]} = 100</math>, pH = pK<sub>a</sub> &ndash; 2
* when the acid is 50% dissociated, that is, when <math>\frac{[HA]}{[A^-]} = 1</math>, pH = pK<sub>a</sub>
* when the acid is 99% dissociated, that is, when <math>\frac{[HA]}{[A^-]} = 0.01</math>, pH = pK<sub>a</sub> + 2
It follows that the range of pH within which there is partial dissociation of the acid is about pK<sub>a</sub> <math>\pm</math> 2.This is shown graphically at the right.

A '''weak acid''' may be '''defined''' as an acid with pK<sub>a</sub> greater than about -2. An acid with pK<sub>a</sub> = -2 would be 99% dissociated at pH 0, that is, in a 1M HCl solution. Any acid with a pK<sub>a</sub> less than about -2 is said to be a '''strong acid'''. Strong acids are said to be fully dissociated. There is no precise pK<sub>a</sub> value that distinguishes between strong and weak acids because strong acids, such as [[sulfuric acid]], are associated in very concentrated solution.

On the pK<sub>a</sub> scale of acid strength, a large value indicates a very weak acid, and a small value indicates a not so weak one.  

The pH of a solution of a weak acid can be easily calculated when the [[analytical concentration]] of the acid is known. See [[ICE table]] for details.

Some polyprotic acids can be treated as a set of individual acids. This is possible when successive pK values differ by 4 or more. For example with phosphoric acid
:H<sub>3</sub>PO<sub>4</sub> <math>\rightleftharpoons</math> H<sub>2</sub>PO<sub>4</sub><sup>-</sup> +H<sup>+</sup>, pK<sub>a1</sub> = 2.15
:H<sub>2</sub>PO<sub>4</sub><sup>-</sup> <math>\rightleftharpoons</math> HPO<sub>4</sub><sup>2-</sup> +H<sup>+</sup>, pK<sub>a2</sub> = 7.20
:HPO<sub>4</sub><sup>2-</sup> <math>\rightleftharpoons</math> PO<sub>4</sub><sup>3-</sup> +H<sup>+</sup>, pK<sub>a3</sub> = 12.37
Both the hydrogenphosphate and dihydrogenphosphate ions can be treated as acids in their own right. On the other hand, the two pKs for malonic acid are 2.51 and 5.05, so there are pH values at which both malonic acid and the hydrogenmalonate ion co-exist. More elaborate calculations are needed to calculate the composition of solutions of malonic acid.

==Factors that determine the relative strengths of acids==

Being an [[equilibrium constant]], the acid dissociation constant ''K''<sub>a</sub> is determined 
by the standard free energy difference ΔG<sup><s>o</s></sup> between the reactants and products, specifically, between the protonated (HA) and deprotonated (A<sup>−</sup>) forms of the substance.

Pauling's second rule<ref name="pauling"/> states that the value of the first pK for acids of the formula XO<sub>m</sub>(OH) <sub>n</sub> is approximately independent of n and X and is approximately 8 for m=0, 2 for m=1, -3 for m=2 and <-10 for m=3. This correlates with the oxidation state of the central atom, X: the higher the oxidation state the stronger the oxyacid. For example, pK<sub>a</sub> for HClO is 7.2, for HClO<sub>2</sub> is 2.0, for HClO<sub>3</sub> is -1 and HClO<sub>4</sub> is a strong acid.    

With organic acids [[inductive effects]] and [[mesomeric effect]]s affect the pKs. The effects are summarised in the [[Hammett equation]] and subsequent extensions. 

Structural effects can also be important. The difference between [[fumaric acid]] and [[maleic acid]] is a classic example. Fumaric acid is (E)-1,4-but-2-enedioic acid, a ''trans'' [[isomer]], whereas maleic acid is the corresponding ''cis'' isomer, i.e. (Z)-1,4-but-2-enedioic acid (see [[cis-trans isomerism]]). Fumaric acid has pK<sub>a</sub>s of approximately 3.5 and 4.5. By contrast, maleic acid has pK<sub>a</sub>s of approximately 1.5 and 6.5. The reason for this large difference is that when one proton is removed from the cis- isomer (maleic acid) a strong [[intramolecular]] [[hydrogen bond]] is formed with the nearby remaining carboxyl group. This favors the formation of the maleate H<sup>+</sup>, and it opposes the removal of the second proton from that species. In the ''trans'' isomer, the two carboxyl groups are always far apart, so hydrogen bonding is not observed.

== Importance of p''K''<sub>a</sub> values ==
The p''K''<sub>a</sub> value(s) of a compound influence many characteristics of the compound such as its reactivity, and spectral properties (colour). In [[biochemistry]] the p''K''<sub>a</sub> values of [[protein]]s and [[amino acid]] side chains are of major importance for the activity of [[enzyme]]s and the stability of proteins.  This property is of general importance in chemistry because ionization of a compound alters its physical behavior and macro properties such as [[solubility]] and [[partition coefficient|lipophilicity]].  For example ionization of any compound will increase the solubility in water, but decrease the lipophilicity.  This can be exploited in drug development to increase the concentration of a compound in the blood by adjusting the pKa of an ionizable group. This must be done with caution, however, since an ionized compound will pass less easily through cell membranes.  
{{further|[[Protein pKa calculations]]}}

==Acidity in nonaqueous solutions==
Three properties of a solvent strongly affect acids and bases.
#A [[Protic solvent]] can form hydrogen bonds and will promote ionisation.
#A solvent with a high [[donor number]] is a strong [[Lewis base]].
#A solvent with a high [[dielectric constant]] (Relative permittivity) will promote ionisation.
{| class="wikitable"
|-
! 
! Donor number
! dielectric constant
|-
| [[Acetonitrile]]
| 14.1
| 37.5
|-
| [[Dimethyl sulfoxide]]
| 29.8
| 45
|-
| [[Ethanol]]
| 31.5
| 24.3
|-
| [[Pyridine]]
| 33.1
| 12.3
|-
| [[Water]]
| 18
| 81.7
|}

For a given acid, pK<sub>a</sub> values will vary depending on solvent. The degree of dissociation of any acid increases with the increasing [[basicity]] of the solvent.  On the other hand, dissociation is relatively less for solvents of low dielectric constant. An acidic solvent will also suppress dissociation of an acid. For example, hydrogen chloride is a weak acid, i.e. poorly ionised, when dissolved in the acidic solvent acetic acid.

Acidity scales have been developed for solvents aside from water, notably for [[dimethyl sulfoxide]] and [[acetonitrile]].<ref>March, J. “Advanced Organic Chemistry” 4th Ed.  J. Wiley and Sons, 1992: New York.  ISBN 0-471-60180-2.</ref>  It can be seen in the table above that DMSO is more basic than water, but its dielectric constant is less. DMSO is widely used as an alternative to water in evaluating acids and bases.

In solvents of low dielectric constant, ions tend to associate, which complicates the interpretation of pK<sub>a</sub>s. In particular, in aprotic solvents the process of homoconjugation occurs when the conjugate base forms a [[hydrogen bond]] with the parent acid as in the following equilibrium
:HA  +  A<sup>-</sup> <math>\rightleftharpoons</math> HA<sub>2</sub><sup>-</sup>
Typically HA<sub>2</sub><sup>-</sup> would have the structure A---H---A. This process does not occur in water because H<sub>2</sub>O molecules are strong hydrogen bond donors and acceptors.

In acetonitrile solution, [[p-Toluenesulfonic acid|para-toluenesulfonic acid]] has a homoconjugation constant pK<sup>f</sup>, of -2.9.<ref>Coetzee, J. F. and Padmanabhan, G. R., "Proton Acceptor Power and Homoconjugation of Mono- and Diamines", J. Amer. Chem. Soc.,1965, '''87''', 5005-5010.</ref> This indicates that the toluenesulfonate anion has a strong tendency to form a hydrogen bond with the parent acid.  Homoconjugation has the effect of enhancing the acidity of acids, lowering their effective pK<sub>a</sub>s, by stabilizing the conjugate base.  Due to homoconjugation, the proton-donating power of toluenesulfonic acid in acetonitrile solution is enhanced by a factor of nearly 800.

==p''K''<sub>a</sub> of some common substances==
Measurements are at 25ºC in water for those with a pKa at or above -1.76:
{{col-begin-small}}
{{col-2}}
* - 25.00: [[Fluoroantimonic acid]]
* - 15.00: [[Magic acid]]
* - 10.00: [[Fluorosulfuric acid]]
* - 10.00: [[Perchloric acid]]
* - 10.00: [[Hydroiodic acid]]
* - 9.00: [[Hydrobromic acid]]
* - 8.00: [[Hydrochloric acid]]
* - 3.00, 1.99: [[Sulfuric acid]]
* - 2.00: [[Nitric acid]]
* - 1.76: [[Hydronium ion]]
* 3.15: [[Hydrofluoric acid]]
* 3.60: [[Carbonic acid]]
* 3.75: [[Formic acid]]
* 4.04: [[Ascorbic acid]] ([[Vitamin C]])
* 4.19: [[Succinic acid]]
* 4.20: [[Benzoic acid]]
* 4.63: [[Aniline]]*
* 4.76: [[Acetic acid]]
* 4.76: Dihydrogencitrate ion ([[Citrate]])
* 5.21: [[Pyridine]]*
* 6.40: Monohydrogencitrate ion [[Citrate]]
* 6.99: [[Ethylenediamine]]*
{{col-2}}
* 7.00: [[Hydrogen sulfide]], [[Imidazole]]* (as an acid) 
* 7.50: [[Hypochlorous acid]]
* 9.25: [[Ammonia]]*
* 9.33: [[Benzylamine]]*
* 9.81: [[Trimethylamine]]*
* 9.99: [[Phenol]]
* 10.08: [[Ethylenediamine]]*
* 10.66: [[Methylamine]]*
* 10.73: [[Dimethylamine]]*
* 10.81: [[Ethylamine]]*
* 11.01: [[Triethylamine]]*
* 11.09: [[Diethylamine]]*
* 11.65: [[Hydrogen peroxide]]
* 12.50: [[Guanidine]]*
* 12.67: Monohydrogenphosphate ion ([[Phosphate]])
* 14.58: [[Imidazole]] (as a base)
* 15.76: [[Water]]
* - 19.00 (pKb) [[Sodium amide]]
* 37.00: [[Lithium diisopropylamide]] (LDA)
* 45.00: [[Propane]]
* 50.00: [[Ethane]]
{{col-end}}
'''*''' Listed values for ammonia and amines are the pKa values for the corresponding ammonium ions.

==Further reading==
*Atkins, Peter, and Loretta Jones. ''Chemical Principles: The Quest for Insight''. 3rd ed. New York: W. H. Freeman and Company, 2005
* Housecroft, Catherine and Sharpe, Alan, ''Inorganic Chemistry'', Prentice Hall, 2nd. edition, 2004. (Non-aqueous solvents) 
==See also==
* [[Hammett equation]]
* [[QSAR]]
* [[Dissociation constant]]
* [[Henderson-Hasselbalch equation]]

==References==
<references/>

==External links==

*[http://www.chem.wisc.edu/areas/reich/pkatable/ Bordwell pKa Table in DMSO]
*[http://www.shodor.org/unchem/basic/ab/ Shodor.org Acid-Base Chemistry]
*[http://library.thinkquest.org/C006669/data/Chem/acidsbases/factors.html Factors that Affect the Relative Strengths of Acids and Bases]
*[http://chemed.chem.purdue.edu/genchem/topicreview/bp/3organic/3org_frame.html Purdue Chemistry]
*[http://www2.iq.usp.br/docente/gutz/Curtipot_.html Distribution diagrams of acids and bases] (generation from p<math>K_{a}</math> values with free spreadsheet)
*[http://sparc.chem.uga.edu SPARC Physical/Chemical property calculator]
*[http://www.jesuitnola.org/upload/clark/Refs/aqueous.htm List of Aqueous-Equilibrium Constants]
*[http://www.raell.demon.co.uk/chem/logp/logppka.htm Free guide to pKa & logP interpretation and measurement]

<br>

{{chemical solutions}}
[[Category:Acids]]
[[Category:Analytical chemistry]]
[[Category:Thermodynamics]]



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