Editing Acid dissociation constant (section)

== Factors that affect p''K''<sub>a</sub> values ==
Pauling's second rule is that the value of the first p''K''<sub>a</sub> for acids of the formula XO<sub>''m''</sub>(OH)<sub>''n''</sub> depends primarily on the number of oxo groups ''m'', and is approximately independent of the number of hydroxy groups ''n'', and also of the central atom X. Approximate values of p''K''<sub>a</sub> are 8 for ''m''&nbsp;=&nbsp;0, 2 for ''m''&nbsp;=&nbsp;1, −3 for ''m''&nbsp;=&nbsp;2 and <&nbsp;−10 for ''m''&nbsp;=&nbsp;3.<ref name="pauling" /> Alternatively, various numerical formulas have been proposed including p''K''<sub>a</sub>&nbsp;=&nbsp;8&nbsp;−&nbsp;5''m'' (known as [[Ronnie Bell (chemist)|Bell's]] rule),<ref name=Miessler2ed /><ref name=House2>{{cite book
| last1 = Housecroft
| first1 = Catherine E.
| last2 = Sharpe
| first2 = Alan G.
| title = Inorganic chemistry
| date = 2005
| publisher = Pearson Prentice Hall
| location = Harlow, U.K.
| isbn = 0-13-039913-2
| pages = 170–171
| edition = 2nd
}}</ref> p''K''<sub>a</sub>&nbsp;=&nbsp;7&nbsp;−&nbsp;5''m'',<ref name=Huheey /><ref name=Douglas>Douglas B., McDaniel D.H. and Alexander J.J. ''Concepts and Models of Inorganic Chemistry'' (2nd ed. Wiley 1983) p.526 {{ISBN|0-471-21984-3}}</ref> or p''K''<sub>a</sub>&nbsp;=&nbsp;9&nbsp;−&nbsp;7''m''.<ref name=Miessler2ed /> The dependence on ''m'' correlates with the oxidation state of the central atom, X: the higher the oxidation state the stronger the oxyacid.

For example, p''K''<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 ({{nowrap|p''K''<sub>a</sub> ≪ 0}}).<ref name="SA" /> The increased acidity on adding an oxo group is due to stabilization of the conjugate base by delocalization of its negative charge over an additional oxygen atom.<ref name="House2" /> This rule can help assign molecular structure: for example, [[phosphorous acid]], having [[molecular formula]] H<sub>3</sub>PO<sub>3</sub>, has a p''K''<sub>a</sub> near 2, which suggested that the structure is HPO(OH)<sub>2</sub>, as later confirmed by [[Nuclear magnetic resonance spectroscopy|NMR spectroscopy]], and not P(OH)<sub>3</sub>, which would be expected to have a p''K''<sub>a</sub> near 8.<ref name="Douglas" />

[[File:Chloroacetic pka.png|thumb|300x300px|pKa values for acetic, chloroacetic, dichloroacetic and trichloroacetic acids.]]

[[Inductive effects]] and [[mesomeric effect]]s affect the p''K''<sub>a</sub> values. A simple example is provided by the effect of replacing the hydrogen atoms in acetic acid by the more electronegative chlorine atom. The electron-withdrawing effect of the substituent makes ionisation easier, so successive p''K''<sub>a</sub> values decrease in the series 4.7, 2.8, 1.4, and 0.7 when 0, 1, 2, or 3 chlorine atoms are present.<ref>{{cite book
| last = Pauling
| first = L.
| title = The nature of the chemical bond and the structure of molecules and crystals; an introduction to modern structural chemistry
| url = https://archive.org/details/natureofchemical00paul
| url-access = registration
| publisher = Cornell University Press
| location = Ithaca (NY)
| year = 1960
| edition = 3rd
| page = [https://archive.org/details/natureofchemical00paul/page/277 277]
| isbn = 0-8014-0333-2
}}</ref> The [[Hammett equation]], provides a general expression for the effect of substituents.<ref>{{cite book
| title = Organic Chemistry
| last = Pine
| first = S.H.
| author2 = Hendrickson, J.B.
| author3 = Cram, D.J.
| author4 = Hammond, G.S.
| year = 1980
| publisher = McGraw–Hill
| isbn = 0-07-050115-7
}} Section 13-3: Quantitative Correlations of Substituent Effects (Part B) – The Hammett Equation</ref>
: log(''K''<sub>a</sub>)&nbsp;=&nbsp;log(''K''{{su|b=a|p=0}})&nbsp;+&nbsp;ρσ.

''K''<sub>a</sub> is the dissociation constant of a substituted compound, ''K''{{su|b=a|p=0}} is the dissociation constant when the substituent is hydrogen, ρ is a property of the unsubstituted compound and σ has a particular value for each substituent. A plot of log(''K''<sub>a</sub>) against σ is a straight line with [[y-intercept|intercept]] log(''K''{{su|b=a|p=0}}) and [[slope]] ρ. This is an example of a [[linear free energy relationship]] as log(''K''<sub>a</sub>) is proportional to the standard free energy change. Hammett originally<ref>{{cite journal
| last = Hammett
| first = L.P.
| title = The Effect of Structure upon the Reactions of Organic Compounds. Benzene Derivatives
| journal = J. Am. Chem. Soc.
| volume = 59
| issue = 1
| pages = 96–103
| doi = 10.1021/ja01280a022
| year = 1937
| bibcode = 1937JAChS..59...96H
}}</ref> formulated the relationship with data from [[benzoic acid]] with different substituents in the ''[[Arene substitution patterns|ortho]]-'' and ''[[Arene substitution patterns|para]]-'' positions: some numerical values are in [[Hammett equation]]. This and other studies allowed substituents to be ordered according to their [[inductive effect|electron-withdrawing]] or [[inductive effect|electron-releasing]] power, and to distinguish between inductive and mesomeric effects.<ref>{{cite journal
| last = Hansch
| first = C.
| author2 = Leo, A.
| author3 = Taft, R. W.
| year = 1991
| title = A Survey of Hammett Substituent Constants and Resonance and Field Parameters
| journal = Chem. Rev.
| volume = 91
| issue = 2
| pages = 165–195
| doi = 10.1021/cr00002a004
| s2cid = 97583278
}}</ref><ref>{{cite journal
| last = Shorter
| first = J
| year = 1997
| title = Compilation and critical evaluation of structure-reactivity parameters and equations: Part 2. Extension of the Hammett σ scale through data for the ionization of substituted benzoic acids in aqueous solvents at 25&nbsp;°C (Technical Report)
| volume = 69
| issue = 12
| pages = 2497–2510
| doi = 10.1351/pac199769122497
| journal = Pure and Applied Chemistry
| s2cid = 98814841
| doi-access= free
}}</ref>

[[Alcohol (chemistry)|Alcohol]]s do not normally behave as acids in water, but the presence of a double bond adjacent to the OH group can substantially decrease the p''K''<sub>a</sub> by the mechanism of [[keto–enol tautomerism]]. [[Ascorbic acid]] is an example of this effect. The diketone 2,4-pentanedione ([[acetylacetone]]) is also a weak acid because of the keto–enol equilibrium. In aromatic compounds, such as [[phenol]], which have an OH substituent, [[Conjugated system|conjugation]] with the aromatic ring as a whole greatly increases the stability of the deprotonated form.

{{Multiple image
| direction         = vertical
| image1            = Fumarsäure.svg
| caption1          = [[Fumaric acid]]
| image2            = Maleic-acid-2D-skeletal-A.svg
| caption2          = [[Maleic acid]]
}}

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 p''K''<sub>a</sub> values of approximately 3.0 and 4.5. By contrast, maleic acid has p''K''<sub>a</sub> values 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 force|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.<ref>{{cite book
| title = Organic chemistry
| last = Pine
| first = S.H.
| author2 = Hendrickson, J.B.
| author3 = Cram, D.J.
| author4 = Hammond, G.S.
| year = 1980
| publisher = McGraw–Hill
| isbn = 0-07-050115-7
}} Section 6-2: Structural Effects on Acidity and Basicity</ref>

[[File:Proton sponge.svg|thumb|upright=0.65|alt=Proton sponge is a derivative of naphthalene with dimethylamino groups in the one and ten positions. This brings the two dimethyl amino groups into close proximity to each other.|Proton sponge]]

[[Proton sponge]], 1,8-bis(dimethylamino)naphthalene, has a p''K''<sub>a</sub> value of 12.1. It is one of the strongest amine bases known. The high basicity is attributed to the relief of strain upon protonation and strong internal hydrogen bonding.<ref>{{cite journal
| last = Alder
| first = R.W.
| author2 = Bowman, P.S.
| author3 = Steele, W.R.S.
| author4 = Winterman, D.R.
| journal = Chem. Commun.
| issue = 13
| year = 1968
| doi = 10.1039/C19680000723
| title = The Remarkable Basicity of 1,8-bis(dimethylamino)naphthalene
| pages = 723–724
}}</ref><ref>{{cite journal
| last = Alder
| first = R.W.
| journal = Chem. Rev.
| year = 1989
| volume = 89
| issue = 5
| pages = 1215–1223
| doi = 10.1021/cr00095a015
| title = Strain Effects on Amine Basicities
}}</ref>

Effects of the solvent and solvation should be mentioned also in this section. It turns out, these influences are more subtle than that of a dielectric medium mentioned above. For example, the expected (by electronic effects of methyl substituents) and observed in gas phase order of basicity of methylamines, Me<sub>3</sub>N > Me<sub>2</sub>NH > MeNH<sub>2</sub> > NH<sub>3</sub>, is changed by water to Me<sub>2</sub>NH > MeNH<sub>2</sub> > Me<sub>3</sub>N > NH<sub>3</sub>. Neutral methylamine molecules are hydrogen-bonded to water molecules mainly through one acceptor, N–HOH, interaction and only occasionally just one more donor bond, NH–OH<sub>2</sub>. Hence, methylamines are stabilized to about the same extent by hydration, regardless of the number of methyl groups. In stark contrast, corresponding methylammonium cations always utilize '''all''' the available protons for donor NH–OH<sub>2</sub> bonding. Relative stabilization of methylammonium ions thus decreases with the number of methyl groups explaining the order of water basicity of methylamines.<ref name="RMC_2013" />

=== Thermodynamics ===
An equilibrium constant is related to the standard [[Gibbs free energy|Gibbs energy]] change for the reaction, so for an acid dissociation constant
: <math>\Delta G^\ominus = -RT \ln K_\text{a} \approx 2.303 RT\ \mathrm{p}K_\text{a}</math>.

''R'' is the [[gas constant]] and ''T'' is the [[kelvin|absolute temperature]]. Note that {{nowrap|p''K''<sub>a</sub> {{=}} −log(''K''<sub>a</sub>)}} and {{nowrap|2.303 ≈ [[natural logarithm|ln]](10)}}. At 25&nbsp;°C, Δ''G''{{sup|⊖}} in kJ·mol<sup>−1</sup> ≈ 5.708 p''K''<sub>a</sub> (1 kJ·mol<sup>−1</sup> = 1000 [[joule]]s per [[mole (unit)|mole]]). Free energy is made up of an [[enthalpy]] term and an [[entropy]] term.<ref name=Goldmine />
:<math>\Delta G^\ominus = \Delta H^\ominus - T \Delta S^\ominus</math>

The standard enthalpy change can be determined by [[calorimetry]] or by using the [[van 't Hoff equation]], though the calorimetric method is preferable. When both the standard enthalpy change and acid dissociation constant have been determined, the standard entropy change is easily calculated from the equation above. In the following table, the entropy terms are calculated from the experimental values of p''K''<sub>a</sub> and Δ''H''{{sup|⊖}}. The data were critically selected and refer to 25&nbsp;°C and zero ionic strength, in water.<ref name="Goldmine">{{cite journal
| title = Thermodynamic Quantities for the Ionization Reactions of Buffers
| last = Goldberg
| first = R.
| author2 = Kishore, N.
| author3 = Lennen, R.
| journal = J. Phys. Chem. Ref. Data
| volume = 31
| issue = 2
| pages = 231–370
| year = 2002
| url = https://www.nist.gov/data/PDFfiles/jpcrd615.pdf
| doi = 10.1063/1.1416902
| bibcode = 2002JPCRD..31..231G
| url-status = dead
| archive-url = https://web.archive.org/web/20081006062140/https://www.nist.gov/data/PDFfiles/jpcrd615.pdf
| archive-date = 2008-10-06
}}</ref>

{| class="wikitable" style="text-align:center;"
|+ Acids
! Compound
! Equilibrium
! p''K''<sub>a</sub>
! Δ''G''{{sup|⊖}} (kJ·mol{{sup|−1}}){{efn|{{nowrap|Δ''G''{{sup|⊖}} ≈ 2.303''RT''p''K''{{sub|a}}}}}}
! Δ''H''{{sup|⊖}} (kJ·mol{{sup|−1}})
! −''T''Δ''S''{{sup|⊖}} (kJ·mol{{sup|−1}}){{efn|Computed here, from Δ''H'' and Δ''G'' values supplied in the citation, using {{nowrap|−''T''Δ''S''{{sup|⊖}} {{=}} Δ''G''{{sup|⊖}} − Δ''H''{{sup|⊖}}}}}}
|-
| style="text-align:left;" | HA = [[Acetic acid]]
| style="text-align:left;" | HA {{eqm}} H<sup>+</sup> + A<sup>−</sup>
|  4.756
| 27.147
| −0.41
| 27.56
|-
| style="text-align:left;" | H<sub>2</sub>A<sup>+</sup> = [[Glycine]]H<sup>+</sup>
| style="text-align:left;" | H<sub>2</sub>A<sup>+</sup> {{eqm}} HA + H<sup>+</sup>
|  2.351
| 13.420
|  4.00
|  9.419
|-
| style="text-align:left;" |
| style="text-align:left;" | HA {{eqm}} H<sup>+</sup> + A<sup>−</sup>
| 9.78
| 55.825
| 44.20
| 11.6
|-
| style="text-align:left;" | H<sub>2</sub>A = [[Maleic acid]]
| style="text-align:left;" | H<sub>2</sub>A {{eqm}} HA<sup>−</sup> + H<sup>+</sup>
|  1.92
| 10.76
|  1.10
|  9.85
|-
| style="text-align:left;" |
| style="text-align:left;" | HA<sup>−</sup> {{eqm}} H<sup>+</sup> + A<sup>2−</sup>
|  6.27
| 35.79
| −3.60
| 39.4
|-
| style="text-align:left;" | H<sub>3</sub>A = [[Citric acid]]
| style="text-align:left;" | H<sub>3</sub>A {{eqm}} H<sub>2</sub>A<sup>−</sup> + H<sup>+</sup>
|  3.128
| 17.855
|  4.07
| 13.78
|-
|
| style="text-align:left;" | H<sub>2</sub>A<sup>−</sup> {{eqm}} HA<sup>2−</sup> + H<sup>+</sup>
|  4.76
| 27.176
|  2.23
|  24.9
|-
| style="text-align:left;" |
| style="text-align:left;" | HA<sup>2−</sup> {{eqm}} A<sup>3−</sup> + H<sup>+</sup>
|  6.40
| 36.509
| −3.38
| 39.9
|-
| style="text-align:left;" | H<sub>3</sub>A = [[Boric acid]]
| style="text-align:left;" | H<sub>3</sub>A {{eqm}} H<sub>2</sub>A<sup>−</sup> + H<sup>+</sup>
| 9.237
| 52.725
| 13.80
| 38.92
|-
| style="text-align:left;" | H<sub>3</sub>A = [[Phosphoric acid]]
| style="text-align:left;" | H<sub>3</sub>A {{eqm}} H<sub>2</sub>A<sup>−</sup> + H<sup>+</sup>
|  2.148
| 12.261
| −8.00
| 20.26
|-
| style="text-align:left;" |
| style="text-align:left;" | H<sub>2</sub>A<sup>−</sup> {{eqm}} HA<sup>2−</sup> + H<sup>+</sup>
|  7.20
| 41.087
|  3.60
|  37.5
|-
| style="text-align:left;" |
| style="text-align:left;" | HA<sup>2−</sup> {{eqm}} A<sup>3−</sup> + H<sup>+</sup>
| 12.35
| 80.49
| 16.00
| 54.49
|-
| style="text-align:left;" | HA<sup>−</sup> = [[Bisulfate|Hydrogen sulfate]]
| style="text-align:left;" | HA<sup>−</sup> {{eqm}} A<sup>2−</sup> + H<sup>+</sup>
|   1.99
|  11.36
| −22.40
|  33.74
|-
| style="text-align:left;" | H<sub>2</sub>A = [[Oxalic acid]]
| style="text-align:left;" | H<sub>2</sub>A {{eqm}} HA<sup>−</sup> + H<sup>+</sup>
|   1.27
|   7.27
|  −3.90
| 11.15
|-
| style="text-align:left;" |
| style="text-align:left;" | HA<sup>−</sup> {{eqm}} A<sup>2−</sup> + H<sup>+</sup>
|  4.266
| 24.351
| −7.00
| 31.35
|}

{| class="wikitable"
|+ Conjugate acids of bases
! Compound
! Equilibrium
! p''K''<sub>a</sub>
! ΔH{{sup|⊖}} (kJ·mol{{sup|−1}})
! −''T''Δ''S''{{sup|⊖}} (kJ·mol{{sup|−1}})
|-
| style="text-align:left;" | B = [[Ammonia]]
| style="text-align:left;" | HB<sup>+</sup> {{eqm}} B + H<sup>+</sup>
| 9.245
| 51.95
| 0.8205
|-
| style="text-align:left;" | B = [[Methylamine]]
| style="text-align:left;" | HB<sup>+</sup> {{eqm}} B + H<sup>+</sup>
| 10.645
| 55.34
| 5.422
|-
| style="text-align:left;" | B = [[Triethylamine]]
| style="text-align:left;" | HB<sup>+</sup> {{eqm}} B + H<sup>+</sup>
| 10.72
| 43.13
| 18.06
|}

The first point to note is that, when p''K''<sub>a</sub> is positive, the standard free energy change for the dissociation reaction is also positive. Second, some reactions are [[exothermic]] and some are [[endothermic]], but, when Δ''H''{{sup|⊖}} is negative ''T''ΔS{{sup|⊖}} is the dominant factor, which determines that Δ''G''{{sup|⊖}} is positive. Last, the entropy contribution is always unfavourable ({{nowrap|Δ''S''{{sup|⊖}} < 0}}) in these reactions. Ions in aqueous solution tend to orient the surrounding water molecules, which orders the solution and decreases the entropy. The contribution of an ion to the entropy is the [[partial molar quantity|partial molar]] entropy which is often negative, especially for small or highly charged ions.<ref>{{cite book
| last1 = Atkins
| first1 = Peter William
| last2 = De Paula
| first2 = Julio
| title = Atkins' physical chemistry.
| url = https://archive.org/details/atkinsphysicalch00pwat
| url-access = registration
| date = 2006
| publisher = W H Freeman
| location = New York
| isbn = 978-0-7167-7433-4
| page = [https://archive.org/details/atkinsphysicalch00pwat/page/94 94]
}}</ref> The ionization of a neutral acid involves formation of two ions so that the entropy decreases ({{nowrap|Δ''S''{{sup|⊖}} < 0}}). On the second ionization of the same acid, there are now three ions and the anion has a charge, so the entropy again decreases.

Note that the ''standard'' free energy change for the reaction is for the changes ''from'' the reactants in their standard states ''to'' the products in their standard states. The free energy change ''at'' equilibrium is zero since the [[chemical potential]]s of reactants and products are equal at equilibrium.