The weak acid acetic acid donates a proton to water in an equilibrium reaction to give the acetate ion and the hydronium ion. Key: Hydrogen is white, oxygen is red, carbon is gray. Lines are chemical bonds.

HA \rightleftharpoons A⁻ + H⁺

K_a is defined, subject to certain conditions, as

K_a = \mathrm{\frac{[A^-][H^+]}{[HA]}}

where [HA], [A⁻] and [H⁺] are equilibrium concentrations of the reactants.

The term acid dissociation constant is also used for pK_a, which is equal to ?log₁₀ K_a. The term pK_b is used in relation to bases, though pK_b has faded from modern use due to the easy relationship available between the strength of an acid and the strength of its conjugate base. Though discussions of this topic typically assume water as the solvent, particularly at introductory levels, the Brønsted?Lowry acid-base theory is versatile enough that acidic behavior can now be characterized even in non-aqueous solutions.

The value of pK_a indicates the strength of an acid: the larger the value the weaker the acid. In aqueous solution, simple acids are partially dissociated to an appreciable extent in in the pH range pK_a ± 2. The actual extent of the dissociation can be calculated if the acid concentration and pH are known.

A knowledge of pK_a values is essential for the understanding of the behaviour of acids and bases in solution. For example, many compounds used for medication are weak acids or bases, so a knowledge of the pK_a and log p values is essential for an understanding of how the compound enters (or does not enter) the blood stream. Other applications include aquatic chemistry, chemical oceanography, buffer solutions, acid-base homeostasis and certain kinds of enzyme kinetics, such as Michaelis?Menten kinetics, which involve a pre-equilibrium step. Also, knowledge of pK_a values is a prerequisite for a quantitative understanding of the interaction between acids or bases and metal ions to form complexes in solution.

Definitions

According to Arrhenius's original definition, an acid is a substanstance which dissociates in aqueous solution, releasing the hydrogen ion.^[1]

HA \rightleftharpoons A⁻ + H⁺

The equilibrium constant for this "dissociation" reaction is known as a dissociation constant. However, since the liberated proton combines with a water molecule to give an hydronium ion, Arrhenius proposed that the "dissociation" reaction should be written as an acid-base reaction.

HA + H₂O \rightleftharpoons A⁻ + H₃O⁺

Brønsted and Lowry generalized this definition as a proton exchange reaction, as follows.^[1]

acid + base \rightleftharpoons conjugate base + conjugate acid

The acid donates a proton to the base. The conjugate base is what is left after the acid has lost a proton and the conjugate acid is created when the base gains a proton. For aqueous solutions an acid, HA, reacts with the base, water, donating a proton to it, creating the conjugate base, A⁻, and the conjugate acid, the hydronium ion. The Brønsted?Lowry definition is particularly useful when the solvent is a substance other than water, such as DMSO; in that case the solvent, S, acts as a base, accepting a proton and forming the conjugate acid SH⁺. It also puts acids and bases on the same footing as being, respectively, donors or acceptors of protons. The conjugate acid of a base, B, "dissociates" according to

BH⁺ + OH⁻ \rightleftharpoons B + H₂O

For example:

H₂CO₃ + H₂O \rightleftharpoons HCO₃⁻ + H₃O⁺

The bicarbonate ion is the conjugate base of carbonic acid.

HCO₃⁻ + OH⁻ \rightleftharpoons CO₃²⁻ + H₂O

and the bicarbonate ion is also the conjugate acid of the base, the carbonate ion. In fact the bicarbonate ion is amphiprotic. These reactions are important for acid-base homeostasis in the human body (see carbonic acid).

It is important to note that, in the context of solution chemistry, a "proton" is understood to mean a solvated hydrogen ion. In aqueous solution the "proton" is a solvated hydronium ion.^[2][3]

Equilibrium Constant

An acid dissociation constant is a particular example of an equilibrium constant. For the specific equilibrium betwen a monoprotic acid, HA and its conjugate base A^?, in water,

HA + H₂O \rightleftharpoons A⁻ + H₃O⁺

the thermodynamic equilibrium constant, K^t can be defined by^[4]

K^\mbox{t}=\mathrm{\frac {\{H_3O^+\}}} {\{H_2O\}}}}

where {A} is the activity of the chemical species A etc (activity is a dimensionless quantity). Activities of the products are placed in the numerator, activities of the reactants are placed in the denominator. See Chemical equilibrium for a derivation of this expression.

Variation of pKa of acetic acid with ionic strength

Since activity is the product of concentration and activity coefficient the definition could also be written as

K^\mbox{t} = \mathrm{\frac{[A^-][H_3O^+]}{[HA][H_2O]}\times \frac{\gamma_{A^-}\gamma_{H_3O^+}}{\gamma_{HA}\gamma_{H_2O}} =\mathrm{\frac{[A^-][H_3O^+]}{[HA][H_2O]}}\times\Gamma}

where [HA] represents the concentration of HA and Γ is a quotient of activity coefficients.

In order to avoid the complications involved in using activities, dissociation constants are determined, where possible, in a medium of high ionic strength, that is, under conditions in which Γ can be assumed to be always constant.^[4] For example, the medium might be a solution of 0.1 M sodium nitrate or 3 M potassium perchlorate. Furthermore, in all but the most concentrated solutions it can be assumed that the concentration of water, [H₂O], is constant, approximately 55 mol dm^?3, and that the hydration of the proton can also be assumed to be constant.

Leaving out the constant terms, the acid dissociation constant can be defined as a concentration quotient.

K_a = \mathrm{\frac{[A^-][H^+]}{[HA]}}

This is the definition in common use. pK_a is defined as ?log₁₀ K_a. 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.

When operating under the assumption that ? is constant, the equilibrium constant does not change upon the addition of other chemicals to the solution. This assumption holds true when the concentration of spectator ions is low relative to the concentrations of other ions in the system. This allows, for example, for the behaviour of various ions to be explored at various pH values without worry that the equilibrium constant will also change. By exploiting this property, it is possible to obtain very complicated buffer solutions composed of many protonations of the same anion. This is accomplished with the addition of a strong acid to a solution of the anion. The conjugate base of the strong acid will act as a spectator ion, and the weak-base anion will be free to react with the proton as the equilibrium constant dictates.

Variation of the % formation of a monoprotic acid, AH, and its conjugate base, A⁻, with the difference between the pH and the pK_a of the acid

Monoprotic acids

After rearranging the expression defining K_a, and putting pH = ?log₁₀[H⁺], one obtains

pH = pK_a ? log ( [AH]/[A^?] )

This is a form of the Henderson?Hasselbalch equation. It shows how

• if the pH is known the ratio [AH]:[A^?] may be calculated. This ratio is independent of the analytical concentration of the acid.
• if the ratio [AH]:[A^?] is known the pH may be calculated. Thus, at 50% neutralization pH =pK_a ([AH]:[A^?] = 1). The buffer region extends over the range pK_a ± 2, though buffering is weak outside the range pK_a ± 1.

In water, measurable pK_a values range from about ?2 for a strong acid to about 12 for a very weak acid (or strong base). Any acid with a pK_a value of less than -2 is more than 99% dissociated at pH 0 (1M acid). Any base with a pK_a value larger than the upper limit is "fully" de-protonated at all attainable pH values. This is known as solvent leveling.^[5]

An example of a strong acid is hydrochloric acid, HCl, which has has a pK_a value, estimated from thermodynamic quantitities, of ?9.3 in water.^[6] The concentration of undissociated acid in a 1 mol dm^-3 solution will be less than 10^-9 mol dm^-3.

The pH of a solution of a monoprotic acid can be easily calculated when the pK_a and analytical concentration of the acid is known. See ICE table for details.

Polyprotic acids

% species' formation as a function of pH

% species formation calculated with the program HySS for a 10mM solution of citric acid. pK_a1=3.13, pK_a2 = 4.76, pK_a3=6.40.

When the difference between succesive pK values is about four or more, each species may be considered as an acid in its own right;^[7] the pH range of existence of each species is about pK± 2, so there is very little overlap between the ranges for successive species. The case of phosphoric acid illustrates this point. In fact salts of either H₂PO₄^? or HPO₄^2? may be crystallized from solution by adjustment of pH to either 4 or 10.

When the difference between succesive pK values is less than about four there is overlap between the pH range of existence of the species in equilibrium. The smaller the difference, the more the overlap. The case of citric acid is shown at the right; solutions of citric acid are buffered over the whole range of pH 2.5 to 7.5.

It is generally true that successive pK values increase (Pauling's first rule).^[8] For example, for a diprotic acid, H₂A, the two equilibria are

H₂A \rightleftharpoons HA⁻ + H⁺

HA⁻ \rightleftharpoons A²⁻ + H⁺

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; that is the cause of the trend noted above. Phosphoric acid, H₃PO₄, (values below), illustrates this rule, as does vanadic acid. When an exception to the rule is found it indicates that a major change in structure is ocurring. In the case of VO₂⁺_(aq), the vanadium is octahedral, 6-coordinate, whereas all the other species are tetrahedral, 4-coordinate. This explans why pK_a1 > pK_a2 for vanadium(V) oxoacids.

|- | | | VO₂⁺\rightleftharpoons H₃VO₄ + H⁺ | pK_a1 = 4.2 |- | H₃PO₄ \rightleftharpoons H₂PO₄⁻ + H⁺ | pK_a1 = 2.15 | H₃VO₄ \rightleftharpoons H₂VO₄⁻ + H⁺ | pK_a2 = 2.60 |- | H₂PO₄⁻ \rightleftharpoons HPO₄²⁻ + H⁺ | pK_a2 = 7.20 | H₂VO₄⁻ \rightleftharpoons HVO₄²⁻ + H⁺ | pK_a3 = 7.92 |- | HPO₄²⁻ \rightleftharpoons PO₄^3? + H⁺ | pK_{a3 = 12.37 | HVO4²⁻ \rightleftharpoons VO4^3? + H⁺ | pKa4 = 13.27 |}}

Water self-ionization

Water has both acidic and basic properies. The equilibrium constant for the equilibrium

H₂O + H₂O \rightleftharpoons OH⁻ + H₃O⁺

is given by

K_a=\mathrm{\frac{[H^+][OH^-]}{[H_2O]^2}}

Since the concentration of water can be assumed to be constant, this expression simplifies to

K_w =[H^+][OH^-]\,

The self-ionization constant of water, K_w, can thus be seen as a special case of an acid dissociation constant.

Bases

Historically the equilibrium constant K_b for a base was defined as the association constant for protonation of the base, B, to form the conjugate acid, HB⁺.

B + H₂O \rightleftharpoons HB⁺ + OH⁻

Using similar reasoning to that used before

K_b = \mathrm{\frac{[HB^+][OH^-]}{[B]}}

In water, the concentration of the hydroxide ion, [OH⁻], is related to the concentration of the hydrogen ion by K_w = [H⁺][OH^?], therefore

\mathrm{[OH^-] = \frac{\mathit{K}_w}{[H^+]}}

Substitution of the expression for [OH^?] into the expression for K_b gives

\mathrm{\mathit{K}_b = \frac{[HB^+]\mathit{K}_w}{[B] [H^+]} = \frac{\mathit{K}_w}{\mathit{K}_a}}

It follows, taking cologarithms, that pK_b = pK_w ? pK_a. In aqueous solutions at 25 °C, pK_w is 13.9965,^[9] so pK_b ~ 14 ? pK_a.

In effect there is no need to define pK_b separately from pK_a, but it is done here because pK_b values can be found in the older literature.

Temperature dependence

All equilibrium constants vary with temperature according the van 't Hoff equation^[10]

\frac {\operatorname{d} \ln \mathit{K}} {\operatorname{d}T} = \frac {RT^2}

Thus, for exothermic reactions, (?H^O is negative) K decreases with temperature, but for endothermic reactions (?H^O is positive) K increases with temperature.

Acidity in nonaqueous solutions

A solvent will be more likely to promote ionization of a dissolved acidic molecule if:^[11]

1. it is a protic solvent, capable of forming hydrogen bonds
2. it has a high donor number, making it a strong Lewis base.
3. it has a high dielectric constant (relative permittivity), making it a good solvent for ionic species.

Solvents can be polar, protic, donor or non-polar. The data in the following table refer to a temperature at or near 25 °C, unless stated otherwise.^[11]

Ionization of acids is less in an acidic solvent than in water. For example, hydrogen chloride is a weak acid when dissolved in acetic acid. This is because acetic acid is a much weaker base than water.

HCl + CH₃CO₂H \rightleftharpoons Cl⁻ + CH₃C(OH)₂⁺

acid + base \rightleftharpoons conjugate base + conjugate acid

Compare this reaction with what happens when acetic acid is dissolved in the more acidic solvent pure sulphuric acid^[12]

H₂SO₄ + CH₃CO₂H \rightleftharpoons HSO₄⁻ + CH₃C(OH)₂⁺

The apparently unlikely geminal diol species CH₃C(OH)₂⁺ is stable in these environments.

pK_a values of organic compounds are often obtained using solvents other than water, such as dimethyl sulfoxide (DMSO) and acetonitrile.^[13] Water is more basic than DMSO so most acids dissociate to a lesser extent in DMSO than in water. DMSO is widely used as an alternative to water in evaluating acids and bases because it has a lower dielectric constant than water, it is less polar and so dissolves non-polar, hydrophobic substances more easily.

Below is a table of selected pK_a values at 25 °C in acetonitrile (AN)^[14][15][16] and dimethyl sulfoxide (DMSO).^[17] Values for water are included for comparison.

!HA \rightleftharpoons A^? + H⁺!! AN !! DMSO!!water |- |p-Toluenesulfonic acid||8.5||0.9||strong |- |2,4-Dinitrophenol||16.66||5.1||3.9 |- |Benzoic acid||21.51||11.1||4.2 |- |Acetic acid||23.51 ||12.6||4.756 |- |Phenol ||29.14 ||18.0||9.99 |}

|- !BH⁺ \rightleftharpoons B + H⁺!! AN !! DMSO!!water |- |Pyrrolidine||19.56||10.8||11.4 |- |Triethylamine||18.82||9.0||10.72 |- |Proton sponge           ||18.62||7.5||12.1 |- |Pyridine||12.53||3.4||5.2 |- |Aniline||10.62||3.6||9.4 |}

In solvents of low dielectric constant ions tend to associate forming ion pairs and clusters, which complicates the interpretation of pK_a values.

dimerization of a carboxylic acid

Homoconjugation does not occur in aqueous solutions because water forms stronger hydrogen bonds and prevents the oligomers from forming.

Mixed solvents

When a compound has limited solubility in water it is common practice (in the pharmaceutical industry, for example) to determine pK_a values in a solvent mixture such as water/dioxane or water/methanol, in which the compound is more soluble.^[19] However, a pK_a value obtained in a mixed solvent cannot be used directly for aqueous solutions. The reason for this is that when the solvent is in its standard state its activity is defined as one. For example, the standard state of water:dioxane 9:1 is precisely that solvent mixture, with no added solutes. To obtain the pK_a value for use with aqueous solutions it has to be extrapolated to zero co-solvent concentration from values obtained from various co-solvent mixtures.

These facts are obscured by the omission of the solvent from the expression which is normally used to define pK_a, but pK_a values obtained in a given mixed solvent can be compared to each other, giving relative acid strengths. The same is true of pK_a values obtained in a particular non-aqueous solvent such a DMSO.

A universal, solvent-independent, scale for acid dissociation constants has not yet been developed, since there is no known way to compare the standard states of two different solvents.

Factors that determine the relative strengths of acids

Pauling's second rule^[8] states that the value of the first pK_a for acids of the formula XO_m(OH) _n 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_a for HClO is 7.2, for HClO₂ is 2.0, for HClO₃ is ?1 and HClO₄ is a strong acid.

fumaric acid

maleic acid

With organic acids inductive effects and mesomeric effects affect the pK_a values. The effects are summarised in the Hammett equation and subsequent extensions.^[20] 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_a values of approximately 3.5 and 4.5. By contrast, maleic acid has pK_a values of approximately 1.5 and 6.5.^[21] 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⁺, 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.

proton sponge

Thermodynamics

An equilibrium constant is related to the standard Gibbs free energy change for the reaction, so for an acid dissociation constant

?G^O = 2.303 RT pK_a.

Note that pK_a= ?log K_a. At 25 °C ?G^O /kJ mol^-1 = 5.708 pK_a. Free energy is made up of an enthalpy term and an entropy term.^[22]

?G^O = ?H^O ? T?S^O

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 pK_a and ?H^O. The data were critically selected and refer to 25 °C and zero ionic strength, in water.^[22]

The first point to note is that when pK_a is positive, the standard free energy change for the dissociation reaction is also positive, that is, dissociation of a weak acid is not a spontaneous process. Secondly some reactions are exothermic and some are endothermic, but when ?H^O is negative ?T?S^O is the dominant factor which determines that ?G^O is positive. Lastly, the entropy contribution is always unfavourable in these reactions.

Note. 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 potentials of reactants and products are equal at equilibrium.

Experimental determination of pK_a values

A calculated titration curve of oxalic acid titrated with a solution of sodium hydroxide

The total volume of added strong base should be small compared to the initial volume of to keep the ionic strength nearly constant. This will ensure that pK_a remains invariant during the titration.

A calculated titration curve for oxalic acid is shown at the right. Oxalic acid has pK_a values of 1.27 and 4.27. Therefore the buffer regions will be centered at about pH 1.3 and pH 4.3. The buffer regions carry the information necessary to get the pK_a values as the concentrations of acid and conjugate base change along a buffer region.

Between the two buffer regions there is an end-point, or equivalence point, where the pH rises by about two units. This end-point is not sharp and is typical of a diprotic acid whose buffer regions overlap by a small amount: pK_a2 ? pK_a1 is about three in this example. (If the difference in pK values were about two or less, the end-point would not be noticeable.) The second end-point begins at about pH 6.3 and is sharp. This indicates that all the protons have been removed. When this is so, the solution is not buffered and the pH rises steeply on addition of a small amount of strong base. However, the pH does not continue to rise indefinitely. A new buffer region begins at about pH 11 (pK_w ? 3), which is where self-ionization of water becomes important.

It is very difficult to measure pH values of less than two with a glass electrode, because the Nernst equation breaks down at such low pH values. To determine pK values of less than about 2 or more than about 11 spectrophotometric^[24] or NMR^[25] measurements may be used instead of, or combined with pH measurements.^[26]

Importance of pK_a values

A knowledge of pK_a values is important for the quantitative treatment of systems involving acid-base equilibria in solution.

• Buffer solutions^[27][28][29]

A buffer solution is made up of a mixture of an acid and its conjugate base, or a base and its conjugate acid. For any given acid or base the buffer region extends over the pH range about pK_a ± 1. When the concentrations of acidic and basic components are equal the pH is equal to the value of pK_a. Such a solution has maximum buffer capacity, meaning that the pH changes by the smallest amount for the addition of a given amount of strong acid.

• Buffer solutions are used extensively in biochemistry to provide solutions at or near the physiological pH for the study of biochemical reactions.^[30] For example, MOPS provides a solution with pH 7.2. Others are listed in buffer solutions and Good's buffers.

• Buffers such as tricine are used in Gel electrophoresis.^{[31] [32]}

• Isoelectric focussing - a technique used for separation of proteins by 2-D gel polyacrylamide gel electrophoresis.

• Acid base physiology including Acid-base homeostasis^[33] and disorders such as Acid-base imbalance.

• Biochemistry

In biochemistry the pK_a values of proteins and amino acid side chains are of major importance for the activity of enzymes and the stability of proteins.^[34] The reaction that converts adenosine triphosphate to adenosine diphosphate is very pH sensistive.

• Drug development^[35]

Ionization of a compound alters its physical behavior and macro properties such as solubility and lipophilicity (log p). For example ionization of any compound will increase the solubility in water, but decrease the lipophilicity. This is exploited in drug development to increase the concentration of a compound in the blood by adjusting the pK_a of an ionizable group. This must be done with caution, however, since an ionized compound will pass less easily through cell membranes.

• Natural waters

Acid-base equilibria are important for rivers and lakes,^[36][37] and in chemical oceanography.^[38]

• Extraction^[39]

In solvent extraction, the efficiency of extraction of a compound into an organic phase, such as ether, can be optimized by adjusting the pH of the aqueous phase using an appropriate buffer. At the optimum pH, the concentration of the electrically neutral species is maximized; such a species is more soluble in organic solvents having a low dielectric constant than it is in water. This technique is used for the purification of weak acids and bases.

• pH indicator

The transition range of a pH indicator is about pK_a ± 1. This is the range over which the color is intermediate between the colors of the acidic and basic forms of the indicator. Universal indicator is a mixture of indicators whose adjacent pK_a values differ by about two.

• Coordination compounds^[40]

A coordination complex is formed by interaction of a metal ion, acting as a Lewis acid, with a ligand acting as a Lewis base. However, the ligand may also undergo protonation reactions, so the formation of a complex in solution could be represented, symbolically by the reaction

[M(H₂O)_n]^m+ +AH \rightleftharpoons [M(H₂O)_n?1A]^(m?1)+ + H₃O⁺

To determine the equilibrium constant for this reaction, the pK_a of the protonated ligand must be known. In practice, the ligand may be polyprotic; for example EDTA⁴⁻ can accept four protons; in that case, all pK_a values must be known. In addition, the metal ion is subject to hydrolysis, that is, it behaves as a weak acid, so the pK values for the hydrolysis reactions must also be known.

pK_a of some common substances

There are multiple techniques to determine the pK_a of a chemical causing some discrepancy between different sources. Well measured values are typically are within 0.1 units of each other. Data presented here was taken at 25 °C in water.^[21][41] More values can be found in thermodynamics, above.

See also

• Determination of equilibrium constants
• Dissociation constant
• Henderson?Hasselbalch equation
• Hammett equation
• Isoelectric point
• Hydrolysis of metal salts
• QSAR

References

Further reading

• (Non-aqueous solvents)
• (translation editor: Mary R. Masson)
• (Previous edition published as Ionization constants of acids and bases. London: Methuen, 1962)

External links

• Acidity-Basicity Data (pKa Values) in Nonaqueous Solvents Extensive bibliography
• Shodor.org Acid-Base Chemistry
• Factors that Affect the Relative Strengths of Acids and Bases
• Purdue Chemistry
• Distribution diagrams of acids and bases (generation from pK_a values with free spreadsheet)
• SPARC Physical/Chemical property calculator
• List of Aqueous-Equilibrium Constants
• Free guide to pKa & logP interpretation and measurement

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