Acid dissociation constant

File:Acetic-acid-dissociation-3D-balls.png

Acetic acid, a weak acid, donates a proton (hydrogen ion, highlighted in green) to water in an equilibrium reaction to give the acetate ion and the hydronium ion. Red: oxygen, black: carbon, white: hydrogen.

An acid dissociation constant, K_a, (also known as acidity constant, or acid-ionization constant) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction known as dissociation in the context of acid-base reactions. The equilibrium can be written symbolically as:

HA 15px A⁻ + H⁺,

where HA is a generic acid that dissociates by splitting into A⁻, known as the conjugate base of the acid, and the hydrogen ion or proton, H⁺, which, in the case of aqueous solutions, exists as a solvated hydronium ion. In the example shown in the figure, HA represents acetic acid, and A⁻ the acetate ion. The chemical species HA, A⁻ and H⁺ are said to be in equilibrium when their concentrations do not change with the passing of time. The dissociation constant is usually written as a quotient of the equilibrium concentrations (in mol/L), denoted by [HA], [A⁻] and [H⁺]:

<math alt="K_a equals the equilibrium concentration of the deprotonated form A-, times the equilibrium concentration of H+, all divided by the equilibrium concentration of the acid AH." >K_{\mathrm a} = \mathrm{\frac{[A^-] [H^+]}{[HA]}}</math>

Due to the many orders of magnitude spanned by K_a values, a logarithmic measure of the acid dissociation constant is more commonly used in practice. The logarithmic constant, pK_a, which is equal to −log₁₀ K_a, is sometimes also (but incorrectly) referred to as an acid dissociation constant:

\[\ pK_{\mathrm a} = - \log_{10}K_{\mathrm a}\]

The larger the value of pK_a, the smaller the extent of dissociation. A weak acid has a pK_a value in the approximate range −2 to 12 in water. Acids with a pK_a value of less than about −2 are said to be strong acids; a strong acid is almost completely dissociated in aqueous solution, to the extent that the concentration of the undissociated acid becomes undetectable. pK_a values for strong acids can, however, be estimated by theoretical means or by extrapolating from measurements in non-aqueous solvents in which the dissociation constant is smaller, such as acetonitrile and dimethylsulfoxide.

Theoretical background

The acid dissociation constant for an acid is a direct consequence of the underlying thermodynamics of the dissociation reaction; the pK_a value is directly proportional to the standard Gibbs energy change for the reaction. The value of the pK_a changes with temperature and can be understood qualitatively based on Le Chatelier's principle: when the reaction is endothermic, the pK_a decreases with increasing temperature; the opposite is true for exothermic reactions. The underlying structural factors that influence the magnitude of the acid dissociation constant include Pauling's rules for acidity constants, inductive effects, mesomeric effects, and hydrogen bonding.

The quantitative behaviour of acids and bases in solution can be understood only if their pK_a values are known. In particular, the pH of a solution can be predicted when the analytical concentration and pK_a values of all acids and bases are known; conversely, it is possible to calculate the equilibrium concentration of the acids and bases in solution when the pH is known. These calculations find application in many different areas of chemistry, biology, medicine, and geology. For example, many compounds used for medication are weak acids or bases, and a knowledge of the pK_a values, together with the water–octanol partition coefficient, can be used for estimating the extent to which the compound enters the blood stream. Acid dissociation constants are also essential in aquatic chemistry and chemical oceanography, where the acidity of water plays a fundamental role. In living organisms, acid-base homeostasis and enzyme kinetics are dependent on the pK_a values of the many acids and bases present in the cell and in the body. In chemistry, a knowledge of pK_a values is necessary for the preparation of buffer solutions and is also a prerequisite for a quantitative understanding of the interaction between acids or bases and metal ions to form complexes. Experimentally, pK_a values can be determined by potentiometric (pH) titration, but for values of pK_a less than about 2 or more than about 11, spectrophotometric or NMR measurements may be required due to practical difficulties with pH measurements.

Definitions

According to Arrhenius's original definition, an acid is a substance that dissociates in aqueous solution, releasing the hydrogen ion H⁺ (a proton):

HA 15px A⁻ + H⁺.

The equilibrium constant for this dissociation reaction is known as a dissociation constant. The liberated proton combines with a water molecule to give a hydronium (or oxonium) ion H₃O⁺, and so Arrhenius later proposed that the dissociation should be written as an acid–base reaction:

HA + H₂O 15px A⁻ + H₃O⁺.

Brønsted and Lowry generalised this further to a proton exchange reaction:

acid + base 15px conjugate base + conjugate acid.

The acid loses a proton, leaving a conjugate base; the proton is transferred to the base, creating a conjugate acid. For aqueous solutions of an acid HA, the base is water; the conjugate base is A⁻ and the conjugate acid is the hydronium ion. The Brønsted–Lowry definition applies to other solvents, such as dimethyl sulfoxide: the solvent S acts as a base, accepting a proton and forming the conjugate acid SH⁺.

In solution chemistry, it is common to use H⁺ as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution H⁺ denotes a solvated hydronium ion rather than a proton.

The designation of an acid or base as "conjugate" depends on the context. The conjugate acid BH⁺ of a base B dissociates according to

BH⁺ + OH⁻ 15px B + H₂O

which is the reverse of the equilibrium

H₂O (acid) + B (base) 15px OH⁻ (conjugate base) + BH⁺ (conjugate acid).

The hydroxide ion OH⁻, a well known base, is here acting as the conjugate base of the acid water. Acids and bases are thus regarded simply as donors and acceptors of protons respectively.

A broader definition of acid dissociation includes hydrolysis, in which protons are produced by the splitting of water molecules. For example, boric acid (B(OH)₃) produces H₃O⁺ as if it were a proton donor, but it has been confirmed by Raman spectroscopy that this is due to the hydrolysis equilibrium:

B(OH)₃ + 2 H₂O 15px B(OH)₄⁻ + H₃O⁺.

Similarly, metal ion hydrolysis causes ions such as [Al(H₂O)₆]³⁺ to behave as weak acids:

[Al(H₂O)₆]³⁺ +H₂O 15px [Al(H₂O)₅(OH)]²⁺ + H₃O⁺.

Equilibrium constant

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

HA + H₂O 15px A⁻ + H₃O⁺

the thermodynamic equilibrium constant, K^{File:StrikeO.png} can be defined by

<math alt="K standard is a ratio involving the chemical activities of the four species in equilibrium. The numerator of the ratio holds the activity of the deprotonated acid A minus, times that of the hydronium ion H 3 O +. The denominator holds the activity of the acid A H, times that of water, H 2 O." >K^{\ominus} =\mathrm{\frac{\{A^-\} \{H_3O^+\}} {\{HA\} \{H_2O\}}}</math>

where {A} is the activity of the chemical species A etc. K^{File:StrikeO.png} 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

Variation of pK_a 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 alt="K standard can be written equivalently as the product of two ratios. The first ratio involves four chemical concentrations, whereas the second ratio involves four activity coefficients. The numerator of the first ratio holds the concentration of the deprotonated acid A minus, times that of the hydronium ion H 3 O +, whereas its denominator holds the concentration of the acid A H, times that of water, H 2 O. The second ratio has the same form as the first, with activity coefficients in place of concentrations. This second ratio is abbreviated by a capital Gamma." >K^{\ominus} = \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}</math>

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

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. For example, the medium might be a solution of 0.1 M sodium nitrate or 3 M potassium perchlorate (1 M = 1 mol·dm⁻³, a unit of molar concentration). 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. On dividing K^{File:StrikeO.png} by the constant terms and writing [H⁺] for the concentration of the hydronium ion the expression

<math alt="K A equals the equilibrium concentration of the deprotonated acid A minus, times the equilibrium concentration of H +, all divided by the equilibrium concentration of the protonated acid A H." >K_{\mathrm a} = \mathrm{\frac{[A^-] [H^+]}{[HA]}}</math>

is obtained. 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, 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 theory (SIT) and other theories.

Although K_a appears to have the dimension of concentration it must in fact be dimensionless or it would not be possible to take its logarithm. The illusion is the result of omitting the constant term [H₂O] from the defining expression. Nevertheless it is not unusual, particularly in texts relating to biochemical equilibria, to see a value quoted with a dimension as, for example, "K_a = 300 M".

Monoprotic acids

File:Weak acid speciation3.png

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

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

<math alt="p H equals p K A plus the logarithm (base ten) of a ratio of chemical concentrations, namely the concentration of the protonated form A H divided by that of the deprotonated form A minus." >\mathrm{pH} = \mathrm{p}K_{\mathrm a} + \log\mathrm{\frac{[A^-]}{[HA]}}</math>

This is a form of the Henderson–Hasselbalch equation, from which the following conclusions can be drawn.

• At half-neutralization [A⁻]/[HA] = 1; since log(1) =0, the pH at half-neutralization is numerically equal to pK_a. Conversely, when pH = pK_a, the concentration of HA is equal to the concentration of A⁻.
• The buffer region extends over the approximate range pK_a ± 2, though buffering is weak outside the range pK_a ± 1. At pK_a ± 1, [A⁻]/[HA] = 10 or 1/10.
• If the pH is known, the ratio may be calculated. This ratio is independent of the analytical concentration of the acid.

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). All acids with a pK_a value of less than −2 are more than 99% dissociated at pH 0 (1 M acid). This is known as solvent leveling since all such acids are brought to the same level of being strong acids, regardless of their pK_a values. Likewise, all bases with a pK_a value larger than the upper limit are more than 99% protonated at all attainable pH values and are classified as strong bases.

An example of a strong acid is hydrochloric acid, HCl, which has a pK_a value, estimated from thermodynamic quantities, of −9.3 in water. The concentration of undissociated acid in a 1 mol·dm⁻³ solution will be less than 0.01% of the concentrations of the products of dissociation. Hydrochloric acid is said to be "fully dissociated" in aqueous solution because the amount of undissociated acid is imperceptible. When the pK_a and analytical concentration of the acid are known, the extent of dissociation and pH of a solution of a monoprotic acid can be easily calculated using an ICE table.

A buffer solution of a desired pH can be prepared as a mixture of a weak acid and its conjugate base. In practice the mixture can be created by dissolving the acid in water, and adding the requisite amount of strong acid or base. The pK_a of the acid must be less than two units different from the target pH.

See also: Acid#Monoprotic acids

Polyprotic acids

File:Phosphoric3.png

% species' formation as a function of pH

File:Citric acid speciation.png

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

Polyprotic acids are acids that can lose more than one proton. The constant for dissociation of the first proton may be denoted as K_a1 and the constants for dissociation of successive protons as K_a2, etc. Phosphoric acid, H₃PO₄, is an example of a polyprotic acid as it can lose three protons.

equilibrium	pKa value
H3PO4 15px H2PO4− + H+	pKa1 = 2.15
H2PO4− 15px HPO42− + H+	pKa2 = 7.20
HPO42− 15px PO43− + H+	pKa3</sup> = 12.37

When the difference between successive pK values is about four or more, as in this example, each species may be considered as an acid in its own right; In fact salts of H₂PO₄⁻ may be crystallised from solution by adjustment of pH to about 5.5 and salts of HPO₄²⁻ may be crystallised from solution by adjustment of pH to about 10. The species distribution diagram shows that the concentrations of the two ions are maximum at pH 5.5 and 10.

When the difference between successive 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.

In general, it is true that successive pK values increase (Pauling's first rule). For example, for a diprotic acid, H₂A, the two equilibria are

H₂A 15px HA⁻ + H⁺

HA⁻ 15px 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 values (above) illustrate this rule, as do the values for vanadic acid, H₃VO₄. When an exception to the rule is found it indicates that a major change in structure is occurring. In the case of VO₂⁺ (aq), the vanadium is octahedral, 6-coordinate, whereas vanadic acid is tetrahedral, 4-coordinate. This is the basis for an explanation of why pK_a1 > pK_a2 for vanadium(V) oxoacids.

equilibrium	pKa value
[VO2(H2O)4]+ 15px H3VO4 + H+ + 2H2O	pKa1 = 4.2
H3VO4 15px H2VO4− + H+	pKa2 = 2.60
H2VO4− 15px HVO42− + H+	pKa3 = 7.92
HVO42− 15px VO43− + H+	pKa4 = 13.27

Isoelectric point

Main article: isoelectric point

For substances in solution the isoelectric point (pI) is defined as the pH at which the sum, weighted by charge value, of concentrations of positively charged species is equal to the weighted sum of concentrations of negatively charged species. In the case that there is one species of each type, the isoelectric point can be obtained directly from the pK values. Take the example of glycine, defined as AH. There are two dissociation equilibria to consider.

AH₂⁺ 15px AH + H⁺; [AH][H⁺] = K₁[AH₂⁺]

AH 15px A^- + H⁺; [A^-][H⁺] = K₂[AH]

Substitute the expression for [AH] into the first equation

[A^-][H⁺]² = K₁K₂[AH₂⁺]

At the isoelectric point the concentration of the positively charged species, AH₂⁺, is equal to the concentration of the negatively charged species, A^-, so

[H⁺]² = K₁K₂

Therefore, taking cologarithms, the pH is given by \[pI=\frac{pK_1+pK_2}{2}\] pI values for amino acids are listed at Proteinogenic amino acid#Chemical properties. When more than two charged species are in equilibrium with each other a full speciation calculation may be needed.

Water self-ionization

Main article: Self-ionization of water

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

2 H₂O 15px OH⁻ + H₃O⁺

is given by

<math alt="The acidity constant K A for water equals the concentration of H + times that of O H minus divided by the concentration of water, H 2 O." >K_{\mathrm a}=\mathrm{\frac{[H_3O^+] [OH^-]}{[H_2O]^2}}</math>

When, as is usually the case, the concentration of water can be assumed to be constant, this expression may be replaced by

<math alt="The ionization constant of water K w equals the concentration of H + times the concentration of O H minus." >K_{\mathrm w} =[\mathrm{H_3O}^+] [\mathrm{OH}^-]\,</math>

The self-ionization constant of water, K_w, is thus just a special case of an acid dissociation constant.

pK_w values for pure water at various temperatures

T/°C	0	5	10	15	20	25	30	35	40	45	50
pKw	14.943	14.734	14.535	14.346	14.167	13.997	13.830	13.680	13.535	13.396	13.262

From these data it can be deduced that K_w = 10⁻¹⁴ at 24.87 °C. At that temperature both hydrogen and hydroxide ions have a concentration of 10⁻⁷ mol dm⁻³.

Amphoteric substances

An amphoteric substance is one that can act as an acid or as a base, depending on pH. Water (above) is amphoteric. Another example of an amphoteric molecule is the bicarbonate ion HCO₃⁻ that is the conjugate base of the carbonic acid molecule H₂CO₃ in the equilibrium

H₂CO₃ + H₂O 15px HCO₃⁻ + H₃O⁺

but also the conjugate acid of the carbonate ion CO₃²⁻ in (the reverse of) the equilibrium

HCO₃⁻ + OH⁻ 15px CO₃²⁻ + H₂O.

Carbonic acid equilibria are important for acid-base homeostasis in the human body.

An amino acid is also amphoteric with the added complication that the neutral molecule is subject to an internal acid-base equilibrium in which the basic amino group attracts and binds the proton from the acidic carboxyl group, forming a zwitterion.

NH₂CHRCO₂H 15px NH₃⁺CHRCO₂^-

At pH less than about 5 both the carboxylate group and the amino group are protonated. As pH increases the acid dissociates according to

NH₃⁺CHRCO₂H 15px NH₃⁺CHRCO₂^- + H⁺

At high pH a second dissociation may take place.

NH₃⁺CHRCO₂^- 15px NH₂CHRCO₂^- + H⁺

Thus the zwitter ion, NH₃⁺CHRCO₂^-, is amphoteric because it may either be protonated or deprotonated.

Bases

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

B + H₂O 15px HB⁺ + OH⁻

Using similar reasoning to that used before

<math alt="The base association constant K b equals the concentration of the protonated form H B +, times the concentration of the hydroxyl anion O H minus, all divided by the concentration of the base B." >K_{\mathrm b} = \mathrm{\frac{[HB^+] [OH^-]}{[B]}}</math>

K_b is related to K_a for the conjugate acid. In water, the concentration of the hydroxide ion, [OH⁻], is related to the concentration of the hydrogen ion by K_w = [H⁺] [OH⁻], therefore

<math alt="The concentration of the hydroxyl anion O H minus equals the ionization constant of water K w divided by the concentration of H +, by the definition of K w." >\mathrm{[OH^-]} = \frac{K_{\mathrm w}}{\mathrm{[H^+]}}</math>

Substitution of the expression for [OH⁻] into the expression for K_b gives

<math alt="K b can be written as a ratio of four terms. The numerator holds the concentration of the protonated base H B + times the ionization constant of water K w. The denominator holds the concentration of the base B times that of H +. Using the definition for K A, K b equals K w divided by K A." >K_{\mathrm b} = \frac{[\mathrm{HB^+}]K_{\mathrm w}}{\mathrm{[B] [H^+]}} = \frac{K_{\mathrm w}}{K_{\mathrm a}}</math>

When K_a, K_b and K_w are determined under the same conditions of temperature and ionic strength, it follows, taking cologarithms, that pK_b = pK_w − pK_a. In aqueous solutions at 25 °C, pK_w is 13.9965, 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 to the van 't Hoff equation

<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 plimsol 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 {\operatorname{d} \ln K} {\operatorname{d}T} = \frac{{\Delta H}^{\ominus}} {RT^2}</math>

R is the gas constant and T is the absolute temperature . Thus, for exothermic reactions, (the standard enthalpy change, ΔH^{File:StrikeO.png}, is negative) K decreases with temperature, but for endothermic reactions (ΔH^{File:StrikeO.png} is positive) K increases with temperature.

Acidity in nonaqueous solutions

A solvent will be more likely to promote ionization of a dissolved acidic molecule in the following circumstances:

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.

pK_a values of organic compounds are often obtained using the aprotic solvents dimethyl sulfoxide (DMSO) and acetonitrile (ACN).

Solvent properties at 25 °C

Solvent	Donor number	Dielectric constant
Acetonitrile	14	37
Dimethylsulfoxide	30	47
Water	18	78

DMSO is widely used as an alternative to water because it has a lower dielectric constant than water, and is less polar and so dissolves non-polar, hydrophobic substances more easily. It has a measurable pK_a range of about 1 to 30. Acetonitrile is less basic than DMSO, and, so, in general, acids are weaker and bases are stronger in this solvent. Some pK_a values at 25 °C for acetonitrile (ACN) and dimethyl sulfoxide (DMSO) are shown in the following tables. Values for water are included for comparison.

pK_a values of acids

HA 15px A− + H+	ACN	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+ 15px B + H+
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	4.6

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 15px Cl⁻ + CH₃C(OH)₂⁺

acid + base 15px conjugate base + conjugate acid

Compare this reaction with what happens when acetic acid is dissolved in the more acidic solvent pure sulfuric acid

H₂SO₄ + CH₃CO₂H 15px HSO₄⁻ + CH₃C(OH)₂⁺

The unlikely geminal diol species CH₃C(OH)₂⁺ is stable in these environments. For aqueous solutions the pH scale is the most convenient acidity function. Other acidity functions have been proposed for non-aqueous media, the most notable being the Hammett acidity function, H₀, for superacid media and its modified version H₋ for superbasic media.

File:Carboxylic acid dimers.png

Dimerization of a carboxylic acid

In aprotic solvents, oligomers, such as the well-known acetic acid dimer, may be formed by hydrogen bonding. An acid may also form hydrogen bonds to its conjugate base. This process, known as homoconjugation, has the effect of enhancing the acidity of acids, lowering their effective pK_a values, by stabilizing the conjugate base. Homoconjugation enhances the proton-donating power of toluenesulfonic acid in acetonitrile solution by a factor of nearly 800. In aqueous solutions, homoconjugation does not occur, because water forms stronger hydrogen bonds to the conjugate base than does the acid.

Mixed solvents

File:Acetic acid pK dioxane water.png

pK_a of acetic acid in dioxane/water mixtures. Data at 25 °C from Pine et al.

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. In the example shown at the right, the pK_a value rises steeply with increasing percentage of dioxane as the dielectric constant of the mixture is decreasing.

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 that 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.

As of 2008, a universal, solvent-independent, scale for acid dissociation constants has not been developed, since there is no known way to compare the standard states of two different solvents.

Factors that affect pK_a values

Pauling's second rule 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.

File:Fumaric-acid-2D-skeletal.png

Fumaric acid

File:Maleic-acid-2D-skeletal-A.png

Maleic acid

File:Proton sponge.svg

proton sponge

With organic acids inductive effects and mesomeric effects affect the pK_a 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 pK_a values decrease in the series 4.7, 2.8, 1.4 and 0.7 when 0,1, 2 or 3 chlorine atoms are present. The Hammett equation, provides a general expression for the effect of substituents.

log K_a = log K_a⁰ + ρσ.

K_a is the dissociation constant of a substituted compound, K_a⁰ 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_a against σ is a straight line with intercept log K_a⁰ and slope ρ. This is an example of a linear free energy relationship as log K_a is proportional to the standard fee energy change. Hammett originally formulated the relationship with data from benzoic acid with different substiuents in the ortho- and para- positions: some numerical values are in Hammett equation. This and other studies allowed substituents to be ordered according to their electron-withdrawing or electron-releasing power, and to distinguish between inductive and mesomeric effects.

Alcohols do not normally behave as acids in water, but the presence of a double bond adjacent to the OH group can substantially decrease the pK_a 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, conjugation with the aromatic ring as a whole greatly increases the stability of the deprotonated form.

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 3.0 and 4.5. By contrast, maleic acid has pK_a 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 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, 1,8-bis(dimethylamino)naphthalene, has a pK_a 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.

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, Me₃N > Me₂NH > MeNH₂ > NH₃, is changed by water to Me₂NH > MeNH₂ > Me₃N > NH₃. Neutral methylamine molecules are hydrogen-bonded to water molecules mailnly through one acceptor, N-HOH, interaction and only occasionally just one more donor bond, NH-OH₂. 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-OH₂ bonding. Relative stabilization of methylammonium ions thus decreases with the number of methyl groups explaining the order of water basicity of methylamines.

Thermodynamics

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

ΔG^{File:StrikeO.png} = −RT ln K_a ≈ 2.303 RT pK_a.

R is the gas constant and T is the absolute temperature. Note that pK_a= −log K_a and 2.303 ≈ ln 10. At 25 °C ΔG^{File:StrikeO.png} in kJ·mol⁻¹ = 5.708 pK_a (1 kJ·mol⁻¹ = 1000 Joules per mole). Free energy is made up of an enthalpy term and an entropy term.

ΔG^{File:StrikeO.png} = ΔH^{File:StrikeO.png} − TΔS^{File:StrikeO.png}

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 ΔH^{File:StrikeO.png}. The data were critically selected and refer to 25 °C and zero ionic strength, in water.

Acids

Compound	Equilibrium	pKa	ΔGFile:StrikeO.png /kJ·mol−1	ΔHFile:StrikeO.png /kJ·mol−1	—TΔSFile:StrikeO.png /kJ·mol−1
HA = Acetic acid	HA 15px H+ + A−	4.756	27.147†	−0.41	27.56‡
H2A+ = GlycineH+	H2A+ 15px HA + H+	2.351	13.420	4.00	9.419
	HA 15px H+ + A−	9.78	55.825	44.20	11.6
H2A = Maleic acid	H2A 15px HA− + H+	1.92	10.76	1.10	9.85
	HA− 15px H+ + A2−	6.27	35.79	−3.60	39.4
H3A = Citric acid	H3A 15px H2A− + H+	3.128	17.855	4.07	13.78
	H2A− 15px HA2− + H+	4.76	27.176	2.23	24.9
	HA2− 15px A3− + H+	6.40	36.509	−3.38	39.9
H3A = Boric acid	H3A 15px H2A− + H+	9.237	52.725	13.80	38.92
H3A = Phosphoric acid	H3A 15px H2A− + H+	2.148	12.261	−8.00	20.26
	H2A− 15px HA2− + H+	7.20	41.087	3.60	37.5
	HA2− 15px A3− + H+	12.35	80.49	16.00	54.49
HA− = Hydrogen sulfate	HA− 15px A2− + H+	1.99	11.36	−22.40	33.74
H2A = Oxalic acid	H2A 15px HA− + H+	1.27	7.27	−3.90	11.15
	HA− 15px A2− + H+	4.266	24.351	7.00	31.35

^† ΔG^{File:StrikeO.png} = 2.303RT pKa

^‡ Computed here, from ΔH and ΔG values supplied in the citation, using —TΔS^{File:StrikeO.png} = ΔG^{File:StrikeO.png} — ΔH^{File:StrikeO.png}

Conjugate acids of bases

Compound	Equilibrium	pKa	ΔHFile:StrikeO.png /kJ·mol−1	—TΔSFile:StrikeO.png /kJ·mol−1
B = Ammonia	HB+ 15px B + H+	9.245	51.95	0.8205
B = Methylamine	HB+ 15px B + H+	10.645	55.34	5.422
B = Triethylamine	HB+ 15px B + H+	10.72	43.13	18.06

The first point to note is that, when pK_a 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 ΔH^{File:StrikeO.png} is negative −TΔS^{File:StrikeO.png} is the dominant factor, which determines that ΔG^{File:StrikeO.png} is positive. Last, the entropy contribution is always unfavourable (ΔS^{File:StrikeO.png} < 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 entropy which is often negative, especially for small or highly charged ions. The ionization of a neutral acid involves formation of two ions so that the entropy decreases (ΔS^{File:StrikeO.png} < 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 potentials of reactants and products are equal at equilibrium.

Experimental determination

File:Oxalic acid titration grid.png

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

See also: Determination of equilibrium constants

The experimental determination of pK_a values is commonly performed by means of titrations, in a medium of high ionic strength and at constant temperature. A typical procedure would be as follows. A solution of the compound in the medium is acidified with a strong acid to the point where the compound is fully protonated. The solution is then titrated with a strong base until all the protons have been removed. At each point in the titration pH is measured using a glass electrode and a pH meter. The equilibrium constants are found by fitting calculated pH values to the observed values, using the method of least squares.

The total volume of added strong base should be small compared to the initial volume of titrand solution in order 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 in aqueous solution 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 or NMR measurements may be used instead of, or combined with, pH measurements.

When the glass electrode cannot be employed, as with non-aqueous solutions, spectrophotometric methods are frequently used. These may involve absorbance or fluorescence measurements. In both cases the measured quantity is assumed to be proportional to the sum of contributions from each photo-active species; with absorbance measurements the Beer-Lambert law is assumed to apply.

Aqueous solutions with normal water cannot be used for ¹H NMR measurements but heavy water, D₂O, must be used instead. ¹³C NMR data, however, can be used with normal water and ¹H NMR spectra can be used with non-aqueous media. The quantities measured with NMR are time-averaged chemical shifts, as proton exchange is fast on the NMR time-scale. Other chemical shifts, such as those of ³¹P can be measured.

Micro-constants

File:Spermine.svg

spermine

A base such as spermine has a few different sites where protonation can occur. In this example the first proton can go on the terminal -NH₂ group, or either of the internal -NH- groups. The pK_a values for dissociation of spermine protonated at one or other of the sites are examples of micro-constants. They cannot be determined directly by means of pH, absorbance, fluorescence or NMR measurements. Nevertheless, the site of protonation is very important for biological function, so mathematical methods have been developed for the determination of micro-constants.

Applications and significance

A knowledge of pK_a values is important for the quantitative treatment of systems involving acid–base equilibria in solution. Many applications exist in biochemistry; for example, 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. Protein pKa values cannot always be measured directly, but may be calculated using theoretical methods. Buffer solutions are used extensively to provide solutions at or near the physiological pH for the study of biochemical reactions; the design of these solutions depends on a knowledge of the pK_a values of their components. Important buffer solutions include MOPS, which provides a solution with pH 7.2, and tricine, which is used in gel electrophoresis. Buffering is an essential part of acid base physiology including acid-base homeostasis, and is key to understanding disorders such as acid-base imbalance. The isoelectric point of a given molecule is a function of its pK values, so different molecules have different isoelectric points. This permits a technique called isoelectric focusing, which is used for separation of proteins by 2-D gel polyacrylamide gel electrophoresis.

Buffer solutions also play a key role in analytical chemistry. They are used whenever there is a need to fix the pH of a solution at a particular value. Compared with an aqueous solution, the pH of a buffer solution is relatively insensitive to the addition of a small amount of strong acid or strong base. The buffer capacity of a simple buffer solution is largest when pH = pK_a. In acid-base extraction, the efficiency of extraction of a compound into an organic phase, such as an ether, can be optimised by adjusting the pH of the aqueous phase using an appropriate buffer. At the optimum pH, the concentration of the electrically neutral species is maximised; 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.

A pH indicator is a weak acid or weak base that changes colour in the transition pH range, which is approximately pK_a ± 1. The design of a universal indicator requires a mixture of indicators whose adjacent pK_a values differ by about two, so that their transition pH ranges just overlap.

In pharmacology ionization of a compound alters its physical behaviour 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.

Knowledge of pK_a values is important for the understanding of coordination complexes, which are formed by the interaction of a metal ion, M^m+, acting as a Lewis acid, with a ligand, L, acting as a Lewis base. However, the ligand may also undergo protonation reactions, so the formation of a complex in aqueous solution could be represented symbolically by the reaction

[M(H₂O)_n]^m+ +LH 15px [M(H₂O)_n−1L]^(m−1)+ + H₃O⁺

To determine the equilibrium constant for this reaction, in which the ligand loses a proton, 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. Assessing the hazard associated with an acid or base may require a knowledge of pK_a values. For example, hydrogen cyanide is a very toxic gas, because the cyanide ion inhibits the iron-containing enzyme cytochrome c oxidase. Hydrogen cyanide is a weak acid in aqueous solution with a pK_a of about 9. In strongly alkaline solutions, above pH 11, say, it follows that sodium cyanide is "fully dissociated" so the hazard due to the hydrogen cyanide gas is much reduced. An acidic solution, on the other hand, is very hazardous because all the cyanide is in its acid form. Ingestion of cyanide by mouth is potentially fatal, independently of pH, because of the reaction with cytochrome c oxidase.

In environmental science acid–base equilibria are important for lakes and rivers; for example, humic acids are important components of natural waters. Another example occurs in chemical oceanography: in order to quantify the solubility of iron(III) in seawater at various salinities, the pK_a values for the formation of the iron(III) hydrolysis products Fe(OH)²⁺, Fe(OH)₂⁺ and Fe(OH)₃ were determined, along with the solubility product of iron hydroxide.

Values for common substances

There are multiple techniques to determine the pK_a of a chemical, leading to some discrepancies between different sources. Well measured values are typically within 0.1 units of each other. Data presented here were taken at 25 °C in water. More values can be found in thermodynamics, above.

Chemical Name	Equilibrium	pKa
B = Adenine	BH22+ 15px BH+ + H+	4.17
	BH+ 15px B + H+	9.65
H3A = Arsenic acid	H3A 15px H2A− + H+	2.22
	H2A− 15px HA2− + H+	6.98
	HA2− 15px A3− + H+	11.53
HA = Benzoic acid	HA 15px H+ + A−	4.204
HA = Butanoic acid	HA 15px H+ + A−	4.82
H2A = Chromic acid	H2A 15px HA− + H+	0.98
	HA− 15px A2− + H+	6.5
B = Codeine	BH+ 15px B + H+ </sup>	8.17
HA = Cresol	HA 15px H+ + A−	10.29
HA = Formic acid	HA 15px H+ + A−	3.751
HA = Hydrofluoric acid	HA 15px H+ + A−	3.17
HA = Hydrocyanic acid	HA 15px H+ + A−	9.21
HA = Hydrogen selenide	HA 15px H+ + A−	3.89
HA = Hydrogen peroxide (90%)	HA 15px H+ + A−	11.7
HA = Lactic acid	HA 15px H+ + A−	3.86
HA = Propionic acid	HA 15px H+ + A−	4.87
HA = Phenol	HA 15px H+ + A−	9.99
H2A = L-(+)-Ascorbic Acid	H2A 15px HA− + H+	4.17
	HA− 15px A2− + H+	11.57

See also

• Acids in wine: tartaric, malic and citric are the principal acids in wine.
• Ocean acidification: dissolution of atmospheric carbon dioxide affects seawater pH. The reaction depends on total inorganic carbon and on solubility equilibria with solid carbonates such as limestone and dolomite.
• Grotthuss mechanism: how protons are transferred between hydronium ions and water molecules, accounting for the exceptionally high ionic mobility of the proton (animation).
• Predominance diagram: relates to equilibria involving polyoxyanions. pK_a values are needed to construct these diagrams.
• Proton affinity: a measure of basicity in the gas phase.
• Stability constants of complexes: formation of a complex can often be seen as a competition between proton and metal ion for a ligand, which is the product of dissociation of an acid.
• Hammett acidity function: a measure of acidity that is used for very concentrated solutions of strong acids, including superacids.

Further reading

• (Previous edition published as )
• (Non-aqueous solvents)
• (translation editor: Mary R. Masson)
• Chapter 4: Solvent Effects on the Position of Homogeneous Chemical Equilibria.

External links

• Acidity-Basicity Data in Nonaqueous Solvents

Extensive bibliography of pK_a values in DMSO, acetonitrile, THF, heptane, 1,2-dichloroethane, and in the gas phase.

• Curtipot

All-in-one freeware for pH and acid-base equilibrium calculations and for simulation and analysis of potentiometric titration curves with spreadsheets.

• SPARC Physical/Chemical property calculator

Includes a database with aqueous, non-aqueous, and gaseous phase pK_a values than can be searched using SMILES or CAS registry numbers.

• Aqueous-Equilibrium Constants

pK_a values for various acid and bases. Includes a table of some solubility products.

• Free guide to pK_a and log p interpretation and measurement

Explanations of the relevance of these properties to pharmacology.

• Free online prediction tool (Marvin) pK_a, logP, logD etc. From ChemAxon
• Chemicalize.org:List of predicted structure based properties
• Evans pKa Chart http://evans.harvard.edu/pdf/evans_pka_table.pdf

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