Acid-base Equilibrium

In chemistry, an acid dissociation constant (also known as acidity constant, or acid-ionization constant; denoted ) is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction
:HA <=> A^- + H^+
known as dissociation in the context of acid–base reactions. The chemical species HA is an acid that dissociates into , the conjugate base of the acid and a hydrogen ion, . The system is said to be in equilibrium when the concentrations of its components will not change over time, because both forward and backward reactions are occurring at the same rate.

The dissociation constant is defined by
:$K_\text = \mathrm,$ or
:$\mathrmK_\ce = - \log_ K_\text = \log_\frac$

where quantities in square brackets represent the concentrations of the species at equilibrium.

Theoretical background

The acid dissociation constant for an acid is a direct consequence of the underlying thermodynamics of the dissociation reaction; the p''K''_a value is directly proportional to the standard Gibbs free energy change for the reaction. The value of the p''K''_a changes with temperature and can be understood qualitatively based on Le Châtelier's principle: when the reaction is endothermic, ''K''_a increases and p''K''_a decreases with increasing temperature; the opposite is true for exothermic reactions.

The value of p''K''_a also depends on molecular structure of the acid in many ways. For example, Pauling proposed two rules: one for successive p''K''_a of polyprotic acids (see Polyprotic acids below), and one to estimate the p''K''_a of oxyacids based on the number of =O and −OH groups (see Factors that affect p''K''_a values below). Other structural factors that influence the magnitude of the acid dissociation constant include inductive effects, mesomeric effects, and hydrogen bonding. Hammett type equations have frequently been applied to the estimation of p''K''_a.

The quantitative behaviour of acids and bases in solution can be understood only if their p''K''_a values are known. In particular, the pH of a solution can be predicted when the analytical concentration and p''K''_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 p''K''_a values, together with the octanol-water 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 p''K''_a values of the many acids and bases present in the cell and in the body. In chemistry, a knowledge of p''K''_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, p''K''_a values can be determined by potentiometric (pH) titration, but for values of p''K''_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 molecular definition, an acid is a substance that dissociates in aqueous solution, releasing the hydrogen ion (a proton): Chapter 6: Acid–Base and Donor–Acceptor Chemistry
: HA <=> 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 (naked protons do not exist in solution), and so Arrhenius later proposed that the dissociation should be written as an acid–base reaction:
:HA + H2O <=> A- + H3O+

Brønsted and Lowry generalised this further to a proton exchange reaction: Includes discussion of many organic Brønsted acids.
Chapter 5: Acids and Bases
: $\text + \text \ce \text + \text$

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 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 .
:HA + S <=> A- + SH+

In solution chemistry, it is common to use as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution 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 of a base B dissociates according to
:BH+ + OH- <=> B + H2O

which is the reverse of the equilibrium
:$\ce\text + \ce\text \ce\text + \ce\text$

The hydroxide ion , 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 () produces 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 <=> B(OH)4- + H3O+

Similarly, metal ion hydrolysis causes ions such as to behave as weak acids:
Section 9.1 "Acidity of Solvated Cations" lists many p''K''_a values.
: l(H2O)63+ + H2O <=> l(H2O)5(OH)2+ + H3O+

According to Lewis's original definition, an acid is a substance that accepts an electron pair to form a coordinate covalent bond.
p.698

Equilibrium constant

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

The thermodynamic equilibrium constant can be defined by Chapter 2: Activity and Concentration Quotients
:$K^\ominus = \frac$

where represents the activity, at equilibrium, of the chemical species X. 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.

Since activity is the product of concentration and activity coefficient (''γ'') the definition could also be written as
:$K^\ominus = , \quad \Gamma=\frac$
where represents the concentration of HA and 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 can be assumed to be always constant. For example, the medium might be a solution of 0.1  molar (M) sodium nitrate or 3 M potassium perchlorate. With this assumption,
:$K_\text = \frac = \mathrm$
:$\mathrmK_\ce = -\log_\frac = \log_\frac$
is obtained. 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.

Cumulative and stepwise constants

A cumulative equilibrium constant, denoted by is related to the product of stepwise constants, denoted by For a dibasic acid the relationship between stepwise and overall constants is as follows
:H2A <=> A^2- + 2H+
:$\beta_2 = \frac$
:$\log \beta_2 = \mathrmK_\ce + \mathrmK_\ce$

Note that in the context of metal-ligand complex formation, the equilibrium constants for the formation of metal complexes are usually defined as ''association'' constants. In that case, the equilibrium constants for ligand protonation are also defined as association constants. The numbering of association constants is the reverse of the numbering of dissociation constants; in this example $\log \beta_1 = \mathrmK_\ce$

Association and dissociation constants

When discussing the properties of acids it is usual to specify equilibrium constants as acid dissociation constants, denoted by ''K''_a, with numerical values given the symbol p''K''_a.
:$K_\text = \frac: \mathrmK_\text = -\log K_\text$

On the other hand, association constants are used for bases.
:$K_\text = \frac: \mathrmK_\text = -\log K_\text$

However, general purpose computer programs that are used to derive equilibrium constant values from experimental data use association constants for both acids and bases. Because stability constants for a metal-ligand complex are always specified as association constants, ligand protonation must also be specified as an association reaction. The definitions show that the value of an acid dissociation constant is the reciprocal of the value of the corresponding association constant.
: $K_\text = \frac$
: $\log K_\text = - \log K_\text$
: $\mathrmK_\text = \log K_\text$

Notes
# For a given acid or base , the self-ionization constant of water.
# The association constant for the formation of a supramolecular complex may be denoted as K_a; in such cases "a" stands for "association", not "acid".
# For polyprotic acids, the numbering of stepwise association constants is the reverse of the numbering of the dissociation constants. For example, for phosphoric acid (details in #polyprotic acids, below)
::$\begin \log K_ &= \mathrmK_ \\ \log K_ &= \mathrmK_ \\ \log K_ &= \mathrmK_ \end$

Temperature dependence

All equilibrium constants vary with temperature according to the van 't Hoff equation
:absolute temperature