# Acid dissociation constant

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

## Overview

An acid dissociation constant, denoted by Ka, is an equilibrium constant for the dissociation of a weak acid. According to the Brønsted-Lowry theory of acids and bases, an acid is a proton donor (HA, where H represents an acidic hydrogen atom), and a base is a proton acceptor. In aqueous solution, water can function as a base, as in the following general example.

HA + H2O ${\displaystyle \rightleftharpoons }$ A- + H3O+

Acid dissociation constants are also known as the acidity constant or the acid-ionization constant. The term is also used for pKa, which is equal to minus the decimal logarithm of Ka (cologarithm of Ka).

## Definitions

### Monoprotic acids

When an acid, HA, dissolves in water, some molecules of the acid 'dissociate' to form hydronium ions and the conjugate base, (A-), of the acid.

${\displaystyle HA\rightleftharpoons H^{+}+A^{-}}$

It is understood that ${\displaystyle H^{+}}$ stands for the hydronium ion and that each species in this equilibrium may solvate to a greater or lesser extent. The acid dissociation constant is defined as

${\displaystyle K_{a}={\frac {[{\mbox{H}}^{+}][{\mbox{A}}^{-}]}{[{\mbox{HA}}]}}}$

where the square brackets are usually taken to signify concentration. ${\displaystyle H_{2}O}$ is omitted from these expressions because in dilute solution the concentration of water may be assumed to be constant. Values of ${\displaystyle K_{a}}$ vary over many orders of magnitude, so it is common to take the logarithm to base ten of the value.

${\displaystyle \operatorname {p} K_{a}=-\log _{10}K_{a}}$

It is easier to compare the strengths of different acids by comparing ${\displaystyle pK_{a}}$ values as they vary over a much smaller range.

### Polyprotic acids

A polyprotic acid is one that has more than one proton to dissociate. Typical examples are malonic acid, which has two ionizable protons, and phosphoric acid, which has three. The constant for dissociation of the first proton may be denoted as ${\displaystyle pK_{a1}}$ and the constants for dissociation of successive protons as ${\displaystyle pK_{a2}}$, etc.

It is generally true that successive pK values increase (Pauling's first rule).[1] For example, for a diprotic acid, ${\displaystyle H_{2}A}$, the two equilibria are

${\displaystyle H_{2}A\rightleftharpoons HA^{-}+H^{+}}$
${\displaystyle HA^{-}\rightleftharpoons A^{2-}+H^{+}}$

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. Therefore ${\displaystyle pK_{a2}>pK_{a1}}$. There are a few exceptions to this rule which occur when there is a major structural change such as in the sequence

 ${\displaystyle VO_{2}^{+}(aq)\rightleftharpoons H_{3}VO_{4}+H^{+}}$ ${\displaystyle pK_{a1}=4.2}$ ${\displaystyle H_{3}VO_{4}\rightleftharpoons H_{2}VO_{4}^{-}+H^{+}}$ ${\displaystyle pK_{a2}=2.60}$ ${\displaystyle H_{2}VO_{4}^{-}\rightleftharpoons HVO_{4}^{2-}+H^{+}}$ ${\displaystyle pK_{a3}=7.92}$ ${\displaystyle HVO_{4}^{2-}\rightleftharpoons VO_{4}^{3-}+H^{+}}$ ${\displaystyle pK_{a4}=13.27}$

The pKs of vanadic acid, ${\displaystyle H_{3}VO_{4}}$, follow Pauling's rule just like phosphoric acid (values below). All species in this series are tetrahedral, but ${\displaystyle VO_{2}\left(H_{2}O\right)_{4}^{+}}$ is octahedral and ${\displaystyle pK_{a2}. [2]

### Bases

Historically the equilibrium constant ${\displaystyle K_{b}}$ for a base was defined as the dissociation constant of ${\displaystyle BH}$, the acid conjugate to the base, ${\displaystyle B}$. Ionic charges on ${\displaystyle B}$ and/or ${\displaystyle BH}$ are omitted here because the base may be a neutral species such as ammonia or a charged species such as the ethanoate ion (acetate).

${\displaystyle B+H_{2}O\rightleftharpoons BH+OH^{-}}$

Using similar reasoning to that used before

${\displaystyle K_{b}={\frac {[{\mbox{BH}}][{\mbox{OH}}^{-}]}{[{\mbox{B}}]}}}$
${\displaystyle \operatorname {p} K_{b}=-\log _{10}K_{b}\,}$

Now, the concentration of the hydroxide ion is related to the concentration of the hydronium by ${\displaystyle K_{w}=\left[H^{+}\right]\left[OH^{-}\right]}$, therefore

${\displaystyle [{\mbox{OH}}^{-}]={\frac {K_{w}}{[{\mbox{H}}^{+}]}}}$;

${\displaystyle K_{w}}$ is the constant for the self-ionization of water. Substituting the expression for ${\displaystyle \left[OH^{-}\right]}$ into the expression for ${\displaystyle K_{b}}$

${\displaystyle K_{b}=K_{w}{\frac {\left[{\mbox{BH}}\right]}{\left[{\mbox{B}}\right]\left[{\mbox{H}}^{+}\right]}}={\frac {K_{w}}{K_{a}}}}$

It follows that

${\displaystyle pK_{b}=pK_{w}-pK_{a}}$

In water at 25 °C ${\displaystyle pK_{w}}$ is 14 so then ${\displaystyle pK_{b}=14-pK_{a}}$.

In effect there is no need to define ${\displaystyle pK_{b}}$ separately from ${\displaystyle pK_{a}}$, but it is done because ${\displaystyle pK_{b}}$ values can be found in literature.

### Protonation constants

The protonation constant for a substance ${\displaystyle S}$, is given by

${\displaystyle S+H^{+}\rightleftharpoons SH}$
${\displaystyle K_{assoc}={\frac {[{\mbox{SH}}]}{[{\mbox{S}}][{\mbox{H}}^{+}]}}}$

Again, ionic charges are omitted from ${\displaystyle S}$ and ${\displaystyle SH}$. ${\displaystyle K_{assoc}}$ is an association constant.

• For any substance ${\displaystyle \log _{10}\left(K_{assoc}\right)=pK_{a}}$

### Temperature dependence

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

${\displaystyle {\frac {d\ln K}{dT}}={\frac {{\Delta H_{m}}^{\Theta }}{RT^{2}}}.}$

Thus, for exothermic reactions, (ΔH is negative) K decreases with temperature, but for endothermic reactions (ΔH is positive) K increases with temperature.

## Usage

The pH of a solution of weak acid can be expressed in terms of the extent of dissociation. After rearranging the expression defining the dissociation constant, and putting pH = -log10[H+], one obtains

${\displaystyle {\mbox{pH}}={\mbox{pK}}_{a}-\log {\frac {[HA]}{[A^{-}]}}}$

This is a form of the Henderson-Hasselbalch equation. It can be deduced from this expression that

• when the acid is 1% dissociated, that is, when ${\displaystyle {\frac {[HA]}{[A^{-}]}}=100}$, pH = pKa – 2
• when the acid is 50% dissociated, that is, when ${\displaystyle {\frac {[HA]}{[A^{-}]}}=1}$, pH = pKa
• when the acid is 99% dissociated, that is, when ${\displaystyle {\frac {[HA]}{[A^{-}]}}=0.01}$, pH = pKa + 2

It follows that the range of pH within which there is partial dissociation of the acid is about pKa ${\displaystyle \pm }$ 2.This is shown graphically at the right.

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

On the pKa scale of acid strength, a large value indicates a very weak acid, and a small value indicates a not so weak one.

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

Some polyprotic acids can be treated as a set of individual acids. This is possible when successive pK values differ by 4 or more. For example with phosphoric acid

H3PO4 ${\displaystyle \rightleftharpoons }$ H2PO4- +H+, pKa1 = 2.15
H2PO4- ${\displaystyle \rightleftharpoons }$ HPO42- +H+, pKa2 = 7.20
HPO42- ${\displaystyle \rightleftharpoons }$ PO43- +H+, pKa3 = 12.37

Both the hydrogenphosphate and dihydrogenphosphate ions can be treated as acids in their own right. On the other hand, the two pKs for malonic acid are 2.51 and 5.05, so there are pH values at which both malonic acid and the hydrogenmalonate ion co-exist. More elaborate calculations are needed to calculate the composition of solutions of malonic acid.

## Factors that determine the relative strengths of acids

Being an equilibrium constant, the acid dissociation constant Ka is determined by the standard free energy difference ΔGo between the reactants and products, specifically, between the protonated (HA) and deprotonated (A) forms of the substance.

Pauling's second rule[1] states that the value of the first pK for acids of the formula XOm(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, pKa for HClO is 7.2, for HClO2 is 2.0, for HClO3 is -1 and HClO4 is a strong acid.

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

Structural effects can also be important. The difference between fumaric acid and maleic acid is a classic example. Fumaric acid is (E)-1,4-but-2-enedioic acid, a trans isomer, whereas maleic acid is the corresponding cis isomer, i.e. (Z)-1,4-but-2-enedioic acid (see cis-trans isomerism). Fumaric acid has pKas of approximately 3.5 and 4.5. By contrast, maleic acid has pKas 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.

## Importance of pKa values

The pKa value(s) of a compound influence many characteristics of the compound such as its reactivity, and spectral properties (colour). In biochemistry the pKa values of proteins and amino acid side chains are of major importance for the activity of enzymes and the stability of proteins. This property is of general importance in chemistry because ionization of a compound alters its physical behavior and macro properties such as solubility and lipophilicity. For example ionization of any compound will increase the solubility in water, but decrease the lipophilicity. This can be exploited in drug development to increase the concentration of a compound in the blood by adjusting the pKa of an ionizable group. This must be done with caution, however, since an ionized compound will pass less easily through cell membranes.

## Acidity in nonaqueous solutions

Three properties of a solvent strongly affect acids and bases.

1. A Protic solvent can form hydrogen bonds and will promote ionisation.
2. A solvent with a high donor number is a strong Lewis base.
3. A solvent with a high dielectric constant (Relative permittivity) will promote ionisation.
Donor number dielectric constant
Acetonitrile 14.1 37.5
Dimethyl sulfoxide 29.8 45
Ethanol 31.5 24.3
Pyridine 33.1 12.3
Water 18 81.7

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

Acidity scales have been developed for solvents aside from water, notably for dimethyl sulfoxide and acetonitrile.[3] It can be seen in the table above that DMSO is more basic than water, but its dielectric constant is less. DMSO is widely used as an alternative to water in evaluating acids and bases.

In solvents of low dielectric constant, ions tend to associate, which complicates the interpretation of pKas. In particular, in aprotic solvents the process of homoconjugation occurs when the conjugate base forms a hydrogen bond with the parent acid as in the following equilibrium

HA + A- ${\displaystyle \rightleftharpoons }$ HA2-

Typically HA2- would have the structure A---H---A. This process does not occur in water because H2O molecules are strong hydrogen bond donors and acceptors.

In acetonitrile solution, para-toluenesulfonic acid has a homoconjugation constant pKf, of -2.9.[4] This indicates that the toluenesulfonate anion has a strong tendency to form a hydrogen bond with the parent acid. Homoconjugation has the effect of enhancing the acidity of acids, lowering their effective pKas, by stabilizing the conjugate base. Due to homoconjugation, the proton-donating power of toluenesulfonic acid in acetonitrile solution is enhanced by a factor of nearly 800.

## pKa of some common substances

Measurements are at 25ºC in water for those with a pKa at or above -1.76:

 - 25.00: Fluoroantimonic acid - 15.00: Magic acid - 10.00: Fluorosulfuric acid - 10.00: Perchloric acid - 10.00: Hydroiodic acid - 9.00: Hydrobromic acid - 8.00: Hydrochloric acid - 3.00, 1.99: Sulfuric acid - 2.00: Nitric acid - 1.76: Hydronium ion 3.15: Hydrofluoric acid 3.60: Carbonic acid 3.75: Formic acid 4.04: Ascorbic acid (Vitamin C) 4.19: Succinic acid 4.20: Benzoic acid 4.63: Aniline* 4.76: Acetic acid 4.76: Dihydrogencitrate ion (Citrate) 5.21: Pyridine* 6.40: Monohydrogencitrate ion Citrate 6.99: Ethylenediamine* 7.00: Hydrogen sulfide, Imidazole* (as an acid) 7.50: Hypochlorous acid 9.25: Ammonia* 9.33: Benzylamine* 9.81: Trimethylamine* 9.99: Phenol 10.08: Ethylenediamine* 10.66: Methylamine* 10.73: Dimethylamine* 10.81: Ethylamine* 11.01: Triethylamine* 11.09: Diethylamine* 11.65: Hydrogen peroxide 12.50: Guanidine* 12.67: Monohydrogenphosphate ion (Phosphate) 14.58: Imidazole (as a base) 15.76: Water - 19.00 (pKb) Sodium amide 37.00: Lithium diisopropylamide (LDA) 45.00: Propane 50.00: Ethane

* Listed values for ammonia and amines are the pKa values for the corresponding ammonium ions.