The equilibrium constant of a chemical reaction is the value of its
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
at
chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the sy ...
, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the
composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and
ionic strength
The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such as ...
may all influence the value of the equilibrium constant.
A knowledge of equilibrium constants is essential for the understanding of many chemical systems, as well as biochemical processes such as oxygen transport by
hemoglobin
Hemoglobin (haemoglobin BrE) (from the Greek word αἷμα, ''haîma'' 'blood' + Latin ''globus'' 'ball, sphere' + ''-in'') (), abbreviated Hb or Hgb, is the iron-containing oxygen-transport metalloprotein present in red blood cells (erythrocyte ...
in blood and
acid–base homeostasis
Acid–base homeostasis is the homeostatic regulation of the pH of the body's extracellular fluid (ECF). The proper balance between the acids and bases (i.e. the pH) in the ECF is crucial for the normal physiology of the body—and for cellu ...
in the human body.
Stability constants, formation constants,
binding constant
The binding constant, or affinity constant/association constant, is a special case of the equilibrium constant ''K'', and is the inverse of the dissociation constant. It is associated with the binding and unbinding reaction of receptor (R) and li ...
s, association constants and
dissociation constant
In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex fa ...
s are all types of equilibrium constants.
Basic definitions and properties
For a system undergoing a
reversible reaction
A reversible reaction is a reaction in which the conversion of reactants to products and the conversion of products to reactants occur simultaneously.
: \mathit aA + \mathit bB \mathit cC + \mathit dD
A and B can react to form C and D or, in the ...
described by the general
chemical equation
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities on the right-hand side with a plus sign between ...
:
a thermodynamic equilibrium constant, denoted by
, is defined to be the value of the
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
''Q
t'' when forward and reverse reactions occur at the same rate. At
chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the sy ...
, the chemical composition of the mixture does not change with time and the
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
change
for the reaction is zero. If the composition of a mixture at equilibrium is changed by addition of some reagent, a new equilibrium position will be reached, given enough time. An equilibrium constant is related to the composition of the mixture at equilibrium by
:
:
where denotes the
thermodynamic activity
In chemical thermodynamics, activity (symbol ) is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depen ...
of reagent X at equilibrium,
the numerical value
[Atkins, P.; Jones, L.; Laverman, L. (2016).''Chemical Principles'', 7th edition, pp. 399 & 461. Freeman. ISBN 978-1-4641-8395-9] of the corresponding
concentration in moles per liter, and γ the corresponding
activity coefficient
In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same ( ...
. If X is a gas, instead of
the numerical value of the partial pressure
in bar is used.
If it can be assumed that the quotient of activity coefficients,
, is constant over a range of experimental conditions, such as pH, then an equilibrium constant can be derived as a quotient of concentrations.
:
An equilibrium constant is related to the standard
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
change of reaction
by
:
where ''R'' is the
universal gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, ''T'' is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
(in
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
s), and is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. This expression implies that
must be a pure number and cannot have a dimension, since
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s can only be taken of pure numbers.
must also be a pure number. On the other hand, the
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
at equilibrium
:
does have the dimension of concentration raised to some power (see , below). Such reaction quotients are often referred to, in the biochemical literature, as equilibrium constants.
For an equilibrium mixture of gases, an equilibrium constant can be defined in terms of
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
or
fugacity
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
.
An equilibrium constant is related to the forward and backward
rate constants, ''k''
f and ''k''
r of the reactions involved in reaching equilibrium:
:
Types of equilibrium constants
Cumulative and stepwise formation constants
A cumulative or overall constant, given the symbol ''β'', is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML
2 is given by
:M + 2 L ML
2;
2">L2= ''β''
12 L]
2
The stepwise constant, ''K'', for the formation of the same complex from ML and L is given by
:ML + L ML
2;
2">L2= ''K''
LL] = ''Kβ''
11 L]
2
It follows that
:''β''
12 = ''Kβ''
11
A cumulative constant can always be expressed as the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such as ''K'' is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.
Competition method
A particular use of a stepwise constant is in the determination of stability constant values outside the normal range for a given method. For example,
EDTA
Ethylenediaminetetraacetic acid (EDTA) is an aminopolycarboxylic acid with the formula H2N(CH2CO2H)2sub>2. This white, water-soluble solid is widely used to bind to iron (Fe2+/Fe3+) and calcium ions (Ca2+), forming water-soluble complexes eve ...
complexes of many metals are outside the range for the potentiometric method. The stability constants for those complexes were determined by competition with a weaker ligand.
:ML + L′ ML′ + L
The formation constant of
4.html" ;"title="Palladium(II) cyanide">[Pd(CN)4">Palladium(II) cyanide, [Pd(CN)4sup>2− was determined by the competition method.
Association and dissociation constants
In organic chemistry and biochemistry it is customary to use p''K''
a values for Acid dissociation constant, acid dissociation equilibria.
:
where ''log'' denotes a logarithm to base 10 or common logarithm, and ''K''
diss is a stepwise
acid dissociation constant
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 ...
. For bases, the
Acid dissociation constant#Bases, base association constant, p''K''
b is used. For any given acid or base the two constants are related by , so p''K''
a can always be used in calculations.
On the other hand, stability constants for
metal complexes, and binding constants for
host–guest complexes are generally expressed as association constants. When considering equilibria such as
:M + HL ML + H
it is customary to use association constants for both ML and HL. Also, in generalized computer programs dealing with equilibrium constants it is general practice to use cumulative constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA,
nitrilotriacetic acid
Nitrilotriacetic acid (NTA) is the aminopolycarboxylic acid with the formula N(CH2CO2H)3. It is a colourless solid that is used as a chelating agent, which forms coordination compounds with metal ions (chelates) such as Ca2+, Co2+, Cu2+, and Fe ...
, N(CH
2CO
2H)
3 is designated as H
3L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.
:
The cumulative association constants can be expressed as
:
Note how the subscripts define the stoichiometry of the equilibrium product.
Micro-constants
When two or more sites in an asymmetrical molecule may be involved in an equilibrium reaction there are more than one possible equilibrium constants. For example, the molecule
-DOPA has two non-equivalent hydroxyl groups which may be deprotonated. Denoting -DOPA as LH
2, the following diagram shows all the species that may be formed (X = ).
:
The concentration of the species LH is equal to the sum of the concentrations of the two micro-species with the same chemical formula, labelled L
1H and L
2H. The constant ''K''
2 is for a reaction with these two micro-species as products, so that
H=
1H">1H+
2H">2Happears in the numerator, and it follows that this macro-constant is equal to the sum of the two micro-constants for the component reactions.
:''K''
2 = ''k''
21 + ''k''
22
However, the constant ''K''
1 is for a reaction with these two micro-species as reactants, and
H=
1H">1H+
2H">2Hin the denominator, so that in this case
:1/''K''
1 =1/ ''k''
11 + 1/''k''
12,
and therefore ''K''
1 =''k''
11 ''k''
12 / (''k''
11 + ''k''
12).
Thus, in this example there are four micro-constants whose values are subject to two constraints; in consequence, only the two macro-constant values, for K
1 and K
2 can be derived from experimental data.
Micro-constant values can, in principle, be determined using a spectroscopic technique, such as
infrared spectroscopy
Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or function ...
, where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include
* Chemical: blocking one of the sites, for example by methylation of a hydroxyl group, followed by determination of the equilibrium constant of the related molecule, from which the micro-constant value for the "parent" molecule may be estimated.
* Mathematical: applying numerical procedures to
13C NMR data.
Although the value of a micro-constant cannot be determined from experimental data, site occupancy, which is proportional to the micro-constant value, can be very important for biological activity. Therefore, various methods have been developed for estimating micro-constant values. For example, the isomerization constant for -DOPA has been estimated to have a value of 0.9, so the micro-species L
1H and L
2H have almost equal concentrations at all
pH values.
pH considerations (Brønsted constants)
pH is defined in terms of the
activity of the hydrogen ion
:pH = −log
10
In the approximation of ideal behaviour, activity is replaced by concentration. pH is measured by means of a glass electrode, a mixed equilibrium constant, also known as a Brønsted constant, may result.
:HL L + H;
It all depends on whether the electrode is calibrated by reference to solutions of known activity or known concentration. In the latter case the equilibrium constant would be a concentration quotient. If the electrode is calibrated in terms of known hydrogen ion concentrations it would be better to write p
rather than pH, but this suggestion is not generally adopted.
Hydrolysis constants
In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by
:
\mathit_W = OH]
:
H\mathit_W
The first step in metal ion
hydrolysis
Hydrolysis (; ) is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution reaction, substitution, elimination reaction, elimination, and solvation reactions in which water ...
can be expressed in two different ways
:
It follows that . Hydrolysis constants are usually reported in the ''β''
* form and therefore often have values much less than 1. For example, if and so that ''β
*'' = 10
−10. In general when the hydrolysis product contains ''n'' hydroxide groups
Conditional constants
Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them. A very common instance is where pH is fixed at a particular value. For example, in the case of iron(III) interacting with EDTA, a conditional constant could be defined by
:
This conditional constant will vary with pH. It has a maximum at a certain pH. That is the pH where the ligand sequesters the metal most effectively.
In biochemistry equilibrium constants are often measured at a pH fixed by means of a
buffer solution
A buffer solution (more precisely, pH buffer or hydrogen ion buffer) is an aqueous solution consisting of a mixture of a weak acid and its conjugate base, or vice versa. Its pH changes very little when a small amount of strong acid or base is ...
. Such constants are, by definition, conditional and different values may be obtained when using different buffers.
Gas-phase equilibria
For equilibria in a
gas phase
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetiza ...
,
fugacity
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
, ''f'', is used in place of activity. However, fugacity has the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, so it must be divided by a standard pressure, usually 1 bar, in order to produce a dimensionless quantity, . An equilibrium constant is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO
2 N
2O
4,
:
Fugacity is related to
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
, ''
'', by a dimensionless fugacity coefficient ''ϕ'': ''
''. Thus, for the example,
:
Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas phase then resemble the expression for solution equilibria with fugacity coefficient in place of activity coefficient and partial pressure in place of concentration.
:
Thermodynamic basis for equilibrium constant expressions
Thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure the
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
is minimum. The slope of the reaction free energy with respect to the
extent of reaction In physical chemistry and chemical engineering, extent of reaction is a quantity that measures the extent to which the reaction has proceeded. Often, it refers specifically to the value of the extent of reaction when equilibrium has been reached. It ...
, ''ξ'', is zero when the free energy is at its minimum value.
:
The free energy change, d''G''
r, can be expressed as a weighted sum of change in amount times the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
, the partial molar free energy of the species. The chemical potential, ''μ
i'', of the ''i''th species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, ''N''
i
:
A general chemical equilibrium can be written as
:
where ''n
j'' are the
stoichiometric coefficient
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities on the right-hand side with a plus sign between t ...
s of the reactants in the equilibrium equation, and ''m
j'' are the coefficients of the products. At equilibrium
:
The chemical potential, ''μ
i'', of the ''i''th species can be calculated in terms of its
activity, ''a
i''.
:
''μ'' is the standard chemical potential of the species, ''R'' is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
and ''T'' is the temperature. Setting the sum for the reactants ''j'' to be equal to the sum for the products, ''k'', so that ''δG''
r(Eq) = 0
:
Rearranging the terms,
:
:
This relates the
standard Standard may refer to:
Symbols
* Colours, standards and guidons, kinds of military signs
* Standard (emblem), a type of a large symbol or emblem used for identification
Norms, conventions or requirements
* Standard (metrology), an object th ...
Gibbs free energy change, Δ''G''
o to an equilibrium constant, ''K'', the
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
of activity values at equilibrium.
:
:
Equivalence of thermodynamic and kinetic expressions for equilibrium constants
At equilibrium the rate of the forward reaction is equal to the backward reaction rate. A simple reaction, such as
ester hydrolysis
In chemistry, an ester is a compound derived from an oxoacid (organic or inorganic) in which at least one hydroxyl group () is replaced by an alkoxy group (), as in the substitution reaction of a carboxylic acid and an alcohol. Glycerides are fa ...
:
AB + H2O <=> AH + B(OH)
has reaction rates given by expressions
:
:
According to
Guldberg and
Waage Waage is a Norwegian surname. Notable people with the surname include:
*Anita Waage (born 1971), Norwegian footballer
*Elsa Waage (born 1959), Icelandic opera singer
*Geir Waage (born 1967), Norwegian politician
*Hilde Waage (born 1959), Norwegian ...
, equilibrium is attained when the forward and backward reaction rates are equal to each other. In these circumstances, an equilibrium constant is defined to be equal to the ratio of the forward and backward reaction rate constants
:
.
The concentration of water may be taken to be constant, resulting in the simpler expression
:
.
This particular concentration quotient,
, has the dimension of concentration, but the thermodynamic equilibrium constant, , is always dimensionless.
Unknown activity coefficient values
![PK acetic acid](https://upload.wikimedia.org/wikipedia/commons/5/53/PK_acetic_acid.png)
It is very rare for activity coefficient values to have been determined experimentally for a system at equilibrium. There are three options for dealing with the situation where activity coefficient values are not known from experimental measurements.
#Use calculated activity coefficients, together with concentrations of reactants. For equilibria in solution estimates of the activity coefficients of charged species can be obtained using
Debye–Hückel theory
The Debye–Hückel theory was proposed by Peter Debye and Erich Hückel as a theoretical explanation for departures from ideality in solutions of electrolytes and plasmas.
It is a linearized Poisson–Boltzmann model, which assumes an extrem ...
, an extended version, or
SIT theory. For uncharged species, the activity coefficient ''γ''
0 mostly follows a "salting-out" model: log
10 ''γ''
0 = ''bI'' where ''I'' stands for
ionic strength
The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such as ...
.
#Assume that the activity coefficients are all equal to 1. This is acceptable when all concentrations are very low.
#For equilibria in solution use a medium of high ionic strength. In effect this redefines the
standard state as referring to the medium. Activity coefficients in the standard state are, by definition, equal to 1. The value of an equilibrium constant determined in this manner is dependent on the ionic strength. 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.
Dimensionality
An equilibrium constant is related to the standard
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
change,
, for the reaction by the expression
:
Therefore, ''K'', must be a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
from which a logarithm can be derived. In the case of a simple equilibrium
:
A + B <=> AB
the thermodynamic equilibrium constant is defined in terms of the
activities, , and , of the species in equilibrium with each other.
:
Now, each activity term can be expressed as a product of a concentration