Equilibrium chemistry is concerned with systems in chemical equilibrium. The unifying principle
is that the free energy of a system at equilibrium is
the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero. This principle,
applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid-base, host-guest, metal-complex, solubility,
partition, chromatography
and redox
equilibria.
Thermodynamic equilibrium
A chemical system is said to be in equilibrium when the
quantities of the chemical entities involved do not and cannot change in
time without the application of an external influence. In this sense a system
in chemical equilibrium is in a stable state. The system at chemical equilibrium will be at a
constant temperature, pressure (or volume) and composition. It will be
insulated from exchange of heat with the surroundings, that is, it is a closed
system. A change of temperature, pressure (or volume) constitutes an
external influence and the equilibrium quantities will change as a result of
such a change. If there is a possibility that the composition might change, but
the rate of change is negligibly slow, the system is said to be in a metastable
state. The equation of chemical equilibrium can be expressed symbolically as
reactant(s) -product(s)
The sign -means
"are in equilibrium with". This definition refers to macroscopic
properties. Changes do occur at the microscopic level of atoms and molecules,
but to such a minute extent that they are not measurable and in a balanced way
so that the macroscopic quantities do not change. Chemical equilibrium is a
dynamic state in which forward and backward reactions proceed at such rates
that the macroscopic composition of the mixture is constant. Thus, equilibrium
sign -symbolizes the fact that reactions
occur in both forward >and backward <directions.
A steady state, on the other hand, is not
necessarily an equilibrium state in the chemical sense. For example, in a
radioactive decay chain the concentrations of intermediate isotopes
are constant because the rate of production is equal to the rate of decay. It
is not a chemical equilibrium because the decay process occurs in one direction
only.
Thermodynamic equilibrium is characterized by the free
energy for the whole (closed) system being a minimum. For systems at constant
volume the Helmholtz free energy is minimum and for
systems at constant pressure the Gibbs
free energy is minimum. Thus a metastable state is one for which the free
energy change between reactants and products is not minimal even though the
composition does not change in time.
The existence of this minimum is due to the free energy of
mixing of reactants and products being always negative. For ideal
solutions the enthalpy of mixing is zero, so the minimum exists because
the entropy of
mixing is always positive. The slope of the reaction free energy, δGr
with respect to the reaction coordinate, ξ, is zero when the free
energy is at its minimum value.
Equilibrium constant
Chemical potential is the partial molar free
energy. The potential, μi, of the ith species in a chemical
reaction is the partial derivative of the free energy with respect to the
number of moles of that species,
It follows that any equilibrium of this kind can be
characterized either by the standard free energy change or by the equilibrium
constant. In practice concentrations are more useful than activities.
Activities can be calculated from concentrations if the activity coefficient are known, but this is
rarely the case. Sometimes activity coefficients can be calculated using, for
example, Pitzer equations or Specific ion interaction theory.
Otherwise conditions must be adjusted so that activity coefficients do not vary
much. For ionic solutions this is achieved by using a background ionic medium
at a high concentration relative to the concentrations of the species in
equilibrium.
If activity coefficients are unknown they may be subsumed
into the equilibrium constant, which becomes a concentration quotient. Each
activity ai is assumed to be the product of a concentration,
[Ai], and an activity coefficient, γi
This definition is much more practical, but an equilibrium
constant defined in terms of concentrations is dependent on conditions. In
particular, equilibrium constants for species in aqueous solution are dependent
on ionic
strength, as the quotient of activity coefficients varies with the ionic
strength of the solution.
The values of the standard free energy change and of the
equilibrium constant are temperature dependent. To a first approximation, the van 't Hoff equation may be used.
This shows that when the reaction is exothermic (ΔH,
the standard enthalpy
change, is negative), then K decreases with increasing temperature, in
accordance with Le Chatelier's principle. The
approximation involved is that the standard enthalpy change, ΔH, is
independent of temperature, which is a good approximation only over a small
temperature range. Thermodynamic arguments can be used to show that
Equilibria involving gases
When dealing with gases, fugacity, f,
is used rather than activity. However, whereas activity is dimensionless,
fugacity has the dimension of pressure. A consequence is that chemical potential has to be
defined in terms of a standard pressure, p
By convention p is usually taken to be 1 bar
Fugacity can be expressed as the product of partial
pressure, p, and a fugacity coefficient,
Fugacity coefficients are dimensionless and can be obtained
experimentally at specific temperature and pressure, from measurements of
deviations from ideal gas behaviour. Equilibrium constants are defined in
terms of fugacity. If the gases are at sufficiently low pressure that they
behave as ideal gases, the equilibrium constant can be defined as a quotient of
partial pressures.
An example of gas-phase equilibrium is provided by the Haber-Bosch
process of ammonia
synthesis.
This reaction is strongly exothermic,
so the equilibrium constant decreases with temperature. However, a temperature
of around 400 °C is required in order to achieve a reasonable rate of
reaction with currently available catalysts.
Formation of ammonia is also favoured by high pressure, as the volume decreases
when the reaction takes place. It is interesting to note that the same
reaction, nitrogen fixation, occurs at ambient temperatures
in nature, when the catalyst is an enzyme such as nitrogenase.
Much energy is needed initially to break the N-N triple bond even though the
overall reaction is exothermic.
Gas-phase equilibria occur during combustion
and were studied as early as 1943 in connection with the development of the V2 rocket
engine.
The calculation of composition for a gaseous equilibrium at
constant pressure is often carried out using ΔG values, rather than equilibrium
constants.
Multiple equilibria
Two or more equilibria can exist at the same time. When this
is so, equilibrium constants can be ascribed to individual equilibria, but they
are not always unique. For example, three equilibrium constants can be defined
for a dibasic acid, H2A.
Speciation
Speciation diagram for a solution of citric acid as a
function of pH.
The concentrations of species in equilibrium are usually
calculated under the assumption that activity coefficients are either known or
can be ignored. In this case, each equilibrium constant for the formation of a
complex in a set of multiple equilibria can be defined as follows
The concentrations of species containing reagent A are
constrained by a condition of mass-balance, that is, the total (or
analytical) concentration, which is the sum of all species' concentrations,
must be constant. There is one mass-balance equation for each reagent of the
type
There are as many mass-balance equations as there are
reagents, A, B .., so if the equilibrium constant values are known, there are n
mass-balance equations in n unknowns, [A], [B].., the so-called free
reagent concentrations. Solution of these equations gives all the information
needed to calculate the concentrations of all the species.
Thus, the importance of an equilibrium constants lies in the
fact that, once their values have been determined by experiment, they can be
used to calculate the concentrations, known as the speciation, of mixtures that contain the
relevant species.
Determination
There are five main types of experimental data that are used
for the determination of solution equilibrium constants. Potentiometric data
obtained with a glass electrode are the most widely used with
aqueous solutions. The others are Spectrophotometric,
Fluorescence
(luminescence) measurements and NMR chemical shift measurements; simultaneous
measurement of K and H for 1:1 adducts in biological systems is routinely
carried out using Isothermal Titration Calorimetry.
The experimental data will comprise a set of data points. At
the i'th data point, the analytical concentrations of the reactants, etc. will
be experimentally known quantities and there will be one or more measured
quantities, yi, that depend in some way on the analytical
concentrations and equilibrium constants. A general computational procedure has
three main components.
- Definition of a chemical model of the equilibria. The model consists of a list of reagents, A, B, etc. and the complexes formed from them, with stoichiometries ApBq... Known or estimated values of the equilibrium constants for the formation of all complexes must be supplied.
- Calculation of the concentrations of all the chemical species in each solution. The free concentrations are calculated by solving the equations of mass-balance, and the concentrations of the complexes are calculated using the equilibrium constant definitions. A quantity corresponding to the observed quantity can then be calculated using physical principles such as the Nernst potential or Beer-Lambert law which relate the calculated quantity to the concentrations of the species.
- Refinement of the equilibrium constants. Usually a Non-linear least squares procedure is used. A weighted sum of squares, U, is minimized.
The weights, wi and quantities y
may be vectors. Values of the equilibrium constants are refined in an iterative
procedure.
Acid-base equilibria
Brønsted and Lowry characterized an
acid-base equilibrium as involving a proton exchange reaction:
acid + base -conjugate
base + conjugate acid.
An acid is a proton donor; the proton is transferred to the
base, a proton acceptor, 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 solvated hydrogen ion. In solution chemistry, it is usual
to use H+ as an abbreviation for the solvated hydrogen ion,
regardless of the solvent. In aqueous solution H+ denotes a solvated
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+. 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, acts as a weak acid, even though it is not a proton donor,
because of the hydrolysis equilibrium
B(OH)3 + H2O -B(OH)4− + H+.
Similarly, metal ion hydrolysis causes ions such as [Al(H2O)6]3+
to behave as weak acids:
[Al(H2O)6]3+ -[Al(H2O)5(OH)]2+
+ H+.
Acid-base equilibria are important in a very wide range of applications, such as acid-base homeostasis, ocean acidification, pharmacology
and analytical chemistry.
Host-guest equilibria
A host-guest complex, also known as a donor-acceptor
complex, may be formed from a Lewis base, B, and a Lewis acid,
A. The host may be either a donor or an acceptor. In biochemistry
host-guest complexes are known as receptor-ligand complexes; they are formed
primarily by non-covalent bonding. Many host-guest complexes
has 1:1 stoichiometry, but many others have more complex structures. The
general equilibrium can be written as
pA +qB -ApBq
The study of these complexes is important for supramolecular chemistry and molecular recognition. The objective of these
studies is often to find systems with a high binding selectivity of a host (receptor) for a
particular target molecule or ion, the guest or ligand. An application is the
development of chemical sensors. Finding a drug which either
blocks a receptor, an antagonist which forms a strong complex the receptor, or
activate it, an agonist,
is an important pathway to drug discovery.
Complexes of metals
Speciation diagram for aluminium in aqueous solution as a
function of pH. A variety of hydroxo complexes are formed, including aluminium
hydroxide, (Al(OH)3(s), which is insoluble at pH ~6.5
The formation of a complex between a metal ion, M, and a
ligand, L, is in fact usually a substitution reaction. For example, In aqueous
solutions, metal ions will be present as aqua-ions, so the reaction for the
formation of the first complex could be written as
[M(H2O)n] + L -[M(H2O)n-1L] +H2O
However, since water is in vast excess, the concentration of
water is usually assumed to be constant and is omitted from equilibrium
constant expressions. Often, the metal and the ligand are in competition for
protons. For the equilibrium
pM + qL +rH -MpLqHr
a stability constant can be defined as follows.
The definition can easily be extended to include any number
of reagents. It includes hydroxide complexes because the concentration of the
hydroxide ions is related to the concentration of hydrogen ions by the self-ionization of water
[OH-] = KW [H+]-1
Stability constants defined in this way, are association
constants. This can lead to some confusion as pKa values are dissociation
constants. In general purpose computer programs it is customary to define all
constants as association constants. The relationship between the two types of
constant is given in association and dissociation constants.
In biochemistry, an oxygen molecule can bind to an iron
(II) atom in a heme prosthetic
group in hemoglobin. The equilibrium is usually written, denoting
hemoglobin by Hb, as
Hb + O2 -HbO2
but this representation is incomplete as the Bohr effect
shows that the equilibrium concentrations are pH-dependent. A better
representation would be
[HbH]+ + O2 -HbO2 + H+
as this shows that when hydrogen ion concentration increases
the equilibrium is shifted to the left in accordance with Le Chatelier's principle. Hydrogen ion
concentration can be increased by the presence of carbon dioxide, which behaves
as a weak acid.
H2O + CO2 -HCO3- + H+
The iron atom can also bind to other molecules such as carbon
monoxide. Cigarette smoke contains some carbon monoxide so the equilibrium
HbO2 + CO -Hb(CO)
+ O2
is established in the blood of cigarette smokers.
Chelation therapy is based on the principle of
using chelating ligands with a high binding selectivity for a particular metal to
remove that metal from the human body.
Complexes with polyamino carboxylic acids find a wide
range of applications. EDTA in particular is used extensively.
Redox equilibria
The concentration of free electrons is effectively zero as
the electrons are transferred directly from the reductant to the oxidant. The standard electrode potential, E0
for the each half-reaction is related to the standard free energy change by
This is an example of the Nernst
equation. The potential is known as a reduction potential. Standard
electrode potentials are available in a table of values. Using
these values, the actual electrode potential for a redox couple can be
calculated as a function of the ratio of concentrations.
Use of this expression allows the effect of a species not
involved in the redox reaction, such as the hydrogen ion in a half-reaction
such as
MnO4- + 8H+ +5e-
-Mn2+ + 4H2O
to be taken into account.
The equilibrium constant for a full redox reaction can be
obtained from the standard redox potentials of the constituent half-reactions.
At equilibrium the potential for the two half-reactions must be equal to each
other and, of course, the number of electrons exchanged must be the same in the
two half reactions.
Redox equilibria play an important role in the electron transport chain. The various cytochromes
in the chain have different standard redox potentials, each one adapted for a
specific redox reaction. This allows, for example, atmospheric oxygen to be
reduced in photosynthesis. A distinct family of cytochromes, the cytochrome P450 oxidases, are involved in steroidogenesis
and detoxification.
Solubility
When a solute forms a saturated solution in a solvent, the
concentration of the solute, at a given temperature, is determined by the
equilibrium constant at that temperature.
Concentrations, indicated by [..], are usually used in place
of activities, but activity must be taken into account of the presence of
another salt with no ions in common, the so-called salt effect. When another
salt is present that has an ion in common, the common-ion
effect comes into play, reducing the solubility of the primary solute.
Partition
When a solution of a substance in one solvent is brought
into equilibrium with a second solvent that is immiscible with the first
solvent, the dissolved substance may be partitioned between the two solvents.
The ratio of concentrations in the two solvents is known as a partition coefficient or distribution coefficient. The partition
coefficient is defined as the ratio of the analytical concentrations of the solute in
the two phases. By convention the value is reported in logarithmic form.
The partition coefficient is defined at a specified
temperature and, if applicable, pH of the aqueous phase. Partition coefficients
are very important in pharmacology because they determine the extent to which
a substance can pass from the blood (an aqueous solution) through a cell wall
which is like an organic solvent. They are usually measured using water and octanol as
the two solvents. Many pharmaceutical compounds are weak acids
or weak
bases. Such a compound may exist with a different extent of protonation
depending on pH and the acid dissociation constant. Because the
organic phase has a low dielectric constant the species with no
electrical charge will be the most likely one to pass from the aqueous phase to
the organic phase. Even at pH 7-7.2, the range of biological pH values, the
aqueous phase may support an equilibrium between more than one protonated form.
Log p is determined from the analytical concentration of the substance
in the aqueous phase, that is, the sum of the concentration of the different
species in equilibrium.
An organic MTBE solution is extracted with aqueous sodium
bicarbonate solution. This base removes benzoic
acid as benzoate
but leaves non-acidic benzil (yellow) behind in the upper organic phase.
Solvent extraction is used extensively in separation and
purification processes. In its simplest form a reaction is performed in an
organic solvent and unwanted by-products are removed by extraction into water
at a particular pH.
A metal ion may be extracted from an aqueous phase into an
organic phase in which the salt is not soluble, by adding a ligand. The ligand,
La-, forms a complex with the metal ion, Mb+, [MLx](b-ax)+
which has a strongly hydrophobic outer surface. If the complex has no
electrical charge it will be extracted relatively easily into the organic
phase. If the complex is charged, it is extracted as an ion pair. The
additional ligand is not always required. For example, uranyl
nitrate, UO2(NO3)2, is soluble in diethyl
ether because the solvent itself acts as a ligand. This property was used
in the past for separating uranium from other metals whose salts are not
soluble in ether. Currently extraction into kerosene is
preferred, using a ligand such as tri-n-butyl phosphate, TBP. In the PUREX process, which
is commonly used in nuclear reprocessing, uranium(VI) is extracted
from strong nitric acid as the electrically neutral complex [UO2(TBP)2(NO3)2].
The strong nitric acid provides a high concentration of nitrate ions which
pushes the equilibrium in favour of the weak nitrato complex. Uranium is
recovered by back-extraction (stripping) into weak nitric acid. Plutonium(IV)
forms a similar complex, [PuO2(TBP)2(NO3)2]
and the plutonium in this complex can be reduced to separate it from uranium.
Another important application of solvent extraction is in
the separation of the lanthanoids. This process also uses TBP and the complexes
are extracted into kerosene. Separation is achieved because the stability constant for the
formation of the TBP complex increases as the size of the lanthanoid ion
decreases.
An instance of ion-pair extraction is in the use of a ligand
to enable oxidation by potassium permanganate, KMnO4, in
an organic solvent. KMnO4 is not soluble in organic solvents. When a
ligand, such as a crown ether is added to an aqueous solution of KMnO4,
it forms a hydrophobic complex with the potassium cation which allows the
uncharged ion-pair, {[KL]+[MnO4]-} to be
extracted into the organic solvent. See also: phase-transfer catalysis.
More complex partitioning problems (i.e. 3 or more phases
present) can sometimes be handled with a fugacity
capacity approach.
Chromatography
In chromatography substances are separated by partition
between a stationary phase and a mobile phase. The analyte is dissolved in the
mobile phase, and passes over the stationary phase. Separation occurs because
of differing affinities of the analytes for the stationary phase. A distribution constant, Kd
can be defined as
There is a wide variety of chromatographic techniques,
depending on the nature of the stationary and mobile phases. When the
stationary phase is solid, the analyte may form a complex with it. A water
softener functions by selective complexation with a sulfonate ion exchange resin. Sodium ions form relatively
weak complexes with the resin. When hard water
is passed through the resin, the divalent ions of magnesium and calcium
displace the sodium ions and are retained on the resin, R.
RNa + M2+ -RM+
+ Na+
The water coming out of the column is relatively rich in
sodium ions and poor in calcium and magnesium which are retained on the column.
The column is regenerated by passing a strong solution of sodium chloride
through it, so that the resin- sodium complex is again formed on the column. Ion-exchange chromatography utilizes a
resin such as chelex 100 in which iminodiacetate residues, attached to a polymer
backbone, form chelate
complexes of differing strengths with different metal ions, allowing the ions
such as Cu2+ and Ni2+ to be separated chromatographically.
Another example of complex formation is in chiral chromatography in which is used to separate enantiomers
from each other. The stationary phase is itself chiral and forms complexes
selectively with the enantiomers. In other types of chromatography with a solid
stationary phase, such as thin layer chromatography the analyte is
selectively adsorbed
onto the solid.
In gas-liquid chromatography (GLC) the
stationary phase is a liquid such as polydimethylsiloxane, coated on a glass tube.
Separation is achieved because the various components in the gas have different
solubility in the stationary phase. GLC can be used to separate literally
hundreds of components in a gas mixture such as cigarette
smoke or essential oils, such as lavender oil.
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