In
electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...
, and more generally in solution chemistry, a Pourbaix diagram, also known as a potential/pH diagram, E
H–pH diagram or a pE/pH diagram, is a plot of possible thermodynamically stable phases (''i.e.'', at
chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the Reagent, reactants and Product (chemistry), products are present in concentrations which have no further tendency to change with time, so that there is no observable chan ...
) of an aqueous electrochemical system. Boundaries (50 %/50 %) between the predominant chemical species (aqueous ions in solution, or solid phases) are represented by lines. As such a Pourbaix diagram can be read much like a standard
phase diagram
A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...
with a different set of axes. Similarly to phase diagrams, they do not allow for
reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...
or kinetic effects. Beside potential and pH, the equilibrium concentrations are also dependent upon, e.g., temperature, pressure, and concentration. Pourbaix diagrams are commonly given at room temperature, atmospheric pressure, and molar concentrations of 10
−6 and changing any of these parameters will yield a different diagram.
The diagrams are named after
Marcel Pourbaix
Marcel Pourbaix (16 September 1904 – 28 September 1998) was a Belgian chemist and pianist. He performed his most well known research at the University of Brussels, studying corrosion. His biggest achievement is the derivation of potential-pH ...
(1904–1998), the
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...
n-born Belgian
chemist
A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties. Chemists carefully describe th ...
who invented them.
Naming
Pourbaix diagrams are also known as ''E''
H-pH diagrams due to the labeling of the two axes.
Diagram
The vertical axis is labeled ''E''
H for the
voltage potential with respect to the
standard hydrogen electrode
The standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be at 25 °C, but to form a basis ...
(SHE) as calculated by the
Nernst equation
In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tempe ...
. The "H" stands for hydrogen, although other standards may be used, and they are for room temperature only.
For a reversible redox reaction described by the following
chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the Reagent, reactants and Product (chemistry), products are present in concentrations which have no further tendency to change with time, so that there is no observable chan ...
:
:
With the corresponding
equilibrium constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
:
:
The Nernst equation is:
:
:
sometimes formulated as:
:
or, more simply directly expressed numerically as:
:
where:
*
volt is the
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
or the "Nernst slope" at standard temperature
* ''λ'' = ln(10) ≈ 2.30, so that
volt.
The horizontal axis is labeled
pH for the −log function of the H
+ ion activity.
:
The lines in the Pourbaix diagram show the equilibrium conditions, that is, where the activities are equal, for the species on each side of that line. On either side of the line, one form of the species will instead be said to be predominant.
In order to draw the position of the lines with the Nernst equation, the activity of the chemical species at equilibrium must be defined. Usually, the activity of a species is approximated as equal to the concentration (for soluble species) or partial pressure (for gases). The same values should be used for all species present in the system.
For soluble species, the lines are often drawn for concentrations of 1 M or 10
−6 M. Sometimes additional lines are drawn for other concentrations.
If the diagram involves the equilibrium between a dissolved species and a gas, the pressure is usually set to ''P''
0 = 1 atm = , the minimum pressure required for gas evolution from an aqueous solution at standard conditions.
In addition, changes in temperature and concentration of solvated ions in solution will shift the equilibrium lines in accordance with the Nernst equation.
The diagrams also do not take kinetic effects into account, meaning that species shown as unstable might not react to any significant degree in practice.
A simplified Pourbaix diagram indicates regions of "immunity", "corrosion" and "passivity", instead of the stable species. They thus give a guide to the stability of a particular metal in a specific environment. Immunity means that the metal is not attacked, while
corrosion
Corrosion is a natural process that converts a refined metal into a more chemically stable oxide. It is the gradual deterioration of materials (usually a metal) by chemical or electrochemical reaction with their environment. Corrosion engine ...
shows that general attack will occur.
Passivation occurs when the metal forms a stable coating of an oxide or other salt on its surface, the best example being the relative stability of
aluminium
Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. I ...
because of the
alumina layer formed on its surface when exposed to air.
Applicable chemical systems
While such diagrams can be drawn for any chemical system, it is important to note that the addition of a metal binding agent (
ligand
In coordination chemistry, a ligand is an ion or molecule (functional group) that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's electr ...
) will often modify the diagram. For instance,
carbonate
A carbonate is a salt of carbonic acid (H2CO3), characterized by the presence of the carbonate ion, a polyatomic ion with the formula . The word ''carbonate'' may also refer to a carbonate ester, an organic compound containing the carbonate g ...
() has a great effect upon the diagram for uranium. (See diagrams at right). The presence of trace amounts of certain species such as chloride ions can also greatly affect the stability of certain species by destroying passivating layers.
Limitations
Even though Pourbaix diagrams are useful for a metal corrosion potential estimation they have, however, some important limitations:
# Equilibrium is always assumed, though in practice it may differ.
# The diagram does not provide information on actual corrosion rates.
# Does not apply to alloys.
# Does not indicate whether passivation (in the form of oxides or hydroxides) is protective or not. Diffusion of oxygen ions through thin oxide layers are possible.
# Excludes corrosion by chloride ions (, etc.).
# Usually applicable only to temperature of , which is assumed by default. The Pourbaix diagrams for higher temperatures exist.
Expression of the Nernst equation as a function of pH
The
and
pH of a solution are related by the Nernst equation as commonly represented by a Pourbaix diagram .
explicitly denotes
expressed versus the
standard hydrogen electrode
The standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be at 25 °C, but to form a basis ...
(SHE). For a
half cell
In electrochemistry, a half-cell is a structure that contains a conductive electrode and a surrounding conductive electrolyte separated by a naturally occurring Helmholtz double layer. Chemical reactions within this layer momentarily pump electri ...
equation, conventionally written as a reduction reaction (''i.e.'', electrons accepted by an oxidant on the left side):
:
The equilibrium constant of this reduction reaction is:
:
where curly braces indicate
activities (), rectangle braces
nbsp;
In word processing and digital typesetting, a non-breaking space, , also called NBSP, required space, hard space, or fixed space (though it is not of fixed width), is a space character that prevents an automatic line break at its position. In ...
denote
molar or
molal
Molality is a measure of the number of moles of solute in a solution corresponding to 1 kg or 1000 g of solvent. This contrasts with the definition of molarity which is based on a specified volume of solution.
A commonly used unit for molali ...
concentrations (),
represent the
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 ( ...
s, and the
stoichiometric
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions.
Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equ ...
coefficients are shown as exponents.
Activities correspond to thermodynamic concentrations and take into account the electrostatic interactions between ions present in solution. When the concentrations are not too high, the activity (
) can be related to the measurable concentration (
) by a linear relationship with the activity coefficient (
):
:
The half-cell
standard reduction potential is given by
:
where
is 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, is the number of electrons involved, and is the
Faraday's constant
In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of ...
. The Nernst equation relates pH and
as follows:
:
In the following, the Nernst slope (or
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
) is used, which has a value of 0.02569... V at
STP
STP may refer to:
Places
* São Tomé and Príncipe (ISO 3166-1 alpha-3 code, IOC country code, and FIFA country code STP)
* St Pancras railway station, London St Pancras (Domestic) railway station (National Rail code STP)
* St. Paul Downtown Air ...
. When base-10 logarithms are used, ''V
T λ'' = 0.05916... V at STP where ''λ'' = ln
0= 2.3026.
:
This equation is the equation of a straight line for
as a function of pH with a slope of
volt (pH has no units).
This equation predicts lower
at higher pH values. This is observed for the reduction of O
2 into H
2O, or OH
−, and for reduction of H
+ into H
2.
is then often noted as
to indicate that it refers to the
standard hydrogen electrode
The standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be at 25 °C, but to form a basis ...
(SHE) whose
= 0 by convention under standard conditions (T = 298.15 K = 25 °C = 77 F, P
gas = 1 atm (1.013 bar), concentrations = 1 M and thus pH = 0).
Calculation of a Pourbaix diagram
When the activities (
) can be considered as equal to the
molar, or the
molal
Molality is a measure of the number of moles of solute in a solution corresponding to 1 kg or 1000 g of solvent. This contrasts with the definition of molarity which is based on a specified volume of solution.
A commonly used unit for molali ...
, concentrations (
) at sufficiently diluted concentrations when the
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 ( ...
s (
) tend to one, the term regrouping all the activity coefficients is equal to one, and the
Nernst equation
In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tempe ...
can be written simply with the concentrations (
) denoted here with square braces
nbsp;
In word processing and digital typesetting, a non-breaking space, , also called NBSP, required space, hard space, or fixed space (though it is not of fixed width), is a space character that prevents an automatic line break at its position. In ...
:
There are three types of line boundaries in a Pourbaix diagram: Vertical, horizontal, and sloped.
Vertical boundary line
When no electrons are exchanged (''z'' = 0), the equilibrium between , , , and only depends on and is not affected by the electrode potential. In this case, the reaction is a classical
acid-base reaction involving only
protonation
In chemistry, protonation (or hydronation) is the adding of a proton (or hydron, or hydrogen cation), (H+) to an atom, molecule, or ion, forming a conjugate acid. (The complementary process, when a proton is removed from a Brønsted–Lowry acid, ...
/deprotonation of dissolved species. The boundary line will be a vertical line at a particular value of pH. The reaction equation may be written:
:
and the energy balance is written as
, where is the
equilibrium constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
:
:
Thus:
:
or, in base-10 logarithms,
:
which may be solved for the particular value of pH.
For example
consider the iron and water system, and the equilibrium line between the
ferric
In chemistry, iron(III) refers to the element iron in its +3 oxidation state. In ionic compounds (salts), such an atom may occur as a separate cation (positive ion) denoted by Fe3+.
The adjective ferric or the prefix ferri- is often used to sp ...
ion Fe
3+ ion and
hematite
Hematite (), also spelled as haematite, is a common iron oxide compound with the formula, Fe2O3 and is widely found in rocks and soils. Hematite crystals belong to the rhombohedral lattice system which is designated the alpha polymorph of . ...
Fe
2O
3. The reaction equation is:
:
2 Fe^(aq) + 3 H_2 O (l) <=> Fe_2 O_3 (s) + 6 H^+ (aq)
which has
.
The pH of the vertical line on the Pourbaix diagram can then be calculated:
:
Because the activities (or the concentrations) of the solid phases and water are equal to unity:
2O3">e2O3=
2O">2O= 1, the pH only depends on the concentration in dissolved :
:
At STP, for
3+">e3+= 10
−6, this yields pH = 1.76.
Horizontal boundary line
When H
+ and OH
− ions are not involved in the reaction, the boundary line is horizontal and independent of pH.
The reaction equation is thus written:
:
As, 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 ...
:
:
Using the definition of the electrode potential ''∆G'' = ''-zFE'', where ''F'' is the
Faraday constant
In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of S ...
, this may be rewritten as a Nernst equation:
:
or, using base-10 logarithms:
:
For the equilibrium /, taken as example here, considering the boundary line between Fe
2+ and Fe
3+, the half-reaction equation is:
:
Fe^3+ (aq) + e^- <=> Fe^2+ (aq)
Since H
+ ions are not involved in this
redox
Redox (reduction–oxidation, , ) is a type of chemical reaction in which the oxidation states of substrate (chemistry), substrate change. Oxidation is the loss of Electron, electrons or an increase in the oxidation state, while reduction ...
reaction, it is independent of pH.
''E
o'' = 0.771 V with only one electron involved in the redox reaction.
The potential E
h is a function of temperature via the
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
and directly depends on the ratio of the concentrations of the and ions:
:
For both ionic species at the same concentration (e.g.,
) at STP, log 1 = 0, so,
, and the boundary will be a horizontal line at ''E
h'' = 0.771 volts. The potential will vary with temperature.
Sloped boundary line
In this case, both electrons and H
+ ions are involved and the electrode potential is a function of pH. The reaction equation may be written:
:
Using the expressions for the free energy in terms of potentials, the energy balance is given by a Nernst equation:
:
For the iron and water example, considering the boundary line between the
ferrous ion
In chemistry, the adjective Ferrous indicates a compound that contains iron(II), meaning iron in its +2 oxidation state, possibly as the divalent cation Fe2+. It is opposed to "ferric" or iron(III), meaning iron in its +3 oxidation state, such a ...
Fe
2+ and
hematite
Hematite (), also spelled as haematite, is a common iron oxide compound with the formula, Fe2O3 and is widely found in rocks and soils. Hematite crystals belong to the rhombohedral lattice system which is designated the alpha polymorph of . ...
Fe
2O
3, the reaction equation is:
:
Fe2O3(s) + 6 H+(aq) + 2 e^- <=> 2 Fe^(aq) + 3 H2O(l)
: with
.
The equation of the boundary line, expressed in base-10 logarithms is:
:
As, the activities, or the concentrations, of the solid phases and water are always taken equal to unity by convention in the definition of the
equilibrium constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
:
2O3">e2O3=
2O">2O= 1.
The Nernst equation thus limited to the dissolved species and is written as:
:
For,
2+">e2+= 10
−6 M, this yields:
:
Note the negative slope (-0.1775) of this line in a E
h–pH diagram.
The stability region of water
In many cases, the possible conditions in a system are limited by the stability region of water. In the Pourbaix diagram for uranium presented here above, the limits of stability of water are marked by the two dashed green lines, and the stability region for water falls between these two lines. It is also depicted here beside by the two dashed red lines in the simplified Pourbaix diagram restricted to the water stability region only.
Under highly reducing conditions (low ''E''
H), water is reduced to hydrogen according to:
:
2 H+ + 2e^- -> H2(g) (at low pH)
and,
:
2 H2O + 2e^- -> H2(g) + 2 OH^- (at high pH)
Using the Nernst equation, setting ''E''
0 = 0 V as defined by convention for the
standard hydrogen electrode
The standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be at 25 °C, but to form a basis ...
(SHE, serving as reference in the reduction potentials series) and the hydrogen gas
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 ...
(corresponding to
chemical activity for a gas) at 1, the equation for the lower stability line of water in the Pourbaix diagram at standard temperature and pressure is:
:
:
Below this line, water is reduced to hydrogen, and it will usually not be possible to pass beyond this line as long as there is still water present in the system to be reduced.
Correspondingly, under highly oxidizing conditions (high ''E''
H) water is oxidized into oxygen gas according to:
:
2 H2O -> 4 H+ + O2(g) + 4e^- (at low pH)
and,
:
4 OH^- -> O2(g) + 2 H_2O + 4e^- (at high pH)
Using the Nernst equation as above, but with E
0 = −ΔG
0H2O/2''F'' = 1.229 V for water oxidation, gives an upper stability limit of water as a function of the pH value:
:
:
at standard temperature and pressure. Above this line, water is oxidized to form oxygen gas, and it will usually not be possible to pass beyond this line as long as there is still water present in the system to be oxidized.
The two upper and lower stability lines having the same negative slope (−59 mV/pH unit), they are parallel in a Pourbaix diagram and the reduction potential decreases with pH.
Applications
Pourbaix diagrams have many applications in different fields dealing with ''e.g.'',
corrosion
Corrosion is a natural process that converts a refined metal into a more chemically stable oxide. It is the gradual deterioration of materials (usually a metal) by chemical or electrochemical reaction with their environment. Corrosion engine ...
problems,
geochemistry
Geochemistry is the science that uses the tools and principles of chemistry to explain the mechanisms behind major geological systems such as the Earth's crust and its oceans. The realm of geochemistry extends beyond the Earth, encompassing the e ...
, and
environmental science
Environmental science is an interdisciplinary academic field that integrates physics, biology, and geography (including ecology, chemistry, plant science, zoology, mineralogy, oceanography, limnology, soil science, geology and physical geograp ...
s. Using the Pourbaix diagram correctly will help shedding light not only on the nature of the species present in
aqueous solution
An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be re ...
, or in the
solid phases, but may also help to understand the
reaction mechanism.
Concept of in environmental chemistry
Pourbaix diagrams are widely used to describe the behaviour of chemical species in the
hydrosphere
The hydrosphere () is the combined mass of water found on, under, and above the surface of a planet, minor planet, or natural satellite. Although Earth's hydrosphere has been around for about 4 billion years, it continues to change in shape. This ...
. In this context,
reduction potential
Redox potential (also known as oxidation / reduction potential, ''ORP'', ''pe'', ''E_'', or E_) is a measure of the tendency of a chemical species to acquire electrons from or lose electrons to an electrode and thereby be reduced or oxidised respe ...
is often used instead of .
The main advantage is to directly work with a
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 ...
scale.
is a dimensionless number and can easily be related to by the equation:
:
Where,
is the
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, with , 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 ...
(), , 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 ...
(298.15 K = 25 °C = 77 °F), and , the
Faraday constant
In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of S ...
(96 485 coulomb/mol of ). Lambda, λ = ln(10) ≈ 2.3026.
Moreover,
: