Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water.
They were first described by
physical chemist
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical me ...
Kenneth Pitzer
Kenneth Sanborn Pitzer (January 6, 1914 – December 26, 1997) was an American physical and theoretical chemist, educator, and university president. He was described as "one of the most influential physical chemists of his era" whose work "sp ...
.
The parameters of the Pitzer equations are linear combinations of parameters, of a
virial expansion
The classical virial expansion expresses the pressure P of a many-particle system in equilibrium as a power series in the density:
Z \equiv \frac = A + B\rho + C\rho^2 + \cdots
where Z is called the compressibility factor. This is the virial e ...
of the excess
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 ...
, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the
osmotic coefficient
An osmotic coefficient \phi is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of ...
, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion
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 (or ...
s and water activities in solutions of high ionic strength for which the
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 ...
is no longer adequate. They are more rigorous than the equations of
specific ion interaction theory
In theoretical chemistry, Specific ion Interaction Theory (SIT theory) is a theory used to estimate single-ion activity coefficients in electrolyte solutions at relatively high concentrations. It does so by taking into consideration ''interaction ...
(SIT theory), but Pitzer parameters are more difficult to determine experimentally than SIT parameters.
Historical development
A starting point for the development can be taken as the
virial equation of state for a gas.
:
where
is the pressure,
is the volume,
is the temperature and
... are known as
virial coefficients
Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potenti ...
. The first term on the right-hand side is for an
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
. The remaining terms quantify the departure from the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
with changing pressure,
. It can be shown by
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
that the second virial coefficient arises from the intermolecular forces between ''pairs'' of molecules, the third virial coefficient involves interactions between three molecules, etc. This theory was developed by McMillan and Mayer.
Solutions of uncharged molecules can be treated by a modification of the McMillan-Mayer theory. However, when a solution contains
electrolyte
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon dis ...
s,
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
interactions must also be taken into account. The
Debye-Hückel theory was based on the assumption that each ion was surrounded by a spherical "cloud" or
ionic atmosphere Ionic Atmosphere is a concept employed in Debye-Hückel theory which explains the electrolytic conductivity behaviour of solutions. It can be generally defined as the area at which a charged entity is capable of attracting an entity of the opposit ...
made up of ions of the opposite charge. Expressions were derived for the variation of single-ion
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 (or ...
s as a function of
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 ...
. This theory was very successful for dilute solutions of 1:1 electrolytes and, as discussed below, the Debye-Hückel expressions are still valid at sufficiently low concentrations. The values calculated with Debye-Hückel theory diverge more and more from observed values as the concentrations and/or ionic charges increases. Moreover, Debye-Hückel theory takes no account of the specific properties of ions such as size or shape.
Brønsted had independently proposed an empirical equation,
:
:
in which the activity coefficient depended not only on ionic strength, but also on the concentration, ''m'', of the specific ion through the parameter ''β''. This is the basis of
SIT theory. It was further developed by Guggenheim.
Scatchard extended the theory to allow the interaction coefficients to vary with ionic strength. Note that the second form of Brønsted's equation is an expression for the
osmotic coefficient
An osmotic coefficient \phi is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of ...
. Measurement of osmotic coefficients provides one means for determining mean activity coefficients.
The Pitzer parameters
The exposition begins with a virial expansion of the excess
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 ...
:
''W
w'' is the mass of the water in kilograms,'' b
i, b
j'' ... are 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 ...
ities of the ions and ''I'' is the ionic strength. The first term, ''f(I)'' represents the Debye-Hückel limiting law. The quantities ''λ
ij(I)'' represent the short-range interactions in the presence of solvent between solute particles ''i'' and ''j''. This binary interaction parameter or second virial coefficient depends on ionic strength, on the particular species ''i'' and ''j'' and the temperature and pressure. The quantities ''μ''
''ijk'' represent the interactions between three particles. Higher terms may also be included in the virial expansion.
Next, the free energy is expressed as the sum of
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 ...
s, or partial molal free energy,
:
and an expression for the activity coefficient is obtained by differentiating the virial expansion with respect to a molality b.
: