The Debye–Hückel theory was proposed by
Peter Debye
Peter Joseph William Debye ( ; born Petrus Josephus Wilhelmus Debije, ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry.
Biography
Early life
Born in Maastricht, Neth ...
and
Erich Hückel as a theoretical explanation for departures from ideality in solutions of
electrolyte
An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
s and
plasmas.
It is a linearized
Poisson–Boltzmann model, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean
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 for ions in dilute solution. The Debye–Hückel equation provides a starting point for modern treatments of non-ideality of electrolyte solutions.
Overview
In the
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
of
electrolyte
An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
solutions, an
ideal solution
An ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing. The vapor pressures of all components obey R ...
is a solution whose
colligative properties
In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present. The numb ...
are proportional to the
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
of the
solute
In chemistry, a solution is defined by IUPAC as "A liquid or solid phase containing more than one substance, when for convenience one (or more) substance, which is called the solvent, is treated differently from the other substances, which are ...
. Real solutions may show departures from this kind of ideality. In order to accommodate these effects in the
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of solutions, the concept of
activity was introduced: the properties are then proportional to the activities of the ions. Activity ''a'' is proportional to concentration ''c'', with the proportionality constant known as an
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 ( ...
:
In an ideal electrolyte solution the activity coefficients for all the ions are equal to one. Ideality of an electrolyte solution can be achieved only in very dilute solutions. Non-ideality of more concentrated solutions arises principally (but not exclusively) because ions of opposite charge attract each other due to
electrostatic
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
forces, while ions of the same charge repel each other. In consequence, ions are not randomly distributed throughout the solution, as they would be in an ideal solution.
Activity coefficients of single ions cannot be measured experimentally because an electrolyte solution must contain both positively charged ions and negatively charged ions. Instead, a mean activity coefficient
is defined. For example, with the electrolyte NaCl,
In general, the mean activity coefficient of a fully dissociated electrolyte of formula ''A
nB
m'' is given by
Activity coefficients are themselves functions of concentration, since the amount of inter-ionic interaction increases as the concentration of the electrolyte increases. Debye and Hückel developed a theory with which single-ion activity coefficients could be calculated. By calculating the mean activity coefficients from them, the theory could be tested against experimental data. It was found to give excellent agreement for "dilute" solutions.
The model
An idealized representation of a solution of a 1:1 electrolyte
A description of Debye–Hückel theory includes a very detailed discussion of the assumptions and their limitations as well as the mathematical development and applications.
A snapshot of a 2-dimensional section of an idealized electrolyte solution is shown in the picture. The ions are shown as spheres with unit electrical charge. The solvent (pale blue) is shown as a uniform medium, without structure. On average, each ion is surrounded more closely by ions of opposite charge than by ions of like charge. These concepts were developed into a quantitative theory involving ions of charge ''z''
1''e''
+ and ''z''
2''e''
−, where ''z'' can be any integer. The principal assumption is that departure from ideality is due to electrostatic interactions between ions, mediated by
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
: the force of interaction between two electric charges, separated by a distance, ''r'' in a medium of
relative permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the vacuum permittivity, electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric co ...
is given by
It is also assumed that
*The
solute
In chemistry, a solution is defined by IUPAC as "A liquid or solid phase containing more than one substance, when for convenience one (or more) substance, which is called the solvent, is treated differently from the other substances, which are ...
is completely dissociated; it is a
strong electrolyte.
*Ions are spherical and are not
polarized by the surrounding
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
.
Solvation
Solvations describes the interaction of a solvent with dissolved molecules. Both ionized and uncharged molecules interact strongly with a solvent, and the strength and nature of this interaction influence many properties of the solute, includi ...
of ions is ignored except insofar as it determines the effective sizes of the ions.
*The solvent plays no role other than providing a medium of constant relative permittivity (
dielectric constant
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
).
*There is no
electrostriction
In electromagnetism, electrostriction is a property of all electrical non- conductor or dielectrics. Electrostriction causes these materials to change their shape under the application of an electric field. It is the dual property to magnetostri ...
.
*Individual ions surrounding a "central" ion can be represented by a statistically averaged cloud of continuous charge density, with a minimum distance of closest approach.
The last assumption means that each cation is surrounded by a spherically symmetric cloud of other ions. The cloud has a net negative charge. Similarly each anion is surrounded by a cloud with net positive charge.
Mathematical development
The deviation from ideality is taken to be a function of the potential energy resulting from the electrostatic interactions between ions and their surrounding clouds. To calculate this energy two steps are needed.
The first step is to specify the electrostatic potential for ion ''j'' by means of
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
is the total potential at a distance, , from the central ion and is the averaged charge density of the surrounding cloud at that distance. To apply this formula it is essential that the cloud has spherical symmetry, that is, the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
is a function only of distance from the central ion as this allows the Poisson equation to be cast in terms of
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
with no angular dependence.
The second step is to calculate the charge density by means of a
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...
.
where is
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
and is the temperature. This distribution also depends on the potential and this introduces a serious difficulty in terms of the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
. Nevertheless, the two equations can be combined to produce the
Poisson–Boltzmann equation
The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules ...
.
Solution of this equation is far from straightforward. Debye and Hückel expanded the exponential as a truncated
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
to first order. The zeroth order term vanishes because the solution is on average
electrically neutral
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other an ...
(so that
), which leaves us with only the first order term. The result has the form of the
Helmholtz equation
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
\nabla^2 f = -k^2 f,
where is the Laplace operator, is the eigenvalue, and is the (eigen)fun ...
which has an
analytical solution
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. C ...
. This equation applies to electrolytes with equal numbers of ions of each charge. Nonsymmetrical electrolytes require another term with . For symmetrical electrolytes, this reduces to the modified spherical Bessel equation
with solutions
The coefficients
and
are fixed by the boundary conditions. As
,
must not diverge, so
. At
, which is the distance of the closest approach of ions, the force exerted by the charge should be balanced by the force of other ions, imposing
, from which
is found, yielding
The
electrostatic potential energy
Electric potential energy is a potential energy (measured in joules) that results from conservative force, conservative Coulomb forces and is associated with the configuration of a particular set of point electric charge, charges within a defi ...
,
, of the ion at
is
This is the potential energy of a single ion in a solution. The multiple-charge generalization from electrostatics gives an expression for the potential energy of the entire solution.

The mean activity coefficient is given by the logarithm of this quantity as follows
where ''I'' is the
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 a ...
and ''a''
0 is a parameter that represents the distance of closest approach of ions. For aqueous solutions at 25 °C ''A'' = 0.51 mol
−1/2dm
3/2 and ''B'' = 3.29 nm
−1mol
−1/2dm
3/2
is a constant that depends on temperature. If
is expressed in terms of molality, instead of molarity (as in the equation above and in the rest of this article), then an experimental value for
''of water'' is
at 25 °C. It is common to use a base-10 logarithm, in which case we factor , so ''A'' is
. The multiplier before
in the equation is for the case when the dimensions of
are
. When the dimensions of
are
, the multiplier must be dropped from the equation

The most significant aspect of this result is the prediction that the mean activity coefficient is a function of ''ionic strength'' rather than the electrolyte concentration. For very low values of the ionic strength the value of the denominator in the expression above becomes nearly equal to one. In this situation the mean activity coefficient is proportional to the square root of the ionic strength. This is known as the Debye–Hückel limiting law. In this limit the equation is given as follows
The excess
osmotic pressure
Osmotic pressure is the minimum pressure which needs to be applied to a Solution (chemistry), solution to prevent the inward flow of its pure solvent across a semipermeable membrane.
It is also defined as the measure of the tendency of a soluti ...
obtained from Debye–Hückel theory is in
cgs units:
Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure
. 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
A ...
is then given by
Nondimensionalization
Taking the differential equation from
earlier (as stated above, the equation only holds for low concentrations):
Using the
Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically ...
on this problem results in the following dimensionless groups:
is called the reduced scalar electric potential field.
is called the reduced radius. The existing groups may be recombined to form two other dimensionless groups for substitution into the differential equation. The first is what could be called the square of the reduced inverse
screening length,
. The second could be called the reduced central ion charge,
(with a capital Z). Note that, though
is already dimensionless, without the substitution given below, the differential equation would still be dimensional.
To obtain the nondimensionalized differential equation and initial conditions, use the
groups to eliminate
in favor of
, then eliminate
in favor of
while carrying out the chain rule and substituting
, then eliminate
in favor of
(no chain rule needed), then eliminate
in favor of
, then eliminate
in favor of
. The resulting equations are as follows:
For table salt in 0.01 M solution at 25 °C, a typical value of
is 0.0005636, while a typical value of
is 7.017, highlighting the fact that, in low concentrations,
is a target for a zero order of magnitude approximation such as perturbation analysis. Unfortunately, because of the boundary condition at infinity, regular perturbation does not work. The same boundary condition prevents us from finding the exact solution to the equations. Singular perturbation may work, however.
Limitations and extensions
This equation for
gives satisfactory agreement with experimental measurements for low electrolyte concentrations, typically less than 10
−3 mol/L. Deviations from the theory occur at higher concentrations and with electrolytes that produce ions of higher charges, particularly unsymmetrical electrolytes. Essentially these deviations occur because the model is
oversimplified, so there is little to be gained making small adjustments to the model. The individual assumptions can be challenged in turn.
*Complete dissociation.
Ion association
In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each ...
may take place, particularly with ions of higher charge. This was followed up in detail by
Niels Bjerrum. The
Bjerrum length is the separation at which the electrostatic interaction between two ions is comparable in magnitude to .
*Weak electrolytes. A weak electrolyte is one that is not fully dissociated. As such it has a
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 ...
. The dissociation constant can be used to calculate the extent of dissociation and hence, make the necessary correction needed to calculate activity coefficients.
*Ions are spherical, not
point charge
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
s and are not
polarized. Many ions such as the
nitrate
Nitrate is a polyatomic ion with the chemical formula . salt (chemistry), Salts containing this ion are called nitrates. Nitrates are common components of fertilizers and explosives. Almost all inorganic nitrates are solubility, soluble in wa ...
ion, NO
3−, are not spherical. Polyatomic ions are also polarizable.
*Role of the solvent. The solvent is not a structureless medium but is made up of molecules. The water molecules in aqueous solution are both
dipolar
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
* An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
and polarizable. Both cations and anions have a strong primary
solvation shell
A solvation shell or solvation sheath is the solvent interface of any chemical compound or biomolecule that constitutes the solute in a solution. When the solvent is water it is called a hydration shell or hydration sphere. The number of solvent ...
and a weaker secondary solvation shell.
Ion–solvent interactions are ignored in Debye–Hückel theory.
Moreover, ionic radius is assumed to be negligible, but at higher concentrations, the
ionic radius
Ionic radius, ''r''ion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cati ...
becomes comparable to the radius of the
ionic atmosphere.
Most extensions to Debye–Hückel theory are empirical in nature. They usually allow the Debye–Hückel equation to be followed at low concentration and add further terms in some power of the ionic strength to fit experimental observations. The main extensions are the
Davies equation
The Davies equation is an empirical extension of Debye–Hückel theory which can be used to calculate activity coefficients of electrolyte solutions at relatively high concentrations at 25 °C. The equation, originally published in 1938, wa ...
,
Pitzer equation
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 Kenneth Pitzer. The parameters of the Pitzer equations ar ...
s and
specific ion interaction theory.
One such extended Debye–Hückel equation is given by:
where
as its
common logarithm
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. It is also known as the decadic logarithm, the decimal logarithm and the Briggsian logarithm. The name "Briggsian logarithm" is in honor of the British ...
is the activity coefficient,
is the integer charge of the ion (1 for H
+, 2 for Mg
2+ etc.),
is the ionic strength of the aqueous solution, and
is the size or effective diameter of the ion in
angstrom
The angstrom (; ) is a unit of length equal to m; that is, one ten-billionth of a metre, a hundred-millionth of a centimetre, 0.1 nanometre, or 100 picometres. The unit is named after the Swedish physicist Anders Jonas Ångström (1814–18 ...
. The effective hydrated radius of the ion, a is the radius of the ion and its closely bound water molecules. Large ions and less highly charged ions bind water less tightly and have smaller hydrated radii than smaller, more highly charged ions. Typical values are 3Å for ions such as H
+, Cl
−, CN
−, and HCOO
−. The effective diameter for the
hydronium ion
In chemistry, hydronium (hydroxonium in traditional British English) is the cation , also written as , the type of oxonium ion produced by protonation of water. It is often viewed as the positive ion present when an Arrhenius acid is dissolved in ...
is 9Å.
and
are constants with values of respectively 0.5085 and 0.3281 at 25 °C in water .
The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. For solutions of greater ionic strengths, the
Pitzer equations
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 Kenneth Pitzer. The parameters of the Pitzer equations ar ...
should be used. In these solutions the activity coefficient may actually increase with ionic strength.

The Debye–Hückel equation cannot be used in the solutions of
surfactant
Surfactants are chemical compounds that decrease the surface tension or interfacial tension between two liquids, a liquid and a gas, or a liquid and a solid. The word ''surfactant'' is a Blend word, blend of "surface-active agent",
coined in ...
s where the presence of
micelle
A micelle () or micella () ( or micellae, respectively) is an aggregate (or supramolecular assembly) of surfactant amphipathic lipid molecules dispersed in a liquid, forming a colloidal suspension (also known as associated colloidal system). ...
s influences on the electrochemical properties of the system (even rough judgement overestimates for ~50%).
Electrolytes mixtures
The theory can be applied also to dilute solutions of mixed electrolytes.
Freezing point depression measurements has been used to this purpose.
Conductivity
The treatment given so far is for a system not subject to an external electric field. When
conductivity is measured the system is subject to an oscillating external field due to the application of an
AC voltage to electrodes immersed in the solution. Debye and Hückel modified their theory in 1926 and their theory was further modified by
Lars Onsager
Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian American physical chemist and theoretical physicist. He held the Gibbs Professorship of Theoretical Chemistry at Yale University. He was awarded the Nobel Prize in Chemist ...
in 1927. All the postulates of the original theory were retained. In addition it was assumed that the electric field causes the charge cloud to be distorted away from spherical symmetry. After taking this into account, together with the specific requirements of moving ions, such as
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
and
electrophoretic effects, Onsager was able to derive a theoretical expression to account for the empirical relation known as
Kohlrausch's Law, for the molar conductivity, Λ
m.
is known as the limiting molar conductivity, ''K'' is an empirical constant and ''c'' is the electrolyte concentration. Limiting here means "at the limit of the infinite dilution").
Onsager's expression is
where ''A'' and ''B'' are constants that depend only on known quantities such as temperature, the charges on the ions and the dielectric constant and viscosity of the solvent. This is known as the Debye–Hückel–Onsager equation. However, this equation only applies to very dilute solutions and has been largely superseded by other equations due to Fuoss and Onsager, 1932 and 1957 and later.
Summary of Debye and Hückel's first article on the theory of dilute electrolytes
The English title of the article is "On the Theory of Electrolytes. I. Freezing Point Depression and Related Phenomena". It was originally published in 1923 in volume 24 of a German-language journal . An English translation
of the article is included in a book of collected papers presented to Debye by "his pupils, friends, and the publishers on the occasion of his seventieth birthday on March 24, 1954".
Another English translation was completed in 2019. The article deals with the calculation of properties of electrolyte solutions that are under the influence of ion-induced electric fields, thus it deals with
electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
.
In the same year they first published this article, Debye and Hückel, hereinafter D&H, also released an article that covered their initial characterization of solutions under the influence of electric fields called "On the Theory of Electrolytes. II. Limiting Law for Electric Conductivity", but that subsequent article is not (yet) covered here.
In the following summary (as yet incomplete and unchecked), modern notation and terminology are used, from both chemistry and mathematics, in order to prevent confusion. Also, with a few exceptions to improve clarity, the subsections in this summary are (very) condensed versions of the same subsections of the original article.
Introduction
D&H note that the Guldberg–Waage formula for electrolyte species in
chemical reaction equilibrium in classical form is
where
*
is a notation for
multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
,
*
is a
dummy variable indicating the species,
*
is the number of species participating in the reaction,
*
is the
mole fraction
In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
of species
,
*
is 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 are on the right-hand side with a plus sign be ...
of species
,
* ''K'' 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 ...
.
D&H say that, due to the "mutual electrostatic forces between the ions", it is necessary to modify the Guldberg–Waage equation by replacing
with
, where
is an overall activity coefficient, not a "special"
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 ( ...
(a separate activity coefficient associated with each species)—which is what is used in modern chemistry .
The relationship between
and the special activity coefficients
is
Fundamentals
D&H use the
Helmholtz and Gibbs free entropies and
to express the effect of electrostatic forces in an electrolyte on its thermodynamic state. Specifically, they split most of the
thermodynamic potential
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
s into classical and electrostatic terms:
where
*
is
Helmholtz free entropy,
*
is
entropy
Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
,
*
is
internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
,
*
is
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
,
*
is
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature ( isothermal). The change in the Helmholtz ene ...
.
D&H give the
total differential
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
dy = f'(x)\,dx,
where f'(x) is the derivative of with resp ...
of
as
where
*
is
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 eve ...
,
*
is
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
.
By the definition of the total differential, this means that
which are useful further on.
As stated previously, the internal energy is divided into two parts:
where
*
indicates the classical part,
*
indicates the electric part.
Similarly, the Helmholtz free entropy is also divided into two parts:
D&H state, without giving the logic, that
It would seem that, without some justification,
Without mentioning it specifically, D&H later give what might be the required (above) justification while arguing that
, an assumption that the solvent is incompressible.
The definition of the
Gibbs free entropy is
where
is
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
.
D&H give the total differential of
as
At this point D&H note that, for water containing 1 mole per liter of
potassium chloride
Potassium chloride (KCl, or potassium salt) is a metal halide salt composed of potassium and chlorine. It is odorless and has a white or colorless vitreous crystal appearance. The solid dissolves readily in water, and its solutions have a sa ...
(nominal pressure and temperature aren't given), the electric pressure
amounts to 20 atmospheres. Furthermore, they note that this level of pressure gives a relative volume change of 0.001. Therefore, they neglect change in volume of water due to electric pressure, writing
and put
D&H say that, according to Planck, the classical part of the Gibbs free entropy is
where
*
is a species,
*
is the number of different particle types in solution,
*
is the number of particles of species ''i'',
*
is the particle specific Gibbs free entropy of species ''i'',
*
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
,
*
is the mole fraction of species ''i''.
Species zero is the solvent. The definition of
is as follows, where lower-case letters indicate the particle specific versions of the corresponding extensive properties:
D&H don't say so, but the functional form for
may be derived from the functional dependence of the chemical potential of a component of an
ideal mixture
An ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing. The vapor pressures of all components obey Ra ...
upon its mole fraction.
D&H note that the internal energy
of a solution is lowered by the electrical interaction of its ions, but that this effect can't be determined by using the crystallographic approximation for distances between dissimilar atoms (the cube root of the ratio of total volume to the number of particles in the volume). This is because there is more thermal motion in a liquid solution than in a crystal. The thermal motion tends to smear out the natural lattice that would otherwise be constructed by the ions. Instead, D&H introduce the concept of an
ionic atmosphere or cloud. Like the crystal lattice, each ion still attempts to surround itself with oppositely charged ions, but in a more free-form manner; at small distances away from positive ions, one is more likely to find negative ions and vice versa.
The potential energy of an arbitrary ion solution
Electroneutrality of a solution requires that
where
*
is the total number of ions of species ''i'' in the solution,
*
is the
charge number of species ''i''.
To bring an ion of species ''i'', initially far away, to a point
within the ion cloud requires
interaction energy
In physics, interaction energy is the contribution to the total energy that is caused by an interaction between the objects being considered.
The interaction energy usually depends on the relative position of the objects. For example, Q_1 Q_2 / ...
in the amount of
, where
is the
elementary charge
The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
, and
is the value of the scalar
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
field at
. If electric forces were the only factor in play, the minimal-energy configuration of all the ions would be achieved in a close-packed lattice configuration. However, the ions are in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...
with each other and are relatively free to move. Thus they obey
Boltzmann statistics and form a
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...
. All species'
number densities are altered from their bulk (overall average) values
by the corresponding
Boltzmann factor
Factor (Latin, ) may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, such a factor is a resource used ...
, where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, and
is the temperature. Thus at every point in the cloud
Note that in the infinite temperature limit, all ions are distributed uniformly, with no regard for their electrostatic interactions.
The
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
is related to the number density:
When combining this result for the charge density with the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
from electrostatics, a form of the
Poisson–Boltzmann equation
The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules ...
results:
This equation is difficult to solve and does not follow the principle of
linear superposition
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
for the relationship between the number of charges and the strength of the potential field. It has been solved analyticallt by the Swedish mathematician
Thomas Hakon Gronwall and his collaborators physical chemists V. K. La Mer and Karl Sandved in a 1928 article from
Physikalische Zeitschrift
''Physikalische Zeitschrift'' (English: ''Physical Journal'') was a German scientific journal of physics published from 1899 to 1945 by S. Hirzel Verlag. In 1924, it merged with ''Jahrbuch der Radioaktivität und Elektronik''. From 1944 onwards, t ...
dealing with extensions to Debye–Huckel theory.
However, for sufficiently low concentrations of ions, a first-order
Taylor series expansion approximation for the exponential function may be used (
for
) to create a linear differential equation.
D&H say that this approximation holds at large distances between ions,
which is the same as saying that the concentration is low. Lastly, they claim without proof that the addition of more terms in the expansion has little effect on the final solution.
Thus
The Poisson–Boltzmann equation is transformed to
because the first summation is zero due to electroneutrality.
Factor out the scalar potential and assign the leftovers, which are constant, to
. Also, let
be the
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 a ...
of the solution:
So, the fundamental equation is reduced to a form of the
Helmholtz equation
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
\nabla^2 f = -k^2 f,
where is the Laplace operator, is the eigenvalue, and is the (eigen)fun ...
:
Today,
is called the
Debye screening length. D&H recognize the importance of the parameter in their article and characterize it as a measure of the thickness of the ion atmosphere, which is an
electrical double layer
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
of the Gouy–Chapman type.
The equation may be expressed in
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
by taking
at some arbitrary ion:
The equation has the following general solution (keep in mind that
is a positive constant):
where
,
, and
are undetermined constants
The electric potential is zero at infinity by definition, so
must be zero.
In the next step, D&H assume that there is a certain radius
, beyond which no ions in the atmosphere may approach the (charge) center of the singled out ion. This radius may be due to the physical size of the ion itself, the sizes of the ions in the cloud, and any water molecules that surround the ions. Mathematically, they treat the singled out ion as a
point charge
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
to which one may not approach within the radius
.
The potential of a point charge by itself is
D&H say that the total potential inside the sphere is
where
is a constant that represents the potential added by the ionic atmosphere. No justification for
being a constant is given. However, one can see that this is the case by considering that any spherical static charge distribution is subject to the mathematics of the
shell theorem
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion a ...
. The shell theorem says that no force is exerted on charged particles inside a sphere (of arbitrary charge).
Since the ion atmosphere is assumed to be (time-averaged) spherically symmetric, with charge varying as a function of radius
, it may be represented as an infinite series of concentric charge shells. Therefore, inside the radius
, the ion atmosphere exerts no force. If the force is zero, then the potential is a constant (by definition).
In a combination of the continuously distributed model which gave the Poisson–Boltzmann equation and the model of the point charge, it is assumed that at the radius
, there is a continuity of
and its first derivative. Thus
By the definition of
electric potential energy
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An ''object'' may be sa ...
, the potential energy associated with the singled out ion in the ion atmosphere is
Notice that this only requires knowledge of the charge of the singled out ion and the potential of all the other ions.
To calculate the potential energy of the entire electrolyte solution, one must use the multiple-charge generalization for electric potential energy:
The additional electric term to the thermodynamic potential
Experimental verification of the theory
To verify the validity of the Debye–Hückel theory, many experimental ways have been tried, measuring the activity coefficients: the problem is that we need to go towards very high dilutions.
Typical examples are: measurements of vapour pressure, freezing point, osmotic pressure (indirect methods) and measurement of electric potential in cells (direct method).
Going towards high dilutions good results have been found using liquid membrane cells, it has been possible to investigate aqueous media 10
−4 M and it has been found that for 1:1 electrolytes (as NaCl or KCl) the Debye–Hückel equation is totally correct, but for 2:2 or 3:2 electrolytes it is possible to find negative deviation from the Debye–Hückel limit law: this strange behavior can be observed only in the very dilute area, and in more concentrate regions the deviation becomes positive.
It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn't possible to investigate the 10
−4 M region, so it is possible that during the next years new theories will be born.
See also
*
Electrolyte
An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
*
Chemical activity
In 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 depend on conc ...
*
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 a ...
*
Poisson-Boltzmann equation
*
Debye length
In plasmas and electrolytes, the Debye length \lambda_\text (Debye radius or Debye–Hückel screening length), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each D ...
*
Bjerrum length
*
Bates-Guggenheim Convention
*
Ionic atmosphere
*
Electrical double layer
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
*
Ion association
In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each ...
*
Davies equation
The Davies equation is an empirical extension of Debye–Hückel theory which can be used to calculate activity coefficients of electrolyte solutions at relatively high concentrations at 25 °C. The equation, originally published in 1938, wa ...
*
Pitzer equation
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 Kenneth Pitzer. The parameters of the Pitzer equations ar ...
*
Specific ion Interaction Theory
References
{{DEFAULTSORT:Debye-Huckel Theory
Thermodynamic models
Electrochemistry
Equilibrium chemistry
Peter Debye