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Peter Debye Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry. Biography Early life Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, D ...
and
Erich Hückel Erich Armand Arthur Joseph Hückel (August 9, 1896, Berlin – February 16, 1980, Marburg) was a German physicist and physical chemist. He is known for two major contributions: *The Debye–Hückel theory of electrolytic solutions *The Hückel m ...
noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of 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 of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient \gamma. This factor takes into account the
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 / ( ...
of ions in solution.


Debye–Hückel limiting law

In order to calculate the activity a_C of an
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
C in a solution, one must know 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'', an ...
and the activity coefficient: a_C = \gamma \frac\mathrm\mathrm, where * \gamma is the activity coefficient of C, * \mathrm is the concentration of the chosen ''standard state'', e.g. 1 mol/kg if
molality 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 ...
is used, * \mathrm is a measure of the concentration of C. Dividing \mathrm with \mathrm gives a dimensionless quantity. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a ''dilute'' solution of known ionic strength. The equation is \ln(\gamma_i) = -\frac = -\frac \sqrt = -A z_i^2 \sqrt, where * z_i is the
charge number Charge number (''z'') refers to a quantized value of electric charge, with the quantum of electric charge being the elementary charge, so that the charge number equals the electric charge (''q'') in coulombs divided by the elementary-charge con ...
of ion species ''i'', * q is the elementary charge, * \kappa is the inverse of the
Debye screening length A double layer (DL, also called an electrical double layer, EDL) is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The D ...
(defined below), * \varepsilon_r is the
relative permittivity 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 ...
of the solvent, * \varepsilon_0 is the permittivity of free space, * k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, * T is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of the solution, * N_\mathrm is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
, * I is the ionic strength of the solution (defined below), * A is a constant that depends on temperature. If I 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 A ''of water'' is 1.172\text^\text^ at 25 °C. It is common to use a base-10 logarithm, in which case we factor \ln 10, so ''A'' is 0.509\text^\text^. The multiplier 10^3 before I/2 in the equation is for the case when the dimensions of I are \text/\text^3. When the dimensions of I are \text/\text^3, the multiplier 10^3 must be dropped from the equation. It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient. The excess osmotic pressure obtained from Debye–Hückel theory is in cgs units: P^\text = -\frac = -\frac. Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure P^\text = k_\text T \sum_i c_i. 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 ...
is then given by \phi = \frac = 1 + \frac.


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. 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 \prod_^s x_i^ = K, where * \prod is a notation for multiplication, * i is a dummy variable indicating the species, * s is the number of species participating in the reaction, * x_i is the
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
of species i, * \nu_i 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 on the right-hand side with a plus sign between ...
of species i, * ''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 K with \gamma K, where \gamma 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 \gamma and the special activity coefficients \gamma_i is \log(\gamma) = \sum_^s \nu_i \log(\gamma_i).


Fundamentals

D&H use the Helmholtz and Gibbs free entropies \Phi and \Xi to express the effect of electrostatic forces in an electrolyte on its thermodynamic state. Specifically, they split most of the thermodynamic potentials into classical and electrostatic terms: \Phi = S - \frac = -\frac, where * \Phi is Helmholtz free entropy, * S is
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, * U is
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, * T is
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
, * A 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 In thermodynamics, an isotherma ...
. 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 ...
of \Phi as d \Phi = \frac \,dV + \frac \,dT, where * P 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 e ...
, * V is
volume Volume is a measure of occupied 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). Th ...
. By the definition of the total differential, this means that \frac = \frac, \frac = \frac, which are useful further on. As stated previously, the internal energy is divided into two parts: U = U_k + U_e where * k indicates the classical part, * e indicates the electric part. Similarly, the Helmholtz free entropy is also divided into two parts: \Phi = \Phi_k + \Phi_e. D&H state, without giving the logic, that \Phi_e = \int \frac \,dT. It would seem that, without some justification,\Phi_e = \int \frac \,dV + \int \frac \,dT. Without mentioning it specifically, D&H later give what might be the required (above) justification while arguing that \Phi_e = \Xi_e, an assumption that the solvent is incompressible. The definition of the Gibbs free entropy \Xi is \Xi = S - \frac = \Phi - \frac = -\frac, where G is
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 ...
. D&H give the total differential of \Xi as d\Xi = -\frac \,dP + \frac \,dT. At this point D&H note that, for water containing 1 mole per liter of potassium chloride (nominal pressure and temperature aren't given), the electric pressure P_e 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 \Xi = \Xi_k + \Xi_e, and put \Xi_e = \Phi_e = \int \frac \,dT. D&H say that, according to Planck, the classical part of the Gibbs free entropy is \Xi_k = \sum_^s N_i (\xi_i - k_\text ln(x_i)), where * i is a species, * s is the number of different particle types in solution, * N_i is the number of particles of species ''i'', * \xi_i is the particle specific Gibbs free entropy of species ''i'', * k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, * x_i is the mole fraction of species ''i''. Species zero is the solvent. The definition of \xi_i is as follows, where lower-case letters indicate the particle specific versions of the corresponding extensive properties: \xi_i = s_i - \frac. D&H don't say so, but the functional form for \Xi_k may be derived from the functional dependence of the chemical potential of a component of an
ideal mixture In chemistry, 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 by definition; the closer to zer ...
upon its mole fraction. D&H note that the internal energy U 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 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 ...
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 \sum_^s N_i z_i = 0, where * N_i is the total number of ions of species ''i'' in the solution, * z_i is the
charge number Charge number (''z'') refers to a quantized value of electric charge, with the quantum of electric charge being the elementary charge, so that the charge number equals the electric charge (''q'') in coulombs divided by the elementary-charge con ...
of species ''i''. To bring an ion of species ''i'', initially far away, to a point P 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 z_i q \varphi, where q is the elementary charge, and \varphi is the value of the scalar
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
field at P. 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 ...
with each other and are relatively free to move. Thus they obey
Boltzmann statistics Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermod ...
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 th ...
. All species' number densities n_i are altered from their bulk (overall average) values n^0_i by the corresponding
Boltzmann factor Factor, a Latin word meaning "who/which acts", 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, suc ...
e^, where k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and T is the temperature. Thus at every point in the cloud n_i = \frac e^ = n^0_i e^. 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 ...
is related to the number density: \rho = \sum_i z_i q n_i = \sum_i z_i q n^0_i e^. 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 t ...
from electrostatics, a form of the
Poisson–Boltzmann equation The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric ...
results: \nabla^2 \varphi = -\frac = -\sum_i \frac e^. 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 tha ...
for the relationship between the number of charges and the strength of the potential field. It has been solved by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physicical 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, ...
dealing with extensions to Debye–Huckel theory, which resorted to Taylor series expansion. However, for sufficiently low concentrations of ions, a first-order
Taylor series expansion 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 serie ...
approximation for the exponential function may be used (e^x \approx 1 + x for 0 < x \ll 1) to create a linear differential equation (Hamann, Hamnett, and Vielstich. Electrochemistry. Wiley-VCH. section 2.4.2). 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 -\sum_i \frac e^ \approx -\sum_i \frac \left(1 - \frac\right) = -\left(\sum_i \frac - \sum_i \frac\right). The Poisson–Boltzmann equation is transformed to \nabla^2 \varphi = \sum_i \frac, because the first summation is zero due to electroneutrality. Factor out the scalar potential and assign the leftovers, which are constant, to \kappa^2. Also, let I be the ionic strength of the solution: \kappa^2 = \sum_i \frac = \frac, I = \frac \sum_i z_i^2 n^0_i. So, the fundamental equation is reduced to a form of the Helmholtz equation: \nabla^2 \varphi = \kappa^2 \varphi. Today, \kappa^ is called the
Debye screening length A double layer (DL, also called an electrical double layer, EDL) is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The D ...
. 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 of the Gouy–Chapman type. The equation may be expressed in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
by taking r = 0 at some arbitrary ion: \nabla^2 \varphi = \frac \frac \left( r^2 \frac \right) = \frac + \frac \frac = \kappa^2 \varphi(r). The equation has the following general solution (keep in mind that \kappa is a positive constant): \varphi(r) = A \frac + A' \frac = A \frac + A'' \frac = A \frac, where A, A', and A'' are undetermined constants The electric potential is zero at infinity by definition, so A'' must be zero. In the next step, D&H assume that there is a certain radius a_i, 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 a_i. The potential of a point charge by itself is \varphi_\text(r) = \frac \frac. D&H say that the total potential inside the sphere is \varphi_\text(r) = \varphi_\text(r) + B_i = \frac \frac + B_i, where B_i is a constant that represents the potential added by the ionic atmosphere. No justification for B_i 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. 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 r, it may be represented as an infinite series of concentric charge shells. Therefore, inside the radius a_i, 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 a_i, there is a continuity of \varphi(r) and its first derivative. Thus \varphi(a_i) = A_i \frac = \frac \frac + B_i = \varphi_\text(a_i), \varphi'(a_i) = -\frac = -\frac \frac = \varphi_\text'(a_i), A_i = \frac \frac, B_i = -\frac \frac . By the definition of electric potential energy, the potential energy associated with the singled out ion in the ion atmosphere is u_i = z_i q B_i = -\frac \frac . 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: U_e = \frac \sum_^s N_i u_i = -\sum_^s \frac \frac \frac .


The additional electric term to the thermodynamic potential


Nondimensionalization

The differential equation is ready for solution (as stated above, the equation only holds for low concentrations): \frac + \frac \frac = \frac = \kappa^2 \varphi(r). Using the
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
on this problem results in the following dimensionless groups: \begin \pi_1 &= \frac = \Phi(R(r)) \\ \pi_2 &= \varepsilon_r \\ \pi_3 &= \frac \\ \pi_4 &= a^3 I \\ \pi_5 &= z_0 \\ \pi_6 &= \frac = R(r). \end \Phi is called the reduced scalar electric potential field. R 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, (\kappa a)^2. The second could be called the reduced central ion charge, Z_0 (with a capital Z). Note that, though z_0 is already dimensionless, without the substitution given below, the differential equation would still be dimensional. \frac = \frac = (\kappa a)^2 \frac = \frac = Z_0 To obtain the nondimensionalized differential equation and initial conditions, use the \pi groups to eliminate \varphi(r) in favor of \Phi(R(r)), then eliminate R(r) in favor of r while carrying out the chain rule and substituting (r) = a, then eliminate r in favor of R (no chain rule needed), then eliminate I in favor of (\kappa a)^2, then eliminate z_0 in favor of Z_0. The resulting equations are as follows: \frac\bigg, _ = - Z_0 \Phi(\infty) = 0 \frac + \frac \frac = (\kappa a)^2 \Phi(R). For table salt in 0.01 M solution at 25 °C, a typical value of (\kappa a)^2 is 0.0005636, while a typical value of Z_0 is 7.017, highlighting the fact that, in low concentrations, (\kappa a)^2 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.


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.


Extensions of the theory

A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry One such extended Debye–Hückel equation is given by: - \log_(\gamma) = \frac where \gamma as its common logarithm is the activity coefficient, z is the integer charge of the ion (1 for H+, 2 for Mg2+ etc.), I is the ionic strength of the aqueous solution, and a is the size or effective diameter of the ion in
angstrom The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...
. 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 common name for the aqueous cation , the type of oxonium ion produced by protonation of water. It is often viewed as the positive ion present when an Arrhenius acid is d ...
is 9Å. A and B 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 are ...
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 surfactants where the presence of
micelle A micelle () or micella () (plural micelles 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 coll ...
s influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%).


See also

* Strong electrolyte *
Weak 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 di ...
*
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 ...
*
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 ...
*
Poisson–Boltzmann equation The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric ...


Notes


References

* * * * * Malatesta, F., and Zamboni, R. (1997). Activity and osmotic coefficients from the EMF of liquid membrane cells, VI – ZnSO4, MgSO4, CaSO4 and SrSO4 in water at 25 °C. ''Journal of Solution Chemistry'', 26, 791–815.


External links

* For easy calculation of activity coefficients in (non-micellar) solutions, check out th
IUPAC open project Aq-solutions
(freeware). *
Gold Book The International Union of Pure and Applied Chemistry publishes many books which contain its complete list of definitions. The definitions are divided into seven "colour books": Gold, Green, Blue, Purple, Orange, White, and Red. There is also an e ...
br>definition
{{DEFAULTSORT:Debye-Huckel equation Analytical chemistry Electrochemical equations Peter Debye