- Debye-Hückel equation
The

**Debye-Hückel limiting law**, named for its developersPeter Debye andErich Hückel , provides one way to obtainactivity coefficient s Ref|Harris. Activities, rather thanconcentrations , are needed in many chemical calculations because solutions that contain ionic solutes do not behave ideally even at very low concentrations. The activity is proportional to the concentration by a factor known as the activity coefficient $gamma\; ,$, and takes into account theinteraction energy of ions in the solution.In order to calculate the activity of an

ion in a solution, one must know theconcentration and the activity coefficient, $gamma\; ,$. The activity of some ion species C, $a\_C\; ,$, is equal to a dimensionless measure of the concentration of C, $[C]\; ,$ multiplied by the activity coefficient of C, $gamma\; ,$.:$a\_C\; =\; gamma\; frac\{\; [C]\; \}\{\; [C\_\{Theta\}]\; \},$::$[C\_\{Theta\}]\; ,$ represents the concentration of the chosen "standard state", e.g. 1 mol/kg if we work in

molality .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 (Hamann, Hamnett, and Vielstich. Electrochemistry. Wiley-VCH. section 2.5.2):$log(gamma\_i)\; =\; -\; frac\; \{z\_i^2\; q^2\; kappa\}\{8\; pi\; varepsilon\_r\; varepsilon\_0\; k\_B\; T\}\; =\; -\; frac\{z\_i^2\; q^3\}\{4\; pi\; (varepsilon\_r\; varepsilon\_0\; k\_B\; T)^\{3/2\; sqrt\{frac\{I\}\{2=\; -\; A\; z\_i^2\; sqrt\{I\}$::$z\_i$ is the

charge number of ion species i::$q$ is theelementary charge ::$kappa$ is the Debye screening length, defined below::$varepsilon\_r$ is therelative permittivity of the solvent::$varepsilon\_0$ is thepermittivity of free space ::$k\_B$ isBoltzmann's constant ::$T$ is thetemperature of the solution::$I$ is is theionic strength of the solution, defined below::$A$ is a constant that depends on the solvent. 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 is 1.172 $mbox\{mol\}^\{-1/2\}\; mbox\{kg\}^\{1/2\}$.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.

**ummary of Debye & Hückel's first paper on the theory of dilute electrolytes**The English title of the paper is called "On the Theory of Electrolytes. I. Freezing Point Depression and Related Phenomena." It was originally published in volume 24 of a German-language journal, called Physikalische Zeitschrift, in 1923. An English translationcite book

title=The Collected Papers of Peter J. W. Debye

publisher=Interscience Publishers, Inc.

place=New York, New York

year=1954] rp|217-63of the paper 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."rp|xv The paper deals with the calculation of properties of electrolyte solutions that are not under the influence of net electric fields, thus it deals withelectrostatics .In the same year they first published this paper, Debye and Hückel, hereinafter D&H, also released a paper 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 paper 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 paper.

**Introduction**D&H note that the Guldberg-Waage formula for electrolyte species in chemical reaction equilibrium in classical form isrp|221

:$prod\_\{i=1\}^\{s\}\; x\_\{i\}^\{\; u\_i\}\; =\; K$

::$extstyle\; prod$ 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 of species $i$::$u\_i$ is thestoichiometric coefficient of species $i$::K is theequilibrium constant 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 (a separate activity coefficient associated for each species)—which is what is used in modern chemistryas of 2007 .The relationship between $gamma$ and the special activity coefficients, $gamma\_i$ isrp|248

:$log(\; gamma\; )\; =\; sum\_\{i=1\}^\{s\}\; u\_i\; log(\; gamma\_i\; )$.

**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 potential s into classical and electrostatic terms.:$Phi\; =\; S\; -\; frac\; \{U\}\; \{T\}\; =\; -\; frac\{A\}\{T\}$

::$Phi$ is Helmholtz free entropy::$S$ is

entropy ::$U$ isinternal energy ::$T$ istemperature ::$A$ isHelmholtz free energy D&H give the total differential of $Phi$ asrp|222

:$d\; Phi\; =\; frac\; \{P\}\; \{T\}\; d\; V\; +\; frac\; \{U\}\; \{T^2\}\; d\; T$

::$P$ is

pressure ::$V$ isvolume By the definition of the total differential, this means that

::$frac\; \{P\}\; \{T\}\; =\; frac\; \{\; partial\; Phi\; \}\; \{partial\; V\}$ and

::$frac\; \{U\}\; \{T^2\}\; =\; frac\; \{\; partial\; Phi\; \}\; \{partial\; T\}$,

which are useful further on.

As stated previously, the internal energy is divided into two parts,rp|222

:$U\; =\; U\_k\; +\; U\_e$

::$k$ 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, thatrp|222

:$Phi\_e\; =\; int\; frac\; \{U\_e\}\{T^2\}\; d\; T$

It would seem that, without some justification,

:$Phi\_e\; =\; int\; frac\; \{P\_e\}\; \{T\}\; d\; V\; +\; int\; frac\; \{U\_e\}\{T^2\}\; d\; T$.

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, $Xi$, isrp|222-3

$Xi\; =\; S\; -\; frac\; \{U\; +\; P\; V\}\; \{T\}\; =\; Phi\; -\; frac\; \{P\; V\}\{T\}\; =\; -\; frac\; \{G\}\{T\}$.

::$G$ is

Gibbs free energy D&H give the total differential of $Xi$ asrp|222

:$d\; Xi\; =\; -frac\; \{V\}\; \{T\}\; d\; P\; +\; frac\; \{U\; +\; P\; V\}\; \{T^2\}\; d\; T$.

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, writingrp|223:$Xi\; =\; Xi\_k\; +\; Xi\_e$

and put

:$Xi\_e\; =\; Phi\_e\; =\; int\; frac\; \{U\_e\}\{T^2\}\; d\; T$.

D&H say that, according to Planck, the classical part of the Gibbs free entropy isrp|223

:$Xi\_k\; =\; sum\_\{i=0\}^s\; N\_i\; (xi\_i\; -\; k\_B\; ln\; (x\_i))$.::$i$ 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::$k\_B$ is

Boltzmann's constant ::$x\_i$ is the mole fraction of species iSpecies zero is the solvent. The definition of $xi\_i$ is as follows, where lower case letters indicate the particle specific versions of the the corresponding extensive properties:rp|223

:$xi\_i\; =\; s\_i\; -\; frac\; \{u\_i\; +\; P\; v\_i\}\; \{T\}$.

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 upon its mole fraction.cite web |url=http://www.chem.arizona.edu/~salzmanr/480a/480ants/mixpmqis/mixpmqis.html |title=Mixtures; Partial Molar Quantities; Ideal Solutions |accessdate=2007-10-23 |last=Salzman |first=William R. |date=2001-08-21 |work=Chemical Thermodynamics |publisher=University of Arizona |language=English |archiveurl=http://web.archive.org/web/20070607142802/http://www.chem.arizona.edu/~salzmanr/480a/480ants/mixpmqis/mixpmqis.html |archivedate=2007-06-07]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 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.rp|225

**The Potential Energy of an Arbitrary Ion Solution**Electroneutrality of a solution requires thatrp|233

:$sum\_\{i=1\}^s\; N\_i\; z\_i\; =\; 0$.::$N\_i$ is the total number of ions of species i in the solution::$z\_i$ is the

charge number of species iTo bring an ion of species i, initially far away, to a point $P$ within the ion cloud requires

interaction energy in the amount of $z\_i\; q\; varphi$, where $q$ is theelementary charge and $varphi$ is the value of the scalarelectric potential field at $P$. If electric forces were the only factor in play, the minimum energy configuration of all the ions would be achieved in a close-packed lattice configuration. However, the ions are inthermal equilibrium with each other and they are relatively free to move. Thus they obeyBoltzmann statistics and form aBoltzmann distribution . All species' number densities, $n\_i$, are altered from their bulk (overall average) values, $n^\{0\}\_i$, by the correspondingBoltzmann factor , $e^\{-frac\{z\_i\; q\; varphi\}\{k\_B\; T$, where $k\_B$ is theBoltzmann constant and $T$ is the temperature (http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html, section 19.3). Thus,rp|233:$n\_i\; =\; frac\; \{N\_i\}\{V\}\; e^\{-dfrac\{z\_i\; q\; varphi\}\{k\_B\; T\; =\; n^\{0\}\_i\; e^\{-dfrac\{z\_i\; q\; varphi\}\{k\_B\; T$::V is the volume of the solution

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.rp|227

The

charge density is related to the number density:rp|233:$ho=sum\_i\; z\_i\; q\; n\_i\; =\; sum\_i\; z\_i\; q\; n^\{0\}\_i\; e^\{-frac\{z\_i\; q\; varphi\}\{k\_B\; T$.

When combining this result for the charge density with the

Poisson equation from electrostatics, a form of thePoisson-Boltzmann equation results:rp|233:$\{\; abla\}^2\; varphi\; =\; -frac\{\; ho\}\{varepsilon\_r\; varepsilon\_0\}\; =\; -\; sum\_i\; frac\; \{z\_i\; q\; n^\{0\}\_i\}\{varepsilon\_r\; varepsilon\_0\}\; e^\{-frac\{z\_i\; q\; varphi\}\{k\_B\; T$.

This equation is difficult to solve and does not follow the principle of

linear superposition for the relationship between the number of charges and the strength of the potential field. However, for sufficiently low concentrations of ions, a first order Taylor series approximation for the exponential function may be used ($e^x\; =\; 1+x$ 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,rp|227 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.rp|227 Thus,:$-sum\_i\; frac\{z\_i\; q\; n^\{0\}\_i\}\{varepsilon\_r\; varepsilon\_0\}\; e^\{-frac\{z\_i\; q\; varphi\}\{k\_B\; T\; approx\; -sum\_i\; frac\{z\_i\; q\; n^\{0\}\_i\}\{varepsilon\_r\; varepsilon\_0\}\; (1-frac\{z\_i\; q\; varphi\}\{k\_B\; T\})=-(sum\_i\; frac\{z\_i\; q\; n^\{0\}\_i\}\{varepsilon\_r\; varepsilon\_0\}-sum\_i\; frac\{z\_i^2\; q^2\; n^\{0\}\_i\; varphi\}\{varepsilon\_r\; varepsilon\_0\; k\_B\; T\})$.

The Poisson-Boltzmann equation is transformed torp|233

:$\{\; abla\}^2\; varphi\; =\; sum\_i\; frac\{z\_i^2\; q^2\; n^\{0\}\_i\; varphi\}\{varepsilon\_r\; varepsilon\_0\; k\_B\; T\}$,

because the first summation is zero due to electroneutrality.rp|234

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:rp|234:$kappa^2\; =\; sum\_i\; frac\{z\_i^2\; q^2\; n^\{0\}\_i\}\{varepsilon\_r\; varepsilon\_0\; k\_B\; T\}\; =\; frac\{2\; I\; q^2\}\{varepsilon\_r\; varepsilon\_0\; k\_B\; T\}$,

:$I\; =\; frac\; \{1\}\{2\}\; sum\_i\; z\_i^2\; n^\{0\}\_i$.

So, the fundamental equation is reduced to a form of the

Helmholtz equation :(http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_2004/files/lu.pdf section 3.1):$\{\; abla\}^2\; varphi\; =\; kappa^2\; varphi$.

Today, $kappa^\{-1\}$ is called the Debye screening length. D&H recognize the importance of the parameter in their paper and characterize it as a measure of the thickness of the ion atmosphere, which is (in this paper) an

electrical double layer of the Helmholtz type.rp|229The equation may be expressed in

spherical coordinates by taking $r=0$ at some arbitrary ion (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/laplace.html):rp|229:$\{\; abla\}^2\; varphi\; =\; frac\; \{1\}\{r^2\}\; frac\; \{partial\; \}\{partial\; r\}\; left\; (\; r^2\; frac\; \{partial\; varphi(r)\}\{partial\; r\}\; ight\; )=\; frac\{part^2\; varphi(r)\; \}\{partial\; r^2\}\; +\; frac\{2\}\{r\}\; frac\{part\; varphi(r)\; \}\{partial\; r\}\; =\; kappa^2\; varphi(r)$.

The equation has the following general solution; keep in mind that $kappa$ is a positive constant:rp|229

:$varphi(r)\; =\; A\; frac\{e^\{-sqrt\; \{kappa^2\}\; r\{r\}\; +\; A\text{\'}\; frac\{e^\{sqrt\; \{kappa^2\}\; r\{2\; r\; sqrt\; \{kappa^2\; =\; A\; frac\{e^\{-kappa\; r\{r\}\; +\; A"\; frac\{e^\{kappa\; r\{r\}\; =\; A\; frac\{e^\{-kappa\; r\{r\}$.::$A$, $A\text{'}$, and $A"$ are undetermined constants

The electric potential is zero at infinity by definition, so $A"$ must be zero.rp|229

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 to which one may not approach within the radius $a\_i$.rp|231The potential of a point charge by itself is:

:$varphi\_\{pc\}(r)\; =\; \{1\; over\; 4\; pi\; varepsilon\_r\; varepsilon\_0\}\{z\_i\; q\; over\; r\}$.

D&H say that the total potential inside the sphere isrp|232

:$varphi\_\{sp\}(r)\; =\; varphi\_\{pc\}(r)\; +\; B\_i\; =\; \{1\; over\; 4\; pi\; varepsilon\_r\; varepsilon\_0\}\{z\_i\; q\; over\; r\}\; +\; B\_i$,

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) (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potsph.html). 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,rp|232

:$varphi(a\_i)\; =\; A\_i\; frac\{e^\{-kappa\; a\_i\{a\_i\}\; =\; \{1\; over\; 4\; pi\; varepsilon\_r\; varepsilon\_0\}\{z\_i\; q\; over\; a\_i\}\; +\; B\_i\; =\; varphi\_\{sp\}(a\_i)$,:$varphi\text{'}(a\_i)\; =\; -frac\{A\_i\; e^\{-kappa\; a\_i\}\; (1\; +\; kappa\; a\_i)\}\{a\_i^2\}\; =\; -\; \{1\; over\; 4\; pi\; varepsilon\_r\; varepsilon\_0\}\{z\_i\; q\; over\; a\_i^2\}\; =\; varphi\_\{sp\}\text{'}(a\_i)$,:$A\_i\; =\; frac\; \{z\_i\; q\}\{4\; pi\; varepsilon\_r\; varepsilon\_0\}\; frac\{e^\{kappa\; a\_i\{1\; +\; kappa\; a\_i\}$, and:$B\_i\; =\; -frac\{z\_i\; q\; kappa\}\{4\; pi\; varepsilon\_r\; varepsilon\_0\}\; frac\; \{1\}\{1\; +\; kappa\; a\_i\}$.

By the definition of

electric potential energy , the potential energy associated with the singled out ion in the ion atmosphere isrp|230 & 232:$u\_i\; =\; z\_i\; q\; B\_i\; =\; -frac\{z\_i^2\; q^2\; kappa\}\{4\; pi\; varepsilon\_r\; varepsilon\_0\}\; frac\; \{1\}\{1\; +\; kappa\; a\_i\}$

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.rp|230 & 232

:$U\_e\; =\; frac\; \{1\}\{2\}\; sum\_\{i=1\}^s\; N\_i\; u\_i\; =\; -\; sum\_\{i=1\}^s\; frac\; \{N\_i\; z\_i^2\}\{2\}\; frac\{q^2\; kappa\}\{4\; pi\; varepsilon\_r\; varepsilon\_0\}\; frac\; \{1\}\{1\; +\; kappa\; a\_i\}$

**The Additional Electric Term to the Thermodynamic Potential****Nondimensionalization**This section was created without reference to the original paper and there are some errors in it (for instance, the ionic strength is off by a factor of two). Once these are rectified, this section should probably be moved to the nondimensionalization article and then be linked from here, since the nondimensional version of the Poisson-Boltzmann equation isn't necessary to understand the D&H theory.

The differential equation is ready for solution (as stated above, the equation only holds for low concentrations):

:$frac\{part^2\; varphi(r)\; \}\{partial\; r^2\}\; +\; frac\{2\}\{r\}\; frac\{part\; varphi(r)\; \}\{partial\; r\}\; =\; frac\{I\; q\; varphi(r)\}\{varepsilon\_r\; varepsilon\_0\; k\_b\; T\}\; =\; kappa^2\; varphi(r)$

Using the

Buckingham π theorem on this problem results in the following dimensionless groups::$pi\_1\; =\; frac\{q\; varphi(r)\}\{k\_b\; T\}\; =\; Phi(R(r))$:$pi\_2\; =\; varepsilon\_r$:$pi\_3\; =\; frac\{a\; k\_b\; T\; varepsilon\_0\}\{q^2\}$:$pi\_4\; =\; a^3\; I$:$pi\_5\; =\; z\_0$:$pi\_6\; =\; frac\{r\}\{a\}\; =\; R(r)$

$Phi$ 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.

:$frac\{pi\_4\}\{pi\_2\; pi\_3\}\; =\; frac\{a^2\; q^2\; I\}\{varepsilon\_r\; varepsilon\_0\; k\_b\; T\}\; =\; (kappa\; a)^2$

:$frac\{pi\_5\}\{pi\_2\; pi\_3\}\; =\; frac\{z\_0\; q^2\}\{4\; pi\; a\; varepsilon\_r\; varepsilon\_0\; k\_b\; T\}\; =\; Z\_0$

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 $\{R^prime\}(r)\; =\; a$, 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:

:$frac\{partial\; Phi(R)\; \}\{partial\; R\}igg|\_\{R=1\}\; =\; -\; Z\_0$

:$Phi(infty)\; =\; 0$

:$frac\{part^2\; Phi(R)\; \}\{part\; R^2\}\; +\; frac\{2\}\{R\}\; frac\{part\; Phi(R)\; \}\{part\; R\}\; =\; (kappa\; a)^2\; Phi(R)$

For table salt in 0.01 M solution at 25°C, typical 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.

**Extensions of the theory**Warning: The notation in this section is (presently) different than in the rest of the article.

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:

:$-\; log(gamma)\; =\; frac\{A|z\_+z\_-|sqrt\{mu\{1\; +\; Basqrt\{mu\; ,$

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.), $mu\; ,$ is the ionic strength of the aqueous solution, and $a\; ,$ is the size or effective diameter of the ion in angstrom. 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 is 9Å. $A\; ,$ and $B\; ,$ are constants with values of respectively 0.5085 and 0.3281 at 25°C in water Ref|Skoog.}The Extended Debye-Hückel Equation provides accurate results for μ ≤ 0.1 M. For solutions of greater ionic strengths, the

Pitzer equation 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 s where the presence ofmicelle s influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%).**ee also***

Strong electrolyte

*Weak electrolyte

*Ionic atmosphere **External links*** [

*http://twt.mpei.ac.ru/MAS/Worksheets/Debye-Huckel.mcd Debye-Hückel coefficient on-line calculation*] For easy calculation of activity coefficients in (non-micellar) solutions, check out the [*http://www.iupac.org/projects/2000/2000-003-1-500.html IUPAC open project Aq-solutions*] (freeware)"'.

*Gold Book [*http://www.iupac.org/goldbook/D01534.pdf definition*]**References*** cite journal

author=P. Debye and E. Hückel

title=The theory of electrolytes. I. Lowering of freezing point and related phenomena

journal=Physikalische Zeitschrift

year=1923

volume=24

pages=185–206

*cite book

last = Harris

first = Daniel C.

title = Quantitative Chemical Analysis

edition = 6th edition

publisher = W. H. Freeman & Company

year = 2003

id = ISBN 0-7167-4464-3

*cite book

last = Skoog

first = Douglas A.

title = Fundamentals of Analytical Chemistry

id = ISBN 0534417965

*Wikimedia Foundation.
2010.*