Statistical mechanics, structure of matter Definition

Debye–Hückel theory

Builds on Poisson's equation in electrostatics Boltzmann distribution Debye length

Definition

Consider an electrolyte at absolute temperature $T$ in a solvent of permittivity $\varepsilon$ . It contains ion species $i = 1, \dots, s$ with charges $z_i e$ and bulk number densities $n_i > 0$ , and it is electrically neutral: $\sum_i z_i n_i = 0$ . Here $e$ is the elementary charge and $k_B$ is Boltzmann's constant. Debye–Hückel theory describes the mean electric potential $\varphi$ around a fixed ion by the following assumptions.

The charge density is $\rho = \sum_i z_i e \, n_i \exp(-z_i e \varphi / (k_B T))$ , by the Boltzmann distribution of each species in the potential $\varphi$ .
The potential obeys Poisson's equation $\nabla^2 \varphi = -\rho / \varepsilon$ outside the fixed ion.
The exponentials are linearized, $\exp(-x) \approx 1 - x$ , which is valid when $\abs{z_i e \varphi} \ll k_B T$ .

With neutrality, clauses 1 to 3 give the linearized Poisson–Boltzmann equation $\nabla^2 \varphi = \kappa^2 \varphi, \qquad \kappa^2 = \frac{e^2}{\varepsilon k_B T} \sum_i z_i^2 n_i,$ where $\kappa = 1/\lambda_D$ is the inverse Debye length.

Remarks

Example: around a point ion of charge $Q$ at the origin, the radial solution that decays at infinity and behaves like $Q/(4\pi\varepsilon r)$ near $0$ is $\varphi(r) = \frac{Q}{4 \pi \varepsilon r} e^{-\kappa r}.$ Check: for a radial function, $\nabla^2 \varphi = r^{-1} (r \varphi)''$ , and $r \varphi = C e^{-\kappa r}$ gives $(r\varphi)'' = \kappa^2 r \varphi$ . The zeroth-order term of clause 1 vanishes by neutrality, which is why only the linear term $-\kappa^2 \varepsilon \varphi$ remains. Non-example: in a concentrated solution, or near a highly charged surface, $\abs{z_i e \varphi}$ is not small compared to $k_B T$ , so clause 3 fails and one must solve the nonlinear Poisson–Boltzmann equation. The theory leads to the Debye–Hückel limiting law: $\ln \gamma_\pm$ is proportional to $-\abs{z_+ z_-} \sqrt{I}$ . Here $\gamma_\pm$ is the mean activity coefficient, and $I = \frac12 \sum_i c_i z_i^2$ is the ionic strength for molar concentrations $c_i$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Boltzmann distribution concept
Debye length concept
Poisson's equation in electrostatics concept
Debye–Hückel theory definition

Tag	`F0Z9R0Q31N`
URI	`m2://F0Z9R0Q31N`
Kind	definition
Writing	W0 stub
MSC	82D15, 82D10
Defines	Debye–Hückel theory
Uses	Poisson's equation in electrostatics, Boltzmann distribution, Debye length
Changed	3 November 2026

Debye–Hückel theory

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here