A charged biological molecule attracts counterions and repels ions of the same sign. Thermal diffusion opposes complete localization, producing a mobile ionic atmosphere whose charge partly cancels the molecule's electric field at larger distances. Solvent polarization also reduces the field through its permittivity, but that dielectric response is distinct from the mobile-ion screening cloud.

T

, an ion of valence

z_{i}

n_{i} (r) = n_{i}^{\infty} exp [- \frac{z _{i} e ϕ ( r )}{k _{B} T}] .

For

∣ z_{i} e ϕ ∣ ≪ k_{B} T

, linearization about an electroneutral bulk gives

ρ_{ions} ≃ - ϕ \frac{\sum _{i} n _{i}^{\infty} z _{i}^{2} e ^{2}}{k _{B} T}, \nabla^{2} ϕ = κ^{2} ϕ, λ_{D} = κ^{- 1} = \frac{ϵ k _{B} T}{e ^{2} \sum _{i} n _{i}^{\infty} z _{i}^{2}} .

This is the Debye length. Here

n_{i}^{\infty}

counts each species separately and

ϵ

Q

, outside its immediate molecular environment the linearized solution is

ϕ (r) = \frac{Q}{4 π ϵr} e^{- r / λ_{D}} .

It is a screened potential, not a claim that the molecular charge has disappeared.

I = \frac{1}{2} \sum_{i} c_{i} z_{i}^{2}

with

c_{i}

mol L^{- 1}

, the density sum is

2 N_{A} 1 0^{3} I

, so

λ_{D} = \frac{ϵ k _{B} T}{2 e ^{2} N _{A} 1 0 ^{3} I} .

In water near

298 K

λ_{D} ≃ 0.304 nm / I / (1 mol L^{- 1})

: about

9.6 nm

1 mM

and

0.79 nm

0.15 M

. Increasing ion density or valence shortens the range; temperature and solvent permittivity also matter. The displayed temperature dependence holds at fixed permittivity, which itself changes with temperature in water.

The derivation assumes dilute mobile ions, a continuum uniform dielectric and weak potentials. Near highly charged DNA or protein surfaces, nonlinear Poisson-Boltzmann equation behaviour, counterion condensation, finite ion sizes, specific binding and correlations can matter. At physiological salt the predicted length approaches molecular scales, so the formula is a useful far-field scale with limited microscopic accuracy, rather than a complete account of the first hydration layer.