= Solution
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.
For an ideal <electrolyte> in <equilibrium> at <temperature> $T$, an <ion> of valence $z_i$ has local <number density>
$$
n_i(\mathbf r)=n_i^{\infty}\exp\left[-\frac{z_ie\phi(\mathbf r)}{k_BT}\right].
$$
For $|z_ie\phi|\ll k_BT$, linearization about an electroneutral bulk gives
$$
\rho_{\rm ions}\simeq-\phi\frac{\sum_i n_i^{\infty}z_i^2e^2}{k_BT},\qquad
\nabla^2\phi=\kappa^2\phi,\qquad
\boxed{\lambda_D=\kappa^{-1}=\sqrt{\frac{\epsilon k_BT}{e^2\sum_i n_i^{\infty}z_i^2}}.}
$$
This is the <Debye length>. Here $n_i^{\infty}$ counts \i[each species separately] and $\epsilon$ is the absolute solvent <permittivity>. For a point charge $Q$, outside its immediate molecular environment the linearized solution is
$$
\boxed{\phi(r)=\frac{Q}{4\pi\epsilon r}e^{-r/\lambda_D}.}
$$
It is a screened potential, not a claim that the molecular charge has disappeared.
In terms of <ionic strength> $I=\tfrac12\sum_i c_i z_i^2$ with $c_i$ in $\mathrm{mol\,L^{-1}}$, the density sum is $2N_A10^3I$, so
$$
\lambda_D=\sqrt{\frac{\epsilon k_BT}{2e^2N_A10^3I}}.
$$
In water near $298\,\mathrm K$, $\lambda_D\simeq0.304\,\mathrm{nm}/\sqrt{I/(1\,\mathrm{mol\,L^{-1}})}$: about $9.6\,\mathrm{nm}$ at $1\,\mathrm{mM}$ and $0.79\,\mathrm{nm}$ at $0.15\,\mathrm M$. 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.
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