= Solution
\b[Stabilization and screening.] <Charge stabilization of colloids> means suppressing aggregation through repulsion between similarly charged particles and their <electric double layers>. In <DLVO theory>, this repulsion competes with attractive <Van der Waals forces>. A substantial repulsive barrier compared with thermal energy $k_BT$ can make aggregation slow, even when the dispersed state is only metastable.
The <Debye–Hückel screening length> $\lambda_D=\kappa^{-1}$ is the characteristic range over which a weak electric potential is attenuated by rearrangement of mobile <ions>. It measures the thickness of a diffuse screening cloud, not a molecular diameter. For a plane, a weak potential decays as $e^{-\kappa x}$ into the bulk.
\b[Derivation and the source convention.] Put a planar interface at $x=0$, with electrolyte in $x>0$, bulk potential zero, and absolute <permittivity> $\varepsilon=\varepsilon_0\varepsilon_r$. For a symmetric $z:z$ electrolyte, let $n_\infty$ be the bulk <number density> of each ionic species. The <Boltzmann distribution> gives
$$
n_\pm(x)=n_\infty\exp\left(\mp\frac{ze\psi(x)}{k_BT}\right),
$$
so the mobile <charge density> is
$$
\rho_e=ze(n_+-n_-)=-2zen_\infty\sinh\left(\frac{ze\psi}{k_BT}\right).
$$
Substitution into the SI <Poisson equation>, $\nabla^2\psi=-\rho_e/\varepsilon$, gives the <Poisson-Boltzmann equation>
$$
\nabla^2\psi=\frac{2zen_\infty}{\varepsilon}\sinh\left(\frac{ze\psi}{k_BT}\right).
$$
For $|ze\psi|\ll k_BT$, linearize the <hyperbolic sine>. The <Debye–Hückel approximation> is then
$$
\nabla^2\psi=\kappa^2\psi,\qquad
\boxed{\kappa^2=\frac{2z^2e^2n_\infty}{\varepsilon_0\varepsilon_rk_BT}\quad\text{(SI)}}.
$$
Here $z$ is the ionic valence magnitude, $e$ the elementary <electric charge>, $k_B$ the <Boltzmann constant>, $T$ absolute <temperature>, $\varepsilon_0$ the <vacuum permittivity>, and $\varepsilon_r$ the solvent's <relative permittivity>. For several ionic species, the numerator $2z^2e^2n_\infty$ is replaced by $e^2\sum_i z_i^2n_{i,\infty}$.
The original PDF includes an additional $4\pi$ in the denominator. That factor does not follow from SI <Poisson equation> with number densities and the stated permittivities. The <Bjerrum length> provides a useful check: $\ell_B=e^2/(4\pi\varepsilon k_BT)$ and the consistent result is $\kappa^2=8\pi\ell_Bz^2n_\infty$. A Gaussian-unit convention moves factors of $4\pi$ together with the charge and permittivity conventions; it does not justify inserting this extra factor while retaining all the given SI quantities. Both numerical interpretations are reported below rather than silently changing the source.
\b[Conditions favoring stability.] Increasing the surface-potential or surface-charge magnitude usually strengthens the electrostatic barrier. Lower <ionic strength> increases $\lambda_D$ and generally makes repulsion longer ranged; added salt and especially multivalent <counterions> shorten it and can promote aggregation. Suitable <pH> can keep an ionizable surface charged, while conditions near an isoelectric state weaken charge repulsion. The solvent-dependent <Van der Waals forces>, charge regulation and particle geometry also affect the barrier, so a long screening length is not by itself sufficient. At fixed density and valence, the linear formula gives $\lambda_D\propto\sqrt{\varepsilon T}$, but actual temperature changes can also alter the permittivity and surface chemistry.
\b[Surface charge at small potential.] Take the usual one-sided planar-interface model, with negligible electric displacement on the nonaqueous side. The solution decaying into the solution is
$$
\boxed{\psi(x)=\psi_0e^{-\kappa x}}.
$$
The <electric field> is $E_x=-\psi'=\kappa\psi_0e^{-\kappa x}$. The <electrostatic interface boundary conditions> then give the <surface charge density>
$$
\boxed{\sigma=\varepsilon E_x(0^+)=\varepsilon\kappa\psi_0}.
$$
One can also derive it by integrating the diffuse <charge density>:
$$
\int_0^\infty\rho_e(x)\,dx=-\varepsilon\kappa^2\psi_0\int_0^\infty e^{-\kappa x}\,dx
=-\varepsilon\kappa\psi_0=-\sigma.
$$
Thus the diffuse layer neutralizes the surface charge. This is the <planar Debye-Hückel double layer>. If instead the object is an isolated sheet wetted by identical electrolyte on both sides, each side screens half its total charge and $\sigma_{\mathrm{sheet}}=2\varepsilon\kappa\psi_0$. The one-sided assumption is the conventional local model for the exterior of a large charged colloid; it must be stated because the question does not specify the material on the other side.
\b[Numerical ion density.] For $T=298.15\,\mathrm K$, $z=1$, $\kappa=10^9\,\mathrm{m^{-1}}$ and $\varepsilon=80\varepsilon_0=7.0834\times10^{-10}\,\mathrm{F\,m^{-1}}$, the consistent SI result is
$$
\boxed{n_\infty=\frac{\varepsilon k_BT\kappa^2}{2e^2}=5.68\times10^{25}\,\mathrm{m^{-3}}}.
$$
This is $\boxed{0.0943\,\mathrm{mol\,L^{-1}}}$ for each ion species, and hence also the salt concentration for a monovalent $1:1$ salt. The combined number density of both signs is $2n_\infty$.
Literal substitution into the formula printed in the PDF instead gives
$$
\boxed{n_{\infty,\mathrm{printed}}=4\pi n_{\infty,\mathrm{SI}}=7.14\times10^{26}\,\mathrm{m^{-3}}\simeq1.19\,\mathrm{mol\,L^{-1}}}.
$$
These values are distinct consequences of the source's extra factor; they cannot both describe the same physical screening length in SI with the same definition of $n_\infty$.
\b[Surface potential and validity.] The requested linear estimate in the one-sided model is
$$
\boxed{\psi_{0,\mathrm{linear}}=\frac{\sigma}{\varepsilon\kappa}=\frac{0.25}{(7.0834\times10^{-10})(10^9)}=0.353\,\mathrm V}.
$$
However, the <thermal voltage> is $k_BT/e=25.7\,\mathrm{mV}$, so $e\psi_{0,\mathrm{linear}}/(k_BT)\simeq13.7$. Thus \b[the small-potential approximation is not self-consistent for the supplied surface charge]. The boxed value is its formal extrapolation, not a reliable prediction of the actual surface potential.
For a valid nonlinear comparison within the ideal one-sided planar <Poisson-Boltzmann equation>, multiply the differential equation by $\psi'$ and use zero potential and field in the bulk. This gives
$$
(\psi')^2=\frac{8n_\infty k_BT}{\varepsilon}\sinh^2\left(\frac{e\psi}{2k_BT}\right).
$$
Using the surface boundary condition yields the <Grahame equation> and its inverse:
$$
\sigma=\frac{2\varepsilon\kappa k_BT}{e}\sinh\left(\frac{e\psi_0}{2k_BT}\right),\qquad
\psi_0=\frac{2k_BT}{e}\operatorname{arsinh}\left(\frac{e\sigma}{2\varepsilon\kappa k_BT}\right).
$$
With the SI screening length, the latter gives $\boxed{\psi_{0,\mathrm{PB}}\simeq0.135\,\mathrm V}$. This is a nonlinear mean-field estimate under explicit assumptions, not a unique experimentally determined potential: compact interfacial layers, finite ion size and surface chemistry were not supplied. The source's normalization problem and its use of a linear formula far outside the small-potential regime are separate issues.
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