| Description: |
The classical Debye–Hückel (DH) theory remains the foundational framework for dilute electrolytes, yet itsmathematical structure imposes strict topological and spectral constraints incompatible with experimental dataabove ionic strengths 𝐼 ≳ 10−2 mol kg−1. Here a rigorous operator formulation is developed for the linearisedPoisson–Boltzmann equation,LDH = −Δ + 𝜅2, 𝜅2 = 2𝑒2 𝐼𝜀𝑘𝐵𝑇 ,acting on H = 𝐻1 (R3). The DH prediction for the mean activity coefficient, log 𝛾± ∝ −√𝐼, belongs to aone-dimensional functional manifold whose curvature is identically zero. Any experimental profile with nonzerosecond derivative,𝐾 (𝐼) = d2d𝐼2 log 𝛾± (𝐼) ≠ 0,cannot be generated by LDH under the assumptions of homogeneous permittivity, point-ion structure, and linearisedresponse. A spectral decomposition of L𝜀 with spatially varying dielectric field 𝜀(x) shows that realistic solvationintroduces nontrivial curvature via eigenvalue modulation 𝜆𝑛 (𝐼), producing experimentally observed inflectionpoints and salting-in/out regimes. Thus DH failure is structural: the theory lacks the degrees of freedom requiredto reproduce the topology of measured activity-coefficient curves. |