Beyond Debye–Hückel: A Spectral–Curvature Theory of Activity Coefficients

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.