A halotolerant micro-organism lives in seawater

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch Charges in salt water and the assembly of amphiphiles

Problem

A halotolerant micro-organism lives in seawater, which behaves as a monovalent salt solution of ionic strength 500 mM500\,\mathrm{mM} at 298 K298\,\mathrm{K}, with water's relative permittivity εr=78.4\varepsilon_r = 78.4. ((a)) Starting from Poisson's equation in a uniform dielectric, ∇2ψ=−ρ/(ε0εr)\nabla^2\psi = -\rho/(\varepsilon_0\varepsilon_r), and the Boltzmann-weighted number densities c±(r)=c∞e∓eψ(r)/kBTc_\pm(\mathbf r) = c_\infty e^{\mp e\psi(\mathbf r)/k_BT} of a symmetric 1:11{:}1 electrolyte, derive the Poisson-Boltzmann equation for the dimensionless potential ϕ≡eψ/kBT\phi \equiv e\psi/k_BT in the form ∇2ϕ=κ2sinh⁡ϕ\nabla^2\phi = \kappa^2\sinh\phi, giving κ2\kappa^2 in terms of the Bjerrum length ℓB\ell_B and the bulk salt concentration c∞c_\infty. ((b)) Show that far from any fixed charge, where ∣ϕ∣≪1|\phi|\ll1, this reduces to the linear (Debye-Hückel) equation ∇2ϕ=κ2ϕ\nabla^2\phi = \kappa^2\phi. ((c)) Hence compute the Debye length κ−1\kappa^{-1} of seawater.

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