Problem 1: Electrical double layer
(a) Consider a symmetrical salt in solution and a solid wall which is spontaneously charged
upon contact with an aqueous electrolyte solution. Assuming that the electrical field is
perpendicular to the solid wall and that there are no coions in the electrical double layer,
show that the electrical double layer thickness is scaled as
$\lambda \sim \left(\frac{\epsilon RT}{F^2 z^2 c}\right)^{1/2}$
where $\epsilon$ is the permittivity, R the gas constant, T the temperature, F the Faraday constant
given by $F = N_A e$ ($N_A$ being Avogadro's constant and e the fundamental charge), z the
charge number, and c the ion concentration.
Tip: Use the balance between the electric potential energy (= charge times voltage) and
the thermal (kinetic) energy. You may find the following Poisson equation useful:
$\nabla^2 \phi = -\frac{\rho_E}{\epsilon}$
where $\phi$ is the electrical potential, and the charge density $\rho_E = Fzc$.
(b) Consider a cylindrical glass capillary of a radius a in which an aqueous electrolyte
solution is contained. When the zeta potential of the wall is $\zeta_0$, schematically plot the
electrical potential profile for different values of the Debye length ratio ($\lambda/a$), ranging from
0.01 to 10, with the potential on the vertical axis and the radial distance on the horizontal
axis.
