Fixed-charge response to added divalent chloride ($\delta \varphi =(k_{B}T/e)\delta c(e^{-2v}+2e^v-3)/(c_{1+}+2c_{2+}-c_{1-}-2c_{2-})$)

For an ideal planar electric double layer at fixed surface charge density, let $v=e{\varphi }_0/(k_{B}T)$. Adding bulk divalent chloride concentration $\delta c$ adds two monovalent coions per counterion. Differentiating the squared Grahame equation gives the displayed expression, with all denominator concentrations evaluated at the surface. The numerator concentrations are bulk increments. Dropping depleted coions at strongly negative potential gives the leading counterion approximation. A large finite increment instead requires solving the fixed-charge equation at the new composition; exponentiating a first-order voltage estimate does not make that estimate exact.