Thermodynamic Relation between Voltage-Concentration Dependence and Salt Adsorption in Electrochemical Cells

Electrochemical cells containing two electrodes dipped in an ionic solution are widely used as charge accumulators, either with polarizable (supercapacitor) or nonpolarizable (battery) electrodes. Recent applications include desalination (“capacitive deionization”) and energy extraction from salinity differences (“capacitive mixing”). In this Letter, we analyze a general relation between the variation of the electric potential as a function of the concentration and the salt adsorption. This relation comes from the evaluation of the electrical and mechanical energy exchange along a reversible cycle, which involves salt adsorption and release by the electrodes. The obtained relation thus describes a connection between capacitive deionization and capacitive mixing. We check this relation with experimental data already reported in the literature, and moreover by some classical physical models for electrodes, including polarizable and nonpolarizable electrodes. The generality of the relation makes it very useful in the study of the properties of the electric double layer.

Figure 1

Schematic representation of the (dimensionless) quantities of the EDL that are connected by Eq. (4), which is shown in a simplified version (neglecting steric effects, i.e., assuming $\nu =0$; see main text). The specific (per unit surface) amounts of charge ($q$) and salt ($w$) adsorbed into the EDL are proportional to the integrals over the whole space of $1/2(c_{+}-c_{-})$ and $1/2(c_{+}+c_{-})$, respectively. ($c_{+}$ and $c_{-}$ are the concentrations of counterions and coions for a negatively charged surface, respectively. Far from the surface, they are equal to the bulk concentration $c$.) The voltage difference between the solid surface $\varphi$ and the bulk ${\varphi }_{\mathrm{bulk}}$ is equal to the voltage drop in the EDL, $\Delta {\varphi }_{\mathrm{EDL}}$.

Figure 2

Experimental verification of Eq. (3), in the case of a cell with two activated carbon electrodes in a sodium chloride solution, with concentrations 20 mM and 5 mM. The data shown are the two sides of Eq. (3), calculated from the experimental data reported in Ref. 25.