Capacity of the diffuse electric double layer

The differential capacitance between two regions of separated charges is, in general, defined as dQ/d U. Here, Q is the charge on each electrode and U is the voltage. 

The capacity of an electric double layer per unit area is thus 

The index GC is a reminder that we calculated the capacitance in the Gouy-Ghapman model. For sufHdently small surface potentials, we can expand cosh into a 

It is instructive to compare this to the capacitance per unit area of a plate capacitor EEo/d. Here, d is the separation between the two plates. We see that the electric double layer behaves like a plate capacitor, in which the distance between the plates is given by the Debye length. The capacity of a double layer - that is the ability to store charge - rises with increasing salt concentration because the Debye length decreases. 

To avoid confusion, we should point out that as defined above is the differential capacitance. The integral capacitance per unit area is o/rpo- Experimentally, the differential capacitance is easier to measure. 

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