Potential Energy Due to Electrical Double Layers

The solution of (7.46) is not given in a closed form, but is available from a table. Let us suppose, as a proper approximation, that the electrical potential at 1 = d is a sum of the potentials of the two double layers, when the overlap between the double layers is small (the two plates are far apart). In this case, cd 1, (7.36) can be transformed to the simpler form 

let us consider a balance of forces impinging on a space charge in a volume element of volume d/, unit area, and thickness d/ located in the diffuse layer (Fig. 7.4). The gradient of hydrostatic pressure dP and the force on a space charge p under an electric field should balance each other  

the difference between the hydrostatic pressure and the electric force /e turns out to be constant at every point. The physical condition that P = Poo for d /dl = 0 at the point far from the charged surface leads to 

The next problem is to evaluate the electrostatic force imposed upon the charged surface by an excess opposite charge in the diffuse layer. The electric force expressed by (7.52) presses the space charge toward the charged surface. This force gradually increases from zero to the final value 

We can conclude that uneven charge distribution in the diffuse layers on both sides of a plate leads to an excess force that pushes the plates apart. The same result was also derived by Langmuir, who used the osmotic pressure caused by the excess ions present between the two plates. 

---