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Pictorial representation of two interacting charged plates across an electrolyte solution. The distance between the plates is abbreviated by h. The most popular model to describe the electrical double layer is the Poisson-Boltzmann (PB) model. This model can be equally used to evaluate double layer forces.
Note that since the particles in the ideal gas are non-interacting, the probability of finding a particle at a certain distance from another particle is the same as the probability of finding a particle at the same distance from any other point; we shall use the center of the sphere.
The relation between C, the counter ion concentration at the surface, and , the counter ion concentration in the external solution, is the Boltzmann factor: = where z is the charge on the ion, e is the charge of a proton, k B is the Boltzmann constant and ψ is the potential of the charged surface.
All quantities are in Gaussian units except energy and temperature which are in electronvolts.For the sake of simplicity, a single ionic species is assumed. The ion mass is expressed in units of the proton mass, = / and the ion charge in units of the elementary charge, = / (in the case of a fully ionized atom, equals to the respective atomic number).
He solved the problem in a general way for a transfer of charge between two bodies of arbitrary shape with arbitrary surface and volume charge. For the self-exchange reaction, the redox pair (e.g. Fe(H 2 O) 6 3+ / Fe(H 2 O) 6 2+ ) is substituted by two macroscopic conducting spheres at a defined distance carrying specified charges.
So long as the alpha particle does not penetrate the sphere, there is no difference between a sphere of charge and a point charge, a mathematical result known as the Shell theorem. q g = positive charge of the gold atom = 79 q e = 1.26 × 10 −17 C; q a = charge of the alpha particle = 2 q e = 3.20 × 10 −19 C; R = radius of the gold atom ...
The difficulty lies in the fact that even though the Coulomb force diminishes with distance as 1/r 2, the average number of particles at each distance r is proportional to r 2, assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.
In 1923, Peter Debye and Erich Hückel reported the first successful theory for the distribution of charges in ionic solutions. [7] The framework of linearized Debye–Hückel theory subsequently was applied to colloidal dispersions by S. Levine and G. P. Dube [8] [9] who found that charged colloidal particles should experience a strong medium-range repulsion and a weaker long-range attraction.