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Double layer capacitance Shkkemian peruseet KE-31.4100 Tanja Kallio tanja.kallio@aalto.fi C213 CH 5.3 5.4

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Gouy-Chapman Theory (1/4) D.C. Grahame, Chem. Rev. 41 (1947) 441 Charge density (x) is given by Poisson equation Charge density of the solution is obtained summarizing over all species in the solution Ions are distributed in the solution obeying Boltzmann distribution

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Gouy-Chapman Theory (2/4) Previous eqs can be combined to yield Poisson-Boltzmann eq The above eq is integrated using an auxiliary variable p

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Gouy-Chapman Theory (3/4) Integration constant B is determined using boundary conditions: i) Symmetry requirement: electrostatic field must vanish at the midplane d /dx = 0 ii) electroneutrality: in the bulk charge density must summarize to zero = 0 Thus x =0

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Gouy-Chapman Theory (4/4) Now it is useful to examine a model system containing only a symmetrical electrolyte The above eq is integrated giving potential on the electrode surface, x = 0 where (5.42)

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Gouy-Chapman Theory potential profile The previous eq becomes more pictorial after linearization of tanh 1:1 electrolyte 2:1 electrolyte 1:2 electrolyte linearized

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Gouy-Chapman Theory surface charge Electrical charge q inside a volume V is given by Gauss law In a one dimensional case electric field strength E penetrating the surface S is zero and thus E. dS is zero except at the surface of the electrode (x = 0) where it is (df/dx) 0 dS. Cosequently, double layer charge density is After inserting eq (5.42) for a symmetric electrolyte in the above eq surface charge density of an electrode is

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Gouy-Chapman Theory double layer capacitance (1/2) Capacitance of the diffusion layer is obtained by differentiating the surface charge eq 1:1 electrolyte 2:1 electrolyte 1:2 electrolyte 2:2 electrolyte

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Gouy-Chapman Theory double layer capacitance (1/2)

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Inner layer effect on the capacity (1/2) + + + - + + + - + + x = 0 0 OHL x = x 2 (x 2 ) = 2 + If the charge density at the inner layer is zero potential profile in the inner layer is linear: x = x 2 (x 2 ) = 2 (0) = 0 x = 0

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Inner layer effect on the capacity (2/2) Surface charge density is obtained from the Gauss law relative permeability in the inner layer relative permeability in the bulk solution 2 is solved from the left hand side eqs and inserted into the right hand side eq and C dl is obtained after differentiating

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Surface charge density C dl E C dl,min m(E)m(E) C dl (E) E pzc

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Effect of specific adsorption on the double layer capacitance (1/2) + + + phase phase x = 0 x = x 2 qdqd From electrostatistics, continuation of electric field, for phase boundary Specific adsorbed species are described as point charges located at point x 2. Thus the inner layer is not charged and its potential profile is linear

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Effect of specific adsorption on the double layer capacitance (2/2) H. A. Santos et al., ChemPhysChem8(2007)1540- 1547 So the total capacitance is