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
Slide 3
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
Slide 4
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
Slide 5
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)
Slide 6
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
Slide 7
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
Slide 8
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
Slide 9
Gouy-Chapman Theory double layer capacitance (1/2)
Slide 10
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
Slide 11
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
Slide 12
Surface charge density C dl E C dl,min m(E)m(E) C dl (E) E
pzc
Slide 13
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
Slide 14
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