13.5 Nonequilibrium Electrochemistry
597
other and move to the surface of the electrode in order to be as far from each other
as possible. Similarly, any excess positive charges will be found at the surface of
a positively charged electrode. Ions and neutral molecules can be adsorbed on an
electrode surface. Adsorbed ions are divided into two classes. If an ion is adsorbed
directly on the surface, it is said to be specifically adsorbed. Specifically adsorbed
ions are generally adsorbed without a complete “solvation sphere” of attached solvent
molecules. Every electrolyte solution contains at least one type of cation and one type
of anion, and the cations will generally not be adsorbed to the same extent as the anions,
so that adsorbed ions contribute to the charge at the surface. In addition to specifically
adsorbed ions, a charged electrode surface will attract an excess of oppositely charged
ions in the solution near the electrode. These nonspecifically adsorbed ions are fully
solvated and will not approach so closely to the surface as specifically adsorbed ions.
The region near an electrode surface thus contains two layers of charge. The electrode
surface (including specifically adsorbed ions) has a charge of one sign. This charge
attracts a more diffuse layer of ions of the opposite charge, as shown schematically in
Figure 13.14a. These two regions are called the electrical double layer or sometimes
the diffuse double layer. The layer of specifically adsorbed ions is called the compact
layer, the inner Helmholtz layer, or the Stern layer. The location of the nonspecifically
adsorbed ions is sometimes called the outer Helmholtz plane or the Guoy plane.
Guoy and Chapman developed a theory of the charge distribution in the double layer
about 10 years before Debye and Hückel developed their theory of ionic solutions, which
is quite similar to it. 29 If one neglects nonelectrostatic contributions to the potential
energy of an ion of type i with valence z i , the concentration of ions of type i in a region
of electric potential ϕ is given by the Boltzmann probability formula, Eq. (9.3-41):
c i c i0 e
−z i e ϕ/k B T
c i0 e
−z i F ϕ/RT
(13.5-1)
where c i is the concentration of ions of type i and where c i0 is the concentration at a
location where ϕ 0. In the second version of the equation, F is Faraday’s constant,
equal to N Av e 96485 C mol −1 .
Guoy and Chapman combined Eq. (13.4-1) with the Poisson equation of electrostatics and found that if the potential is taken equal to zero at large distances, the electric
potential in the diffuse double layer is given as a function of x, the distance from the
electrode, by
ϕ ϕ 0 e
−κx
(13.5-2)
where ϕ 0 is the value of the potential at the surface of the electrode. The parameter κ
is given by
κ e
2N Av ρ 1 I
εk B T
1/2
(13.5-3)
where ε is the permittivity of the solvent, ρ 1 is the density of the solvent, I is the
ionic strength as defined in Chapter 7, e is the charge on a proton, k B is Boltzmann’s
constant, and T is the absolute temperature. Figure 13.14b shows this potential as a
function of distance from the electrode surface for an ionic strength of 0.010 mol kg −1
and a temperature of 298.15 K. The Debye length is defined equal to 1/κ. It is the
distance from the surface of the electrode to the location at which the potential is
equal to 1/e (about 0.368) of its value at the surface and is a measure of the effective
thickness of the diffuse double layer.
29 G. Guoy, J. Phys. (Paris), 9, 457 (1910); D. L Chapman, Phil. Mag., 25, 475 (1913).
597
other and move to the surface of the electrode in order to be as far from each other
as possible. Similarly, any excess positive charges will be found at the surface of
a positively charged electrode. Ions and neutral molecules can be adsorbed on an
electrode surface. Adsorbed ions are divided into two classes. If an ion is adsorbed
directly on the surface, it is said to be specifically adsorbed. Specifically adsorbed
ions are generally adsorbed without a complete “solvation sphere” of attached solvent
molecules. Every electrolyte solution contains at least one type of cation and one type
of anion, and the cations will generally not be adsorbed to the same extent as the anions,
so that adsorbed ions contribute to the charge at the surface. In addition to specifically
adsorbed ions, a charged electrode surface will attract an excess of oppositely charged
ions in the solution near the electrode. These nonspecifically adsorbed ions are fully
solvated and will not approach so closely to the surface as specifically adsorbed ions.
The region near an electrode surface thus contains two layers of charge. The electrode
surface (including specifically adsorbed ions) has a charge of one sign. This charge
attracts a more diffuse layer of ions of the opposite charge, as shown schematically in
Figure 13.14a. These two regions are called the electrical double layer or sometimes
the diffuse double layer. The layer of specifically adsorbed ions is called the compact
layer, the inner Helmholtz layer, or the Stern layer. The location of the nonspecifically
adsorbed ions is sometimes called the outer Helmholtz plane or the Guoy plane.
Guoy and Chapman developed a theory of the charge distribution in the double layer
about 10 years before Debye and Hückel developed their theory of ionic solutions, which
is quite similar to it. 29 If one neglects nonelectrostatic contributions to the potential
energy of an ion of type i with valence z i , the concentration of ions of type i in a region
of electric potential ϕ is given by the Boltzmann probability formula, Eq. (9.3-41):
c i c i0 e
−z i e ϕ/k B T
c i0 e
−z i F ϕ/RT
(13.5-1)
where c i is the concentration of ions of type i and where c i0 is the concentration at a
location where ϕ 0. In the second version of the equation, F is Faraday’s constant,
equal to N Av e 96485 C mol −1 .
Guoy and Chapman combined Eq. (13.4-1) with the Poisson equation of electrostatics and found that if the potential is taken equal to zero at large distances, the electric
potential in the diffuse double layer is given as a function of x, the distance from the
electrode, by
ϕ ϕ 0 e
−κx
(13.5-2)
where ϕ 0 is the value of the potential at the surface of the electrode. The parameter κ
is given by
κ e
2N Av ρ 1 I
εk B T
1/2
(13.5-3)
where ε is the permittivity of the solvent, ρ 1 is the density of the solvent, I is the
ionic strength as defined in Chapter 7, e is the charge on a proton, k B is Boltzmann’s
constant, and T is the absolute temperature. Figure 13.14b shows this potential as a
function of distance from the electrode surface for an ionic strength of 0.010 mol kg −1
and a temperature of 298.15 K. The Debye length is defined equal to 1/κ. It is the
distance from the surface of the electrode to the location at which the potential is
equal to 1/e (about 0.368) of its value at the surface and is a measure of the effective
thickness of the diffuse double layer.
29 G. Guoy, J. Phys. (Paris), 9, 457 (1910); D. L Chapman, Phil. Mag., 25, 475 (1913).