Physical Chemistry Third Edition

13.5 Nonequilibrium Electrochemistry 599

κ(1. 602 × 10 −^19 C)

(

2(6. 022 × 1023 mol−^1 )(997 kg m−^3 )(0.0200 mol kg−^1 )

(6. 954 × 10 −^10 C^2 N−^1 m−^2 )(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

) 1 / 2

 4. 64 × 108 m−^1

Debye length

1

κ

 2. 15 × 10 −^9 m 21. 5 A ̊

The electrical double layer resembles a capacitor, which is a pair of parallel conduct-

ing plates separated by a dielectric medium or a vacuum, and which can hold charges

of opposite sign on the two plates. Typical values of the capacitance of the electrical

double layer range from 10 to 40μFcm−^2 (microfarads per square centimeter).^30 The

faradis defined such that a voltage difference of 1 volt on a capacitor with capacitance

1 farad will produce a charge of 1 coulomb on each plate. It is a rather large unit, so

that the microfarad is the practical unit. Thecharge density(charge per unit volume)

ρcis given by

ρcF(z+c++z−c−) (13.5-4)

wherez+andz−are the valences of the cation and anion, respectively, andc+and

c−are the concentrations of the cation and anion, respectively. In a neutral electrolyte

solution far from a charged surface, the charge density equals zero. The charge density

near a charged surface can be obtained by combining Eqs. (13.5-1), (13.5-2), and

(13.5-4).

EXAMPLE13.8

Write the equation for the charge density in the Guoy–Chapman theory.

c+c+ 0 e−z+F φ/RT

c−c− 0 e−z−F φ/RT

φφ 0 e−κx

ρcFz+c+ 0 exp

(

−z+Fφ 0 e−κx

RT

)

+Fz−c− 0 exp

(

−z−Fφ 0 e−κx

RT

)

Figure 13.15 shows the charge density in an aqueous 1-1 electrolyte solution of

molality equal to 0.010 mol kg−^1 at 298.15 K in the vicinity of a positive electrode

withφ 0 10 mV, according to the Guoy–Chapman theory.

0

2 0.2

2 0.4

2 0.6

2 0.8

2 1.0

020406080

x/Å

Charge per unit volume/C cm

23

Figure 13.15 The Distribution of Cha-

rge in the Diffuse Double Layer Accord-

ing to the Guoy–Chapman Theory.

Rates of Electrode Processes

The chemical reaction at an electrode can be an oxidation (anodic) half-reaction or

a reduction (cathodic) half-reaction. Either of these half-reactions if occurring alone

(^30) A. J. Bard and L. R. Faulkner,Electrochemical Methods—Fundamentals and Applications, Wiley,
New York, 1980, p. 8.