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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 663
(ii) If the current is entirely anodic, only the �rst term is needed
j = (2.5 × 10−3 A cm−2) × e(1−0.58)×(38.921 V
−1)×(0.30 V)
= 0.34 A cm−2
�e result con�rms that the current is indeed dominated by the anodic
term, which is the term for which the power of the exponential is positive.
E19D.5(b) �e Butler–Volmer equation is [19D.2–848], j = j0(e(1−α) f η − e−α f η). �e
overpotential is equal to the applied potential E minus the potential developed
by the electrode which, assuming standard conditions, is +0.77 V: η = E −
(0.77 V). With the given value for j0, and assuming α = 0.5
j = (2.5 mA cm−2) × (e(1−0.5)× f×[E−(0.77 V)] − e−0.5× f×[E−(0.77 V)])
j = (2.5 mA cm−2) × (e0.5 f [E−(0.77 V)] − e−0.5 f [E−(0.77 V)])
E19D.6(b) At equilibrium, only the exchange current �ows, therefore for an electrode with
area A the current is j0A, and thus the charge passing in time t is (current ×
time): q = j0At. If each species passing through the double layer carries one
fundamental change, the number of charges is N = q/e = j0At/e. �us the
number per second through an area of 1.0 cm2 is, for H+/Cu,
N/t = j0A/e = (1.0 × 10−6 A cm−2) × (1 cm2)/(1.6022 × 10−19 C)
= 6.24... × 1012 s−1 = 6.2 × 1012 s−1
A similar calculation for Ce4+/Pt gives 2.5 × 1014 s−1 .
�e number of atoms covering 1 cm2 of electrode is (10−4m2)/(260×10−12m)2
= 1.47...×1015.�erefore for H+/Cu the number of times per second that each
atom is involved in a electron transfer event is (number of such events)/(number
of atoms) = (6.24... × 1015 s−1)/(1.47... × 1015) = 4.2 × 10−3 s−1 . A similar
calculation for Ce4+/Pt gives 0.17 s−1 .
E19D.7(b) In the linear region the current density and overpotential are related by [19D.4–
849], η = RT j/F j0, therefore the current density is j = ηF j0/RT . For an
electrode of area A the current is I = jA, and therefore the resistance is
r = η
I
= η
ηF j0A/RT
= RT
F j0A
For H+/Pb
r = (8.3145 JK−1mol−1) × (298 K)
(96485Cmol−1) × (5.0 × 10−12 A cm−2) × (1.0 cm2)
= 5.1 × 109 Ω
�e units are resolved by using (from inside the front cover) 1 V = 1 J C−1 and
1 Ω = 1 V A−1. A similar calculation for Fe3+/Pt gives 10 Ω .