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666 19 PROCESSES AT SOLID SURFACES
passing over this time. Term 1 integrates to zero because the integral of a
cosine wave over a complete period is zero.�e integral of term 2 is
j0β f 2η20 ∫
2π/ω
0
cos2 ωt dt = 1
2 j0β f
2η20 ∫
2π/ω
0
dt+ 12 j0β f
2η20 ∫
2π/ω
0
cos 2ωt dt
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
term 3
where the identity cos2 A = 1
2 (1+cos 2A) is used. Term 3 is zero as it is the
integral of a cosine wave over two complete periods. �e other integral
evaluates to 2π/ω and therefore the charge passing over the complete
period is
1
2 j0β f
2η20
2π
ω
�emean current over the complete period is foundbydividing the charge
passing by the time, which is the period 2π/ω.�erefore
jav = 1
2 j0β f
2η20 = j0 f 2η20 12 (
1
2 − α)
As required, the mean current goes to zero when α = 1
2 .
(b) For H+/Pt j0 = 7.9 × 10−4 A cm−2; with the data given
jav = (7.9 × 10−4 A cm−2) × (38.921V−1)2 × (0.010 V)2 × 1
2 (0.5 − 0.38)
= 7.2 µA cm−2
P19D.8 (a) �is voltammogram is that expected for a reversible process, as described
in Section 19D.3 on page 850, and characteristic of a process with a ‘large’
exchange current density. Only a small overpotential is required to achieve
a signi�cant current.
(b) �e �rst part of the voltammogram shows two successive reductions, at
di�erent potentials. Both of these reductions appear to be reversible as the
current changes direction on the reverse sweep, with two features which
complement those in the �rst part of the sweep.
(c) �is voltammogram is that expected for a irreversible process, as described
in Section 19D.3 on page 850, and characteristic of a process with a ‘small’
exchange current density. In this case a larger overpotential is needed to
generate a signi�cant current and the cathodic process dominates.
(d) �e �rst part of the voltammogram shows two successive reductions, at
di�erent potentials. �e second of these appears to be reversible as the
current changes direction for the �rst part of the reverse sweep, but the
second reduction appears to be irreversible as there is no negative-going
current associated with it.
Answers to integrated activities
I19.2 �e relativistic correction term is
e∆ϕ
2mec2
= (1.6022 × 10−19 C) × (50 × 103 V)
2 × (9.1094 × 10−31 kg) × (2.9979 × 108ms−1)2
= 0.0489...