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644 18 REACTION DYNAMICS
P18E.4 �e variation of the electron-transfer rate constantwith ∆rG−○ is given by [18E.8–
814]
ln ket = −
RT
4∆ER
(∆rG
−○
RT
)
2
− 1
2 (
∆rG−○
RT
) + const.
A plot of ln ket against −∆rG−○ is expected to be an inverted parabola and, as
described in the text, the maximum occurs at −∆rG−○ = ∆ER.�e plot is shown
in Fig 18.6.
−∆rG−○/eV ket/(106 s−1) lg[ket/(106 s−1)]
0.665 0.657 −0.182
0.705 1.52 0.182
0.745 1.12 0.049
0.975 8.99 0.954
1.015 5.76 0.760
1.055 10.1 1.004
0.6 0.7 0.8 0.9 1.0 1.1 1.2
0.0
0.5
1.0
−∆rG−○/eV
lg
[k
et
/(
10
6
s−
1 )
]
Figure 18.6
�e data are rather poor �t to the second-order polynomial
lg[ket/(106 s−1)] = −3.682 × (−∆rG−○/eV)2 + 9.157 × (−∆rG−○/eV) − 4.5983
which is shown on the plot. �e maximum of this function occurs when the
derivative is zero, that is when 2×−3.682×(−∆rG−○/eV)+9.157 = 0; this occurs
at (−∆rG−○/eV) = 1.24....�erefore ∆ER = 1.2 eV .
P18E.6 �e theoretical treatment given in the text applies only at relatively high tem-
peratures. At temperatures above 130 K, the reaction in question is observed to
follow a temperature dependence consistent with [18E.5–812], namely increas-
ing rate with increasing temperature. Below 130 K, the temperature dependent
terms in the equation are replaced by Franck–Condon factors (Topic 11F); that
is, temperature-dependent terms are replaced by temperature-independentwave-
function overlap integrals.