King and Altman [25] were apparently the first to provide an analytic rate
expression for a unicyclic catalytic mechanism. They performed this analysis in
the context of homogeneous enzymatic catalysis, but the main result carries over
directly to the case of metal-mediated homogeneous catalysis. Accordingly, for a
mechanism possessing intermediates of one and the same nuclearity, the analytic
rate expression in terms of the total steady-state concentrations of intermediates
Σ[I i ] is given by Eq. 3 where the turnover frequency term TOF contains terms in just
rate constants k and steady-state concentrations of reactants (but no intermediates).
The approach introduced by King and Altman has been verified and extended by
others [26], particularly for developing rate expressions in terms of any individual
intermediate I i . As a consequence, it can be shown that the rate of product formation
is a linear function of each intermediate concentration (Eq. 4). The importance of
the analytical results is straightforward (Fig. 4a, b); regardless of the nuclearity of
the intermediates, the rate is a linear function of the intermediate concentrations
(at least this is the case observed with multimetallic metalloproteins). A unicyclic
catalytic reaction mechanism possessing dinuclear intermediates does not exhibit a
non-linear rate expression.
r 1 ¼ TOF
X
I i
½
ð3Þ
r 1 ¼ k
app
i
I i
½
ð4Þ
More complex branching occurs in mechanisms where selectivities arise. Therefore,
consider as an example the overall transformation of a symmetric alkene
cyclopentene in the presence of CO and H 2 to both cyclopentane carboxaldehyde
and cyclopentane, facilitated by the addition of a Group 9 (or CAS systems VIIIB)
metal hydride complex HML n capable of performing both hydroformylation [27, 28]
and hydrogenation [29]. Upon addition of the metal hydride to the system, one would
expect the steps HML n ! HML nÀ1 ! H(π-cyclopentene)ML nÀ1 ! C 5 H 9 ML nÀ1
where the bold intermediates should be common to both unicyclic catalytic sequence
of steps. At this point there is a branching in the network of intermediates, and one of
either two paths is followed:
1. Molecular hydrogen activation occurs at the metal centre, eventually resulting in
cyclopentane and HML nÀ1 .
2. CO association and then CO insertion occur, followed by molecular hydrogen
activation at the metal centre, eventually resulting in cyclopentane
carboxaldehyde and HML nÀ1 .
A possible graph representation for this interconnected unicyclic catalytic
mechanism, possessing only mononuclear intermediates, which exhibits
chemoselectivity resulting in two distinct products, is shown in Fig. 5. Again, the
mechanism is colour-coded, representing the three distinct linear sequences of
mononuclear intermediates. The light blue is strictly associated with those intermediates which produce aldehyde, the dark blue is associated strictly with those
196
M. Garland
expression for a unicyclic catalytic mechanism. They performed this analysis in
the context of homogeneous enzymatic catalysis, but the main result carries over
directly to the case of metal-mediated homogeneous catalysis. Accordingly, for a
mechanism possessing intermediates of one and the same nuclearity, the analytic
rate expression in terms of the total steady-state concentrations of intermediates
Σ[I i ] is given by Eq. 3 where the turnover frequency term TOF contains terms in just
rate constants k and steady-state concentrations of reactants (but no intermediates).
The approach introduced by King and Altman has been verified and extended by
others [26], particularly for developing rate expressions in terms of any individual
intermediate I i . As a consequence, it can be shown that the rate of product formation
is a linear function of each intermediate concentration (Eq. 4). The importance of
the analytical results is straightforward (Fig. 4a, b); regardless of the nuclearity of
the intermediates, the rate is a linear function of the intermediate concentrations
(at least this is the case observed with multimetallic metalloproteins). A unicyclic
catalytic reaction mechanism possessing dinuclear intermediates does not exhibit a
non-linear rate expression.
r 1 ¼ TOF
X
I i
½
ð3Þ
r 1 ¼ k
app
i
I i
½
ð4Þ
More complex branching occurs in mechanisms where selectivities arise. Therefore,
consider as an example the overall transformation of a symmetric alkene
cyclopentene in the presence of CO and H 2 to both cyclopentane carboxaldehyde
and cyclopentane, facilitated by the addition of a Group 9 (or CAS systems VIIIB)
metal hydride complex HML n capable of performing both hydroformylation [27, 28]
and hydrogenation [29]. Upon addition of the metal hydride to the system, one would
expect the steps HML n ! HML nÀ1 ! H(π-cyclopentene)ML nÀ1 ! C 5 H 9 ML nÀ1
where the bold intermediates should be common to both unicyclic catalytic sequence
of steps. At this point there is a branching in the network of intermediates, and one of
either two paths is followed:
1. Molecular hydrogen activation occurs at the metal centre, eventually resulting in
cyclopentane and HML nÀ1 .
2. CO association and then CO insertion occur, followed by molecular hydrogen
activation at the metal centre, eventually resulting in cyclopentane
carboxaldehyde and HML nÀ1 .
A possible graph representation for this interconnected unicyclic catalytic
mechanism, possessing only mononuclear intermediates, which exhibits
chemoselectivity resulting in two distinct products, is shown in Fig. 5. Again, the
mechanism is colour-coded, representing the three distinct linear sequences of
mononuclear intermediates. The light blue is strictly associated with those intermediates which produce aldehyde, the dark blue is associated strictly with those
196
M. Garland