aspects, resulting in PR products, how many pathways PW are there for these
product syntheses? One could try to draw out the entire network and then trace
out all independent pathways, but errors are likely to occur for very complex
networks.
Horiuti was faced with a similar problem, but arguably, on a more difficult scale.
He was trying to figure out the number of independent pathways in heterogeneous
systems [31]. The Horiuti criteria are also based on topological arguments, and
therefore, the conclusions obtained can be taken with a great deal of confidence.
The Horiuti criteria are straightforward to implement after writing all the assumed
individual steps in any order. They do not have to be structured. After entering all
the needed information on steps and reactants, an integer number is obtained. This
is the number of independent pathways. The Horiuti criteria seems to be the most
rational approach to take with non-linear and selective reaction mechanisms in
order to understand how many independent pathways PW to the multiple products
PR occur. The most important consequence of PW > PR in complex systems is that,
in the construction of any truly accurate kinetic model, expressions to account for
PW independent pathways and not just the PR products need to be developed.
The unicyclic graphs possessing {M} and {M–M}/{M–M
0 } containing loops in
Fig. 4 have PR ¼ 1 product and PW ¼ 2 independent paths. The chemoselective
graph [M] in Fig. 5 has PR ¼ 2 products and PW ¼ 2 independent paths. The CBER
mechanisms [M] CBER in Figs. 2 and 7 have PR ¼ 1 product and PW ¼ 1 independent paths, and the CBER mechanisms [M] CBER+UNI in Figs. 8 and 9 have PR ¼ 1
product and PW ¼ 2 independent paths.
2.7.3 Detailed Balance, Cycles and Constraints
Wegscheider is generally credited with developing the first detailed balancing for
complex chemical systems and more specifically catalytic systems [84]. In distilled
form, all the steps in a mechanism are constrained, in a non-trial manner, to the
overall organic reaction involved and particularly the Δ r G of the organic reaction.
Hence for a simple unicyclic network, the product of all the forward rate constants
are related to the product of all reverse rate constants for all times. This holds from
reaction start up to end of the reaction when the entire system is equilibrated. The
arguments in the rate constant exponentials sum up to +/À(Δ r G/RT). If the forward
product and reverse product are equal, then all rate constants cannot be independent. Since the metal-mediated homogeneous mechanisms mentioned in this chapter are fairly complex and since there are a number of simultaneous material and
kinetic constraints operating, it is worthwhile to briefly review two cases, one from
[M] and one from [M].
In Sect. 2.1 a chemoselective unicyclic mechanism was presented for a simultaneous hydroformylation and hydrogenation of cyclopentene. This mechanism, by
definition, possesses intermediates of only mononuclear intermediates. The updated
figure which provides emphasis to constraints is shown in Fig. 13 where additional
212
M. Garland
product syntheses? One could try to draw out the entire network and then trace
out all independent pathways, but errors are likely to occur for very complex
networks.
Horiuti was faced with a similar problem, but arguably, on a more difficult scale.
He was trying to figure out the number of independent pathways in heterogeneous
systems [31]. The Horiuti criteria are also based on topological arguments, and
therefore, the conclusions obtained can be taken with a great deal of confidence.
The Horiuti criteria are straightforward to implement after writing all the assumed
individual steps in any order. They do not have to be structured. After entering all
the needed information on steps and reactants, an integer number is obtained. This
is the number of independent pathways. The Horiuti criteria seems to be the most
rational approach to take with non-linear and selective reaction mechanisms in
order to understand how many independent pathways PW to the multiple products
PR occur. The most important consequence of PW > PR in complex systems is that,
in the construction of any truly accurate kinetic model, expressions to account for
PW independent pathways and not just the PR products need to be developed.
The unicyclic graphs possessing {M} and {M–M}/{M–M
0 } containing loops in
Fig. 4 have PR ¼ 1 product and PW ¼ 2 independent paths. The chemoselective
graph [M] in Fig. 5 has PR ¼ 2 products and PW ¼ 2 independent paths. The CBER
mechanisms [M] CBER in Figs. 2 and 7 have PR ¼ 1 product and PW ¼ 1 independent paths, and the CBER mechanisms [M] CBER+UNI in Figs. 8 and 9 have PR ¼ 1
product and PW ¼ 2 independent paths.
2.7.3 Detailed Balance, Cycles and Constraints
Wegscheider is generally credited with developing the first detailed balancing for
complex chemical systems and more specifically catalytic systems [84]. In distilled
form, all the steps in a mechanism are constrained, in a non-trial manner, to the
overall organic reaction involved and particularly the Δ r G of the organic reaction.
Hence for a simple unicyclic network, the product of all the forward rate constants
are related to the product of all reverse rate constants for all times. This holds from
reaction start up to end of the reaction when the entire system is equilibrated. The
arguments in the rate constant exponentials sum up to +/À(Δ r G/RT). If the forward
product and reverse product are equal, then all rate constants cannot be independent. Since the metal-mediated homogeneous mechanisms mentioned in this chapter are fairly complex and since there are a number of simultaneous material and
kinetic constraints operating, it is worthwhile to briefly review two cases, one from
[M] and one from [M].
In Sect. 2.1 a chemoselective unicyclic mechanism was presented for a simultaneous hydroformylation and hydrogenation of cyclopentene. This mechanism, by
definition, possesses intermediates of only mononuclear intermediates. The updated
figure which provides emphasis to constraints is shown in Fig. 13 where additional
212
M. Garland