Research: Science and Education
Appreciating Formal Similarities in the Kinetics of Homogeneous, Heterogeneous, and Enzyme Catalysis
W
Michael T. Ashby Department of Chemistry and Biochemistry, University of Oklahoma, Norman, OK 73019;
[email protected] Catalysts are important to a wide variety of reactions, from small-scale reactions in the laboratory, and biological reactions in vivo, through multi-ton catalysis of industrial chemicals (1). Not surprisingly, the subject of catalysis is studied by widely diverse groups of individuals, including chemists, engineers, and biologists. To a certain extent, the theory of catalysis has been developed independently by such groups, particularly with regard to the topics of homogeneous, heterogeneous, and enzyme catalysis. Despite the fact that the theories that span these disciplines share many common denominators, the nomenclatures, symbols, abbreviations, forms of the equations, and methods that are employed to analyze and present data are often dissimilar. Given the emerging interest in interdisciplinary research and the consequential need to communicate with diverse groups regarding the topic of catalysis, it is desirable to expose students to the subject of catalysis from diverse perspectives. The purpose of this article is to illustrate the similarities of the kinetic models by deriving the rate laws that are commonly employed to describe homogeneous, heterogeneous, and enzyme catalysis.1 Let us consider homogeneous catalysis first.
product, and intermediate are described for the mechanism of eqs 1 and 2 by the following differential equations (2):
d[ A ] = − k1 [ A ][C ] + k −1 [ AC ] dt
d[ AC ] = k1 [ A ][C ] − k −1 [ AC ] − k 2 [ AC ] {[B]} (5) dt
A + C
AC {+ B}
A {+ B}
k1 k−1
k2
C
d[B ] = − k 2 [ AC][B] dt
(6)
d[P] = k 2 [ AC ] {[B]} dt
(7)
Since [C] > α[A] and 1 > α [ A ] ): v ≈ α k 2 [ A ] {[B ]}[C ]0 =
k1 k 2 [ A ] {[B]}[C ]0 k −1 + k 2 {[B]}
(11)
Case 1 can be further analyzed in terms of two limiting conditions. For the situation where k᎑1 k2{[B]}, occurs when the equilibrium of eq 1 lies on the left,
Case 1b (k −1 >> k 2 {[B]} ): k v ≈ 1 k 2 [ A ] {[B ]}[C ]0 = K1 k 2 [ A ] {[B]}[C ]0 (13) k −1 where K1 is the equilibrium constant and equal to k1兾k᎑1. Returning to the limiting conditions for eq 10, the situation where 1 βAPA (e.g., at low pressures of A) and 1
