Skip to main content
Engineering LibreTexts

9.7: Untitled Page 223

  • Page ID
    18356
  • Chapter 9

    Catalysts

    A catalyst is an agent that causes an increase in the reaction rate without undergoing any permanent change, and the enzymes represented by species E in Figure 9‐9 perform precisely that function in the production of intracellular and extracellular material. Enzymes are global proteins that bind substrates (reactants) in particular configurations that enhance the reaction rate. The simplest description of this process is due to Michaelis and Menten (1913) who proposed a two‐step process in which a substrate first binds reversibly with an enzyme and then reacts irreversibly to form a product. In this development we first consider the substrate A, the enzyme E, and the product D. To begin with, the enzyme E forms a complex with substrate A in a reversible manner as indicated by Eqs. 9‐64a and 9‐65a.

    k I

    Elementary chemical kinetic schema I:

    E A

     EA

    (9‐64a)

    I

    I

    I

    I

    Elementary stoichiometry I:

    R

      R

    ,

    R

    R

    (9‐64b)

    E

    EA

    E

    A

    Elementary chemical reaction rate equation I:

    I

    R

      k c c

    (9‐64c)

    E

    I E A

    k II

    Elementary chemical kinetic schema II:

    EA

    

    E A

    (9‐65a)

    Elementary stoichiometry II:

    II

    II

    II

    R

      R ,

    R

      R II

    (9‐65b)

    EA

    E

    EA

    A

    Elementary chemical kinetic rate equation II:

    II

    R

      k c

    (9‐65c)

    EA

    II EA

    In the final step, the complex EA reacts irreversibly to form the product D and the enzyme E according to

    k III

    Elementary chemical kinetic schema III:

    EA

    

    E D

    (9‐66a)

    Elementary stoichiometry III:

    III

    III

    III

    R

      R ,

    R

      R III

    (9‐66b)

    EA

    E

    EA

    D

    Elementary chemical reaction rate equation III:

    III

    R

      k c

    (9‐66c)

    EA

    III EA

    In all these elementary steps we assume that the stoichiometric schemata are identical in form to the chemical kinetic schemata. In the shorthand

    Reaction Kinetics

    415

    nomenclature of biochemical engineering, Michaelis‐Menten kinetics are represented by

    k I

    k III

    E A

    

     EA 

    E D

    (9‐67)

    k II

    Our objective at this point is to develop an expression for the net rate of production of species D in terms of the concentration of species A. The net rate of production for species D takes the form

    I

    II

    III

    R

    R R R

    k c

    (9‐68)

    D

    D

    D

    D

    III EA

    and the net rates of production of the other species are given by I

    II

    III

    R

    R

    R R

      k c c k c

    (9‐69a)

    A

    A

    A

    A

    I E A

    I

    I

    EA

    I

    II

    III

    R

    R

    R R

      k c c k c

    k c

    (9‐69b)

    E

    E

    E

    E

    I E A

    II EA

    III EA

    I

    II

    III

    R

    R

    R

    R

    k c c

    k c

    k c

    (9‐69c)

    EA

    EA

    EA

    EA

    I E A

    II EA

    III EA

    Since a catalyst only facilitates a reaction and is neither consumed nor produced by the reaction, we can assume that the total concentration of the enzyme catalyst is constant. We express this idea as

    o

    c

    c

    c

    (9‐70)

    E

    EA

    E

    in which o

    c is the initial concentration of the enzyme in the reactor. In addition, E

    the net rate of production of the enzyme catalyst should be zero and we express this idea as

    R

     0

    (9‐71)

    E

    Use of Eq. 9‐71 with Eq. 9‐69b leads to a constraint on the rates of reaction given by

    0   k c c

    k c

    k c

    (9‐72)

    I E A

    II EA

    III EA

    This can be arranged in the form

    k c c

    I E A

    cEA

    k

    (9‐73)

    k

    II

    III

    416