McKay, Donald. "Front matter" Multimedia Environmental Models ...

©2001 CRC Press LLC
Basic expression N = VCk
Group C M/(C + C M)
Combined expression N = VCC Mk/(C + C M)
When C is small compared to C M, the rate reduces to VCk. When C is large compared
to C M, it reduces to VC Mk, which is independent of C, is constant, and corresponds
to the maximum, or zero-order rate. The concentration, C M, therefore corresponds
to the concentration that gives the maximum rate using the basic expression. When
C equals C M, the rate is half the maximum value. This can be (and usually is)
expressed in terms of other rate constants for describing the kinetics of the association
of the chemical with the enzyme.
The rate expression is usually written in biochemistry texts in the form
N/V = C v M/(C + k M)
where v M is a maximum rate or velocity equivalent to kC M, and k M is equivalent to
C M and is viewed as a ratio of rate constants. A somewhat similar expression, the
Monod equation, is used to describe cell growth.
If kinetics are not of the first order, it may be necessary to write the appropriate
equations and accept the increased difficulty of solution. A somewhat cunning but
unethical alternative is to guess the concentration, calculate the rate N using the
non-first-order expression, then calculate the pseudo first-order rate constant in the
expression. For example, if a reaction is second order and C is expected to be about
2 mol/m 3 , V is 100 m 3 , and the second-order rate constant, k 2, is 0.01 m 3 /mol·h,
then N equals 4 mol/h. We can set this equal to VCk; then, k is 0.02 h –1 . Essentially,
we have lumped Ck 2 as a first-order rate constant. This approach must be used, of
course, with extreme caution, because k depends on C.
6.3.3 Additivity of Rate Constants
A major advantage of forcing first-order kinetics on all reactions is that, if a
chemical is susceptible to several reactions in the same phase, with rate constants
k A, k B, k C, etc., then the total rate constant for reaction is (k A + k B + k C), i.e., the
rate constants are simply added. Another favorite trick of perverse examiners is to
inform a student that a chemical reacts by one mechanism with a half-life of 10
hours, and by another mechanism with a half-life of 20 hours, and asks for the total
half-life. The correct answer is 6.7 hours, not 30 hours. Half-lives are summed as
reciprocals, not directly.
6.3.4 Level II Reaction Algebra Using Partition Coefficients
We can now perform certain calculations describing the behavior of chemicals
in evaluative environments. The simplest is a Level II equilibrium steady-state