Lecture 18

Derivation of simple integrated rate equations


Introduction


     As with many physicochemical principles, the rules governing rate of reaction were first established empirically, and later subject to extensive theoretical analysis, leading to our present understanding of the controlling factors. The empirical observations centered around establishment of reaction conditions under which the rate could be measured as a function of concentration of reactants. It was found that the rate was related to concentration in a predictable way, leading to the concept of the rate constant, and equations for simple reactions that fell into several classes:
 
  1. 1st-order reactions
  2. 2nd-order reactions of Class I
  3. 2nd-order reactions of Class II
  4. Higher-order reactions
  5. Zero-order reactions
     Before deriving of the equations describing these processes, a brief discussion of order of reaction is necessary.

Reaction equation, stoichiometric coefficients, and order of reaction


     Reactions are described by reaction equations of the sort already discussed at length in earlier lectures. The general properties of the reaction equation are:
For example:
glucose + ATP glucose-6-P + ADP

glucose + 2ADP + 2 Pi 2lactate + 2ATP

In general:

aA + bB pP

     The stoichiometric coefficients are important in the context of this kinetic discussion, because they determine the order of reaction. This term describes how the rate of the reaction depends on the concentrations of the species involved. In general, the rate of a reaction, v, is described by an equation such as the following:
v = k[A]a[B]b[P]p

where k is the rate constant, A and B are reactants, and P is the product, with stoichiometric coefficients a, b, p, respectively. Then the overall order of reaction is given by the sum of the stoichiometric coefficients:

order of reaction = a + b + p + .....n
where n indicates the possible involvement of other species not included in the general mechanism above.

     However, use of the term "overall order of reaction" is a little muddled, because it is generally recognized as an empirical term, so that the value found by experiment might not correspond to that expected by applying the above general rule. Most frequently, for simple chemical processes, the order of reaction found experimentally turns out to be equal to the sum of the stoichiometric coefficients of the reactants. This is generally the case for reactions with a large Keq, starting from the condition of [P] = 0, since [P] will then be negligible during the time of measurement, and the reverse reaction will not be significant. Then, for the general reaction above:

v = k[A]a[B]b

If a and b are both 1, then the overall order of reaction will be 2nd-order (1 + 1 = 2).

For most biochemical processes, enzyme catalysis and the saturation effects resulting from this, determine that the steady-state reaction does not obey the simple rules. However, if the pre-steady-state kinetics are measured, in which the enzyme is considered as a reactant, then the simple rate laws pertain. We will examine this case separately in a later lecture.
     While the overall order of reaction is described as above, a second term is also often used, - the order of reaction with respect to a particular species. For example, in a reaction involving 2A, the reaction is said to be 2nd.-order in A. The order is given simply by the stoichiometric ratio. From this it can be seen that measurement of the order of reaction can provide a value for the coefficient if this is otherwise unknown. A useful protocol for determining the order of reaction with respect to a particular component is to measure the concentration dependence of rate when all other reactants are in great excess. Under these circumstances, their concentrations will not vary significantly during the reaction, and the rate law revealed by experiment will give the order of reaction with respect to the tested component:

v = k[A]aexcess[B]b

v = k'[B]b

k' = k[A]aexcess

Relation between rate constants and the equilibrium constant

An important relation between the forward and reverse rate constants for a process, the equilibrium constant, and DGo' is covered elsewhere.

Derivation of the rate equations