BioPHYSICAL chemistry

The rate of a reaction involving a molecule A, which at any time thas
the concentration A(t), is the change in the concentration or population
of the molecule with time:

(7.2)

For this simple reaction, the rate can be expressed in terms of either the
change in [A] with respect to time or the change in [B] with time. Since
the only source of molecule B is from those molecules of A that convert
to molecule B, the rates at which molecules A and B change with time
must be equal, but with opposite signs:

(7.3)

Rates are typically concentration-, and therefore time-,
dependent. In such cases there is no single rate for
a reaction. To compare rates it would be necessary
to specify all concentrations of each component. For
example, consider the rate at which gumballs come out
of a small hole in a container (Figure 7.1). If the con-
tainer has many gumballs in it, then many gumballs
are likely to hit the hole and pop out in any given
period of time. However, if the number of gumballs
is small, then the rate at which the gumballs will come
out is slower. For simplicity, rather than try to use rates
that are dependent upon the specific concentrations,
parameters are determined that describe the reaction
kinetics but which are concentration-independent.
For the case of the irreversible reaction (eqn 7.1), to relate A(t) to a
rate constant we need to make the fundamental assumption that all of
the A molecules have the same probability of converting to B in a given
period of time regardless of their history. If this is true, then the instan-
taneous probability that any particular molecule of A will change to B is
independent of time. The instantaneous probability is a useful constant
for characterizing irreversible first-order reactions. The fractional change in
the concentration of A(t), dA(t)/A(t), is equal to the instantaneous probability
of A converting to B at a ratek, multiplied by the time inter 9 al dt:

or

(7.4)

dA

d

A

()

()

t

t

=−kt

dA

A

d

()

()

t

t

=−kt

dA

d

dB

d

()t ()

t

t

t

=−

Rate

dA

d

()

=

t

t

CHAPTER 7 KINETICS AND ENZYMES 135

Figure 7.1The

number of gumballs

leaving the dispenser

in a period of time is

dependent on the

total number of

gumballs.