Page 79 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:36 9in x 6in b985-ch07 Elementary Physical Chemistry

64 Elementary Physical Chemistry

7.5. Integrated Rate Law

Rate laws are differential equations which must be integrated to obtain
concentrations as a function of time. The integrated rate laws are directly
related to experimental observables of concentration and time.

7.5.1. First-Order Reaction
Consider the first-order reaction
r = −(1/a)d[A]/dt = k[A] 1 (7.14)

The rate of consumption of A is
−d[A]/dt = ak[A]= k A[A] (7.15)

Note that it is k A = ak and not k that enters the expression for the
integrated rate,

−d[A]/[A]= k A dt (7.16)

Integration between t = 0, when the concentration is [A] 0 and the time t,
when the value is [A], gives

A t
d[A]/[A]= − k A dt (7.17)
A 0 0
Note that dx/x =d ln x and so Eq. (7.17) becomes
ln [A]/[A] 0 = −k A t (7.18a)
or

ln [A] 0/[A]= k At (7.18b)
[A]= [A] 0e −kt (7.18c)

Comment: An important class of first-order reactions are nuclear
reactions.

In a first-order reaction, a plot of ln [A]vs. t gives a straight line
(Fig. 7.1), whose slope is −k A . Conversely, if the order of a reaction is not
known, and a plot of ln[A]vs. t gives a straight line, this indicates that the
reaction is first order.