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Physical chemistry 168
the linear plot of the corresponding
rate law.
with the condition [A]=[A] 0 at t=0 gives:
ln[A]−ln[A] 0=−kt
The equation can also be written in the form:
−kt
[A]=[A] 0e
which emphasizes the fact that all first-order reactions are characterized by an
exponential decay of reactant concentration with time. The larger the value of the first
order rate constant, k, the faster the decay in time. Since the exponent must be
−1
dimensionless all first-order rate constants have units of time . A first order reaction is
identified by linearity in the plot of ln[A] against t (Table 1) and the gradient of this plot
equals −k. Many chemical reactions and other physical processes are characterized by
first order behavior (for example, radioactive decay), and many bimolecular reactions can
be made to exhibit pseudo-first order behavior by ensuring one reactant is in excess. The
advantage of first order kinetics is that the value of the rate constant can be derived from
using only a relative measure of the concentration of A with time. The absolute
concentration of A is not required.
Integrated rate law: second order reactions
The differential equation for a rate law that is second order (or pseudo-second order) in
removal of species A is:
Separating the variables and integrating:
with the condition that [A]=[A] 0 at t=0 gives:
The test for a second-order reaction is linearity in a plot of 1/[A] against t. The second-
order rate constant is equal to the gradient of this plot (Table 1).
The analysis of a second-order reaction is slightly more complicated in the general
case of a rate law that incorporates first-order removal in two separate species A and B
(of equal stoichiometry) of different initial concentration [A] 0 and [B] 0, i.e.