Reaction Rate vs. Time (section 14.4)
The average rate of a simple first-order reaction A ® B can be measured over a time interval; the relevant expression is
. (1)
The rate can also be determined at any moment when the concentration of substance A—symbolized [A]—is known using
. (2)
Neither of the above equations by itself can be used to determine the instantaneous reaction rate based on time alone. One might use a graph of the reaction rate over time, applying equation (1) over successively smaller intervals; as the size of the time interval Δt is reduced, the rate approaches the value of the instantaneous reaction rate at time t.
A graphical approach is cumbersome, however, and requires precise measurement in order to obtain an accurate value. Fortunately, simple calculus will allow us to combine equations (1) and (2) to produce equations useful in determining the reaction rate at any chosen moment. As we will not be responsible for deriving the equation (via calculus, or any other means), we can simply state the first-order concentration-time equation:
, (3)
which can be rearranged to the integrated first-order rate law:
(4)
when the concentration at a particular time is sought.
The latter form is of particular utility in illustrating the linear relationship between the natural logarithm of the reactant concentration (ln [A]) and time; equation (4) is of the form
y = mx + b, with time and ln [A] corresponding to x and y, respectively. The y -intercept (b) is the natural logarithm of the concentration at time zero—the initial concentration. The slope of the line, therefore, is the rate constant k.
graphs of [A] and ln[A] vs. time for a first-order reaction
A second-order reaction, where
does not have a linear relationship between ln [A] and time; instead, the second-order concentration-time expression (second-order integrated rate law) is
,
and a line results when is plotted vs. time.
For a zero-order reaction, the reaction rate is constant over time (independent of reaction concentration) is
The zero-order concentration-time equation (zero-order integrated rate law), indicating the concentration of the reactant at time t , is
.
We can determine, then, the order of a reaction by graphing ln [A], , and [A] vs. time; the linear graph indicates the correct reaction order.
Another measure of the reaction rate, often used in nuclear and medical applications, is the half-life of the reaction—the time required to reach half of the initial reactant concentration. The half-life is given the symbol t½ . Rearranging the first-order reaction expression in terms of time, we obtain
,
the reduction to half the initial concentration means that
[A] = ½[A]0, so
For first-order reactions, the half-life is independent of the initial concentration.
The same is not true, though, for zero- or second-order reactions. The zero-order half-life expression is derived thus:
, so
, and
.