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Kinetics 199
reaction is first order.” In some cases, the order comes indirectly from the problem,
for example, “the rate law is Rate k[A] 2 , which means the reaction is
second order.”
In most problems, it will be to your advantage to find and label as many of the
following terms as possible: [A] 0, [A] t, k, t, and t 1/2. You will not need all these
terms for all problems. However, the ones you have, or do not have, often direct
you towards the solution.
Let’s apply the preceding two paragraphs to an example problem. The firstorder
decomposition of gaseous dinitrogen pentoxide, N 2O 5, to nitrogen dioxide,
NO 2, and oxygen, O 2, has a rate constant of 4.9 10 4 s 1 at a certain temperature.
Calculate the half-life of this reaction.
This is a first-order reaction. The following relationships apply to first-order
reactions:
Rate k[A] ln [A] 0
kt t1/2
[A] t
0.693
k
These are the only equations we have exclusive to first-order reactions. For this
reason, one or more of these equations will be necessary to work the problem.
(In this problem, A N 2O 5.)
Using the list of terms ([A] 0, [A] t, k, t, and t1/2), and the given information, we
find that k = 4.9 104 s1 and t1/2 ? (We are using a “?” to indicate the term
we are seeking.) The only first-order relationship we have that relates k and t1/2 0.693
is t1/2 k . This significantly limits our options to finishing the problem. To finish
the problem, we need to enter k into the relationship:
t1/2 0.693
k
0.693
4.9 104 s1 1414.2857 1.4 103 s
Let’s try another example. A substance undergoes a simple decomposition. The
reaction is first order. In one experiment, at 25°C, the concentration of this substance
decreased from 1.000 M to 0.355 M after 4.25 min. What was the rate constant
for this reaction?
In this case, as in the preceding example, we have a first-order reaction. This limits
us to the same set of three relationships as in the earlier example. Using the list
of terms ([A] 0, [A] t, k, t, and t1/2), and the given information, we find that k ?,
[A] 0 1.000 M, [A] t 0.355 M, and t 4.25 min. (We also know T 25°C, but
since T is not one of the five terms, this is irrelevant.). The only first-order
[A]
0
relationship with these four terms is ln [A] kt. We can either enter the given values
t
into this equation, or we can rearrange the equation before entering the values.