concentration unit.*) Therefore, a plot of ln[A]t—the natural logarithm of the
amount of A at various times—versus the time will give a straight line, as seen
in Figure 20.2. This line will have a slope ofk, the negative of the rate con-
stant, with ln[A] 0 , the logarithm of the original amount, as the y-intercept. A
straight-line plot of ln[A]tversus twill be produced only if the reaction is
indeed first-order with respect to A.
Finally, there is a very popular concept connected to first-order reactions:
half-life. The half-lifeof a first-order reaction is the amount of time necessary
for half of the original amount to react. We can use equation 20.14 to derive a
simple expression for the half-life,t1/2:ln
1
5
0
0
0
%
%
kt1/2
t1/2ln
k^2 0.6
k(^931) (20.17)
Notice that t1/2is independent of the original amount [A] 0! It is related only
to the first-order rate constant of the reaction. Because all natural radioactive
processes are first-order processes, the concept of half-life is a common one.
Not all reactions are first-order. A second-order reactionis defined by the
rate law
d[
d
A
t]
k[A]^2 (20.18)We can do the same thing for this equation as we did for the first-order rate
law: rearrange the variables in A on one side and the variables in time on the
other side:d
[A[A
]^2
]kdt (20.19)Again, integrating both sides of the equation between the final and initial lim-
its, and presuming again that we start at some initial time ti0 so that trep-
resents elapsed time, we get for the integrated second-order rate law
[A1
]t[A
1
] 0
kt (20.20)As the concentration of the reactant species A changes with time, its concen-
tration fits the above equation (as long as the reaction follows second-order ki-
netics with respect to the species A). We can rearrange equation 20.20 into a
form that mimics a straight-line equation:
[A1
]tkt
[A1
] 0
(20.21)
where now yis 1/[A]t,xis time again, the slope mis given by the rate constant
k, and the y-intercept bis represented by 1/[A] 0 , the reciprocal of the initial
amount of species A. Figure 20.3 shows how a plot of a second-order reaction688 CHAPTER 20 Kinetics
Slope kln[A] 0Timeln[A]tFigure 20.2 For a first-order reaction, a plot
of the natural logarithm of [A]tversus time gives
a straight line whose slope is kand y-intercept
is ln [A] 0 , the logarithm of the initial amount.
This is characteristic of a first-order reaction; no
other order reaction gives a straight line when
plotting ln [A]tversus time.
*Equation 20.16 can be written asln con[cAn]tunitln con[cAn]unit^0 kt
as a way of addressing the units issue. This chapter is simply avoiding overcomplicating the
equations.1
[A] 0(^1) [A]t Slope k
Time
Figure 20.3 For a second-order reaction, a
plot of the reciprocal of [A]t, 1/[A]t, versus
time gives a straight line whose slope is kand
y-intercept is 1/[A] 0 , the reciprocal of the initial
amount. This is characteristic of a second-order
reaction; no other order reaction gives a straight
line when plotting 1/[A]tversus time.