Physical Chemistry , 1st ed.

A first-order reactionis one whose kinetics follow a rate law where the order

is 1; that is,



d[

d

A

t

]

k[A]^1 (20.11)

All spontaneous radioactive processes and many chemical processes (including

many metabolic processes in the body) are first-order reactions. We can re-

arrange equation 20.11 to isolate all terms in the species A on one side, and all

other variables on the other:

d

[

[

A

A

]

]kdt (20.12)

In this equation, we are leaving out the 1 exponent, since it is understood to

be present. If we presume that at some initial time ti, the amount of A is rep-

resented as [A]i, and at some final time tfthe amount of A is represented as

[A]f, we can integrate both sides of the above equation between the limits of

the respective variable:



[A]f

[A]i

d

[

[

A

A

]

]

tf

ti

kdt

The integral on the left side is the natural logarithm of A, evaluated between

the two limits. The integral on the right side is simply the variable t(time)

evaluated between its two limits. After performing the evaluations on either

side, we get

ln 

[

[

A

A

]

]

f

i

k(tfti)

The leading negative sign on the right side is usually removed by switching the

numerator and denominator in the logarithm term. (This is a mathematical

property of logarithms.)

ln 

[

[

A

A

]

]

f

ik(t

fti)

The minus signs cancel to yield

ln 

[

[

A

A

]

]

f

ik(tfti) (20.13)

Typically when measuring a process the initial time is set to 0 and the final time

is simply the elapsed time. We will use the symbol [A] 0 to indicate the initial

amount of A at time 0 and [A]tto stand for the amount of A at some time t.

Using these conventions, equation 20.13 is commonly written as

ln 

[

[

A

A

]

]

0

t

kt (20.14)

where trepresents the elapsed time. This is the basic equation that relates

how the concentration of A varies over time for a first-order reaction.

Equation 20.14 is known as the integrated form of a first-order rate law(or

more simply, the integrated rate law). Note that [A] 0 and [A]tmusthave the

same units (and can even be percentages, where [A] 0 is usually considered

100%, that is, whatever the original amount was). Also,kand tmust have the

same units of time.

686 CHAPTER 20 Kinetics