Physical Chemistry , 1st ed.

line, as shown in Figure 20.4. Also, we can integrate equation 20.23 to get the

integrated zeroth-order rate law

[A] 0 [A]tkt (20.24)

where trepresents elapsed time.

Also, we can again determine an expression for the half-life for a zeroth-or-

der reaction. It is

t1/2

2

[A



]

k

^0 (20.25)

Note that the half-life depends on the initial amount, as expected. You need to

recognize something else about a zeroth-order reaction: after two half-lives,all

of reactant A is gone and the reaction is finished.(This assumes that the reac-

tion remains zeroth-order at all concentrations and actually goes to comple-

tion.) Thus, a straight-line plot of a zeroth-order reaction will have an

x-intercept in addition to a y-intercept.

So far, we have focused on reactions whose rate law can easily be written in

terms of a single reactant. Many reactions have rates that are dependent on

more than one concentration, and in the extreme can be very complex. The

simplest of these can be written for the two-species reaction

aA bB →products

and we can write the rate(s) as

rate ^1

a

d[

d

A

t

]^1

b

d[

d

B

t

]k[A][B] (20.26)

This reaction is first-order with respect to A and first-order with respect to B,

but is overall a second-order reaction. To determine the integrated form of this

rate law, we will have to take a double integral over concentrations of A and B.

We present only the final result:



b[A] 0 

1

a[B] 0

ln 

[

[

A

B]

]

0

0

ln 

[

[

A

B]

]

t

t

kt (20.27)

This equation can also be written as a straight-line equation (although some

rearrangement is needed), and by plotting the log of the ratio [B]t/[A]tversus

time, one can determine the rate constant kin terms of the initial amounts as

well as the coefficients of the balanced reaction.

Rate laws can get very complicated, very quickly. (Simply browse through a

textbook on kinetics to get an idea of how complex some of them are.)

Experimentally, it is difficult to determine rate laws of complicated reactions

having many concentration terms in the rate law expression unless the exper-

imenter resorts to some simplifying tactics. One such tactic is to perform the

reaction with a large excess of all but one of the reactants. Consider a reaction

that has a rate law

rate k[A]m[B]n

If the reactant B were present in very large excess with respect to A, then dur-

ing the initial course of the reaction the concentration of B does not change

very much and can be approximated as constant. Any change in the rate of the

reaction is thus related to changes in [A]. We have

rate  k [A]m [B]n

measured constant constant

20.3 Characteristics of Specific Initial Rate Laws 691

t  k

[A] 0

Slope  k

[A] 0

Time

[A]

t

Figure 20.4 For a zeroth-order reaction, a
plot of [A]tversus time gives a straight line with
a slope ofk,a y-intercept of [A] 0 , and an
x-intercept of [A] 0 /k. This is characteristic of a
zeroth-order reaction; no other order reaction
gives a straight line when plotting [A]tversus
time. This is also the only graph (for simple rate
laws) that has an x-intercept.

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