The Foundations of Chemistry

16-4 Concentration versus Time: The Integrated Rate Equation 671


represents the average

rate over a finite time interval t.


involves a change over an

infinitesimally short time interval dt,so

it represents the instantaneousrate.

d[A]



dt

1



a

[A]



t

1



a

aA88nproducts

the rate is expressed as

rate


For a first-order reaction, the rate is proportional to the first power of [A].


k[A]

In calculus terms, we express the change during an infinitesimally short time dtas the deriv-
ative of [A] with respect to time.

k[A]

Separating variables, we obtain

(ak)dt

We integrate this equation with limits: As the reaction progresses from time0 (the start
of the reaction) to timetelapsed, the concentration of A goes from [A] 0 , its starting value,
to [A], the concentration remaining after time t:



[A]

[A] 0

ak

t

0

dt

The result of the integration is

(ln [A]ln [A] 0 )ak(t0) or ln [A] 0 ln [A]akt

Remembering that ln(x)ln(y)ln(x/y), we obtain

ln

[

[

A

A

]

]

^0 akt (first order)

This is the integrated rate equation for a reaction that is first order in reactant A and first
order overall.
Integrated rate equations can be derived similarly from other simple rate laws. For a
reaction aA nproducts that is second order in reactant A and second order overall, we can
write the rate equation as

k[A]^2

Again, using the methods of calculus, we can separate variables, integrate, and rearrange to
obtain the corresponding integrated second-order rate equation.



[A

1

]



[A

1

] 0

akt (second order)

d[A]



adt

d[A]



[A]

d[A]



[A]

d[A]



dt

1



a

[A]



t

1



a

[A]



t

1



a