The Foundations of Chemistry

CONCENTRATION VERSUS TIME:

THE INTEGRATED RATE EQUATION

Often we want to know the concentration of a reactant that would remain after some

specified time, or how long it would take for some amount of the reactants to be used up.

The equation that relates concentrationand timeis the integrated rate equation.

We can also use it to calculate the half-life, t1/2,of a reactant—the time it takes

for half of that reactant to be converted into product. The integrated rate equation

and the half-life are different for reactions of different order.

We will look at relationships for some simple cases. If you know calculus, you may be

interested in the derivation of the integrated rate equations. This development is presented

in the Enrichments at the end of this section.

First-Order Reactions

For reactions involving aA n products that are first order inA and first order overall,the

integrated rate equation is

ln
akt (first order)

[A] 0 is the initial concentration of reactant A, and [A] is its concentration at some time,

t,after the reaction begins. Solving this relationship for tgives

t ln


By definition, [A]^12 [A] 0 at tt1/2. Thus

t1/2 ln  ln 2

t1/2 (first order)

This relates the half-life of a reactant in a first-order reactionand its rate constant, k.In

such reactions, the half-life does not dependon the initial concentration of A. This is not

true for reactions having overall orders other than first order.

EXAMPLE 16-5 Half-Life: First-Order Reaction

Compound A decomposes to form B and C in a reaction that is first order with respect to A

and first order overall. At 25°C, the specific rate constant for the reaction is 0.0450 s^1. What

is the half-life of A at 25°C?

0.693



ak

ln 2



ak

1



ak

[A] 0

 1

2 [A] 0

1



ak

[A] 0



[A]

1



ak

[A] 0



[A]

16-4

arepresents the coefficient of reactant
A in the balanced overall equation.

664 CHAPTER 16: Chemical Kinetics

See the Saunders Interactive
General Chemistry CD-ROM,
Screen 15.6, Concentration-Time
Relationships.

Nuclear decay (Chapter 26) is a very
important first-order process. Exercises
at the end of that chapter involve
calculations of nuclear decay rates.

When time t1/2has elapsed, half of the
original [A] 0 has reacted, so half of it
remains.