Physical Chemistry , 1st ed.

As defined above, a rate of reaction can be expressed in the change in amount

of reactant per some amount of time. Expressed mathematically, this is

rate 

ch

c

a

h

n

a

g

n

e

ge

in

in

am

tim

ou

e

nt



(



a

(

m

tim

ou

e

n

)

t)

 (20.1)

where the Greek letter capital delta implies “change.” If amounts were ex-

pressed in moles and time in seconds, a rate would have units of mol/s.

Moles of what?This is a necessary distinction, but one that is easy to forget.

For example, in the balanced chemical reaction

2H 2 O 2 →2H 2 O (20.2)

there are 2 moles of hydrogen reacting with every mole of oxygen to make

water. If a rate is expressed as 1.00 mol/s, are we talking about 1 mole of

hydrogen gas reacting every second, or 1 mole of oxygen gas? Because of the

stoichiometry in equation 20.2, a rate of 1 mole per second is not specific

enough to communicate what the actual rate of the reaction is.

However, also because of the stoichiometry of the balanced chemical reac-

tion, rates of reactions in terms of individual reactants and products are re-

lated. All one has to do is specify one rate in terms of a single species, and the

rate with respect to any other species in the balanced chemical reaction can be

determined. For a general chemical reaction

aA bB →cC dD

where A and B are the reactants, C and D are the products, and a,b,c, and d

are the coefficients of the balanced reaction, one can express the rate of the re-

action in terms of four different changing amounts:

rate 

1

a







t

[

im

A]

e



1

b







t

[

im

B]

e



or rate 

1

c







t

[

im

C]

e



d

1







t

[

i

D

m

]

e



(20.3)

The convention is to express rates in terms of reactants as negative, since the

concentrations of reactants are decreasing as the reaction progresses.

Conversely, rates expressed in terms of products are positive, since product

amounts are increasing as the reaction proceeds. To remind ourselves of these

facts, we write the and signs explicitly in equation 20.3. The brackets [ ]

imply amounts, usually moles or molarity (that is, concentration) units. The

coefficients a,b,c,or din the denominators are the scaling factors from the

stoichiometry of the balanced chemical equation. These allow a rate to be ex-

pressed as the same numerical value no matter which change in amount is used

to express the rate.

For infinitesimal changes, instead of expressing changes using , we should

use the differential d. The expressions in equation 20.3 become

rate 

1

a



d[

d

A

t

]



1

b



d[

d

B

t

]



or rate 

1

c



d[

d

C

t

]



d

1



d[

d

D

t

]



(20.4)

where we are using the variable tto represent time. Again, numerically, these

different ways to express the rate are the same. They simply refer to the change

in amount of different chemical species.

682 CHAPTER 20 Kinetics