Physical Chemistry , 1st ed.

It is common to express the rate of a reaction in terms of the absolute
change in the amount of one species, then scale the other rates proportionately.
This way, the numerical value of the rate of the reaction differs depending on
the species used to express the rate as well as the coefficients in the balanced
chemical reaction. One of the expressions in equation 20.4 can be rearranged
to get

rate 

d[

d

A

t

]



a

b



d[

d

B

t

]

 (20.5)

However, with this convention, the numerical values of the rates with respect
to the different species are no longer the same.

Example 20.1

Referring to equation 20.2, if the reaction proceeds at a rate of5.00 mol/min

with respect to H 2 , what are the rates with respect to O 2 and H 2 O?

Solution

Using the balanced chemical reaction and equation 20.5, the rate expressed

in terms of oxygen will be ^12 (5.00 mol/min) 2.50 mol/min. The rate ex-

pressed in terms of H 2 O will be ^22 (5.00 mol/min) or 5.00 mol/min. Notice

that we are still, by convention, expressing the rate in terms of reactants as

negative and the rate in terms of products as positive. If we wanted to express

the rate of this reaction as an invariant value, it would be 2.50 mol/min.

(The sign would depend on whether the intent is to express the disappear-

ance of reactants or the appearance of products.)

Rates of reactions can be expressed numerically and usually refer to the rate
at a specific extent of the reaction, typically at the beginning (that is,0).
This numerical rate is accurate only for that point, however. If conditions
change—as the reaction progresses or as the amounts of reactants and prod-
ucts change, or even if the same reaction is set up but with different initial con-
ditions—the numerical value of the rate is usually no longer valid. (We will
discuss an exception to this shortly.) It would be useful to determine the rate
of a reaction in a way that is more applicable to differing conditions, especially
changes in initial concentrations of reactants.
For most reactions, the initial rate is related to the initial amounts of some
or all of the reactants. Experimentally, what is found is that the initial rate is
proportional to the concentration (that is, molarity) of some or all of the re-
actants raised to some exponent. Expressing this concept mathematically, for
some general reaction “aA bB →products”:

rate [A]m[B]n (20.6)

In order to make this proportionality an equality, a proportionality constant k
is introduced:

rate k[A]m[B]n (20.7)

The proportionality constant kis the rate constantfor the reaction, and is usu-
ally independent of the exact concentrations of A and B (or any other species
whose concentration appears in the algebraic expression) but is usually de-
pendent on temperature. The exponents mand nare called orders; the way

20.2 Rates and Rate Laws 683