324 Chapter 11First-order differential equations
and for the H
22 O
2reaction,
The mechanism of a chemical reaction is in general complex, proceeding through
several elementary kinetic steps, that may be consecutive, competitive, or both, and it is
the task of the kineticist to unravel the mechanism and to relate the rate of the overall
chemical reaction to the rates of the elementary steps. It is found by experiment that
the overall rate may depend on the amounts of all the chemical species present
at any time, both reactant and product, as well as on the temperature, the pressure,
and the nature of the container. We consider here only the very simplest reactions,
those whose rates depend only on the amounts of the reactants; we represent such a
reaction by
aA 1 + 1 bB 1 +1-1→products (11.29)
It is found from experiment that the rate of the reaction has the form
v 1 = 1 k[A]
α[B]
β= (11.30)
where kis called the rate constantof the reaction, and the numbers α, β, =, define the
orderof the reaction. The exponent αis called the order with respect to reactant A,
βis the order with respect to B, =, and the sumα 1 + 1 β 1 +1-is called the order of the
reaction. The cases considered are
(1)A 1 → 1 products first order
(2)2A 1 → 1 products second order
(3)A 1 + 1 B 1 → 1 products second order
(1) The first-order process: A 1 → 1 products
The rate equationis
(11.31)
or, if the concentration [A]is represented by the variable x,
(11.32)
This is the differential equation discussed in Examples 3.20 and 11.1, and the solution
is that given in Example 11.4(ii). Thus, separating the variables and integrating,
ln 1 x 1 = 1 −kt 1 + 1 c or ln[A] 1 = 1 −kt 1 + 1 c (11.33)
dx
x
=−kdt,
dx
dt
=−kx
v=− =
d
dt
k
[]
[]
A
A
v=− =− =
1
2
1
2
22 2d
dt
d
dt
d
dt
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