The Chemistry Maths Book, Second Edition

322 Chapter 11First-order differential equations

The general solution of the equation, after substitutingu 1 = 1 y 2 x, is therefore

The result can be checked by implicit differentiation:

and

0 Exercises 23–25

0 Exercises 26–28

11.4 Separable equations in chemical kinetics

A chemical reaction is denoted by a ‘stoichiometric’ equation of the form

aA 1 + 1 bB 1 +1-1= 1 pP 1 + 1 qQ 1 +1- (11.21)

in which the symbols A, B, =(in practice, chemical formulas) represent the chemical

species that are reacting to give the products P, Q, =The presence of the numbers

a,b, =and p,q, =in the equation means that amolecules of Aare reacting with b

molecules of B, =to give pmolecules of P, qmolecules of Q, =For example

2H

2

1 + 1 O

2

1 = 1 2H

2

O

means that two molecules of hydrogen react with one molecule of oxygen to give two

molecules of water.

In thermodynamics and in kinetics, the general chemical reaction is written

(11.22)

in which the sum is over all the species present, both reactants and products, and the

ν

B

, called stoichiometric numbers, are the numbers a, b, p, q, =in (11.21), but with

a change of sign for the reactants. In this form equation (11.21) is

01 = 1 −aA 1 − 1 bB 1 + 1 pP 1 + 1 qQ 1 +1-

= 1 ν

A

1 A 1 + 1 ν

B

1 B 1 + 1 ν

P

1 P 1 + 1 ν

Q

1 Q 1 +1-

0 =

∑

B

B

ν B

dy

dx

x

yx

y

x

x

y

y

x

xy

xy

=+

















=+=

• 
  

22

3

22

1

d

dx

y

x

d

dx

xc

y

x

dy

dx

y

x

x

2

22

2

3

2

1

















=+,(ln ) −=

y

x

xc

2

2

2

=+ln