1550078481-Ordinary_Differential_Equations__Roberts_

158 Ordinary Differential Equations

If we let CA(t), CB(t), Cc(t), ... represent the concentrations of substances
A , B , C, ... at time t expressed in moles per liter and if y is the number of
moles per liter which have reacted at time t, then the law of mass action states
that the velocity of reaction, dy/dt, satisfies the differential equation

(1)

where the constant of proportionality k is called the velocity constant. The
sum of the exponents a + b + c + · · · is called the order of the reaction.

I EXAMPLE 1 An Order 2 Chemical Reaction

Consider the order 2 reaction

A+B-.P

Suppose two reactants, A and B, are combined in solution and have an initial

concentration CA(O) = 7 moles/liter and CB(O) = 3 moles/ liter. Assume the

law of mass action applies and the velo city constant is k = 2 liters/(mole·sec).

a. Numerically solve equation (1) for the number of moles per liter which

have reacted at time t , y(t). (In this case, a= b = 1.)

b. What is the limiting concentration of the product P? (That is, what

is limt_, 00 y(t)?)

c. How long does it take to produce one-half of the limiting concentra-

tion? 90% of the limiting concentration?

d. What is the concentration of substances A and B at time t? That is,

what is CA(t) and CB(t)?

e. What is limt-.oo CA(t) and limt-+oo CB(t)?

SOLUTION

Let y(t) denote the number of moles/liter of the substance Pat time t. The

concentration of substance A at time tis CA (t) = CA(O) -y(t) = 7 -y(t) and

the concentration of substance Bat time tis CB (t) = CB(O) -y(t) = 3 -y(t).
Since the concentrations CA and CB are positive and can only decrease as the
reaction proceeds, 0 :::; CA(t) :::; 7 and 0 :::; CB(t) :::; 3. Thus, we must have
0 :::; 7 - y :::; 7 and 0 :::; 3 - y :::; 3. So we must have both 0 :::; y :::; 7 and
0 :::; y :::; 3 which implies 0 :::; y :::; 3. According to the law of mass action

dy

dt = kCA(t)CB(t) = 2(7 -y)(3 -y).

At time t = 0, we have y(t) = 0, since no reaction has taken place. We ran

SOLVEIVP by setting f(x, y) = 2(7 - y)(3 - y), by inputing the interval of

integration as [O, l], and by inputing the initial condition y(O) = 0.