1550078481-Ordinary_Differential_Equations__Roberts_

160 Ordinary Differential Equations

EXERCISES 3.8

1. Assume that the law of mass action a pplies to the chemical reaction

A+ B ~ P. Suppose the initial concentrations are CA (0) = 6 moles/liter

and CB(O) = 2 moles/liter and that the velocity constant is k =

1 liter/(mole·sec).

a. Find and graph the number of moles per liter , y(t), which have

reacted at time t.

b. What is the limiting concentration of the product P?

c. What is the limiting concentration of the reactant B?

2. The chemical reaction A + B ~ P satisfies the law of mass action.

Suppose CA(O) = CB(O) = 5 moles/liter and k = 1.5 liters/(mole·sec).

a. Find and graph the number of moles per liter, y(t), which have

reacted at time t.

b. What is limt_,= y(t)? limt_,= CA(t)? limt-+<XJ CB(t)?

3. The law of mass action appli es to the chemical reaction A+ B + C ~ P.

If y(t) is the number of moles per liter which h ave reacted at time t ;

if CA(t), CB(t), and Cc(t) are the concentrations of substances A, B,

and C respectively at time t; if CA(O), CB(O), and Cc(O) are the initial

concentrations; and if k is the velocity constant; then y(t) satisfies the

initial value problem

y' = kCA(t)CB(t)Cc(t) = k(CA(O)-y)(CB(O)-y)(Cc(O)-y); y(O) = 0.

Suppose k = .5 liter/(mole-sec), CA(O) = 8 moles/liter, CB(O)

5 moles/liter , and Cc(O) = 3 moles/liter.

a. Find and graph y(t) on the interval [O, 5].

b. Find limt-+= y(t).

c. When is y(t) = .95 x limt-= y(t)?

4. The chemical reaction A + B + C ~ P satisfies the law of mass action.

Find and graph y(t) on [O, 5], if CA (0) = CB (0) = Cc(O) = 6 moles/liter

and k = .3 liter/(mole·sec). (See exercise 3.)

5. The chemical reaction 2A + B ~ 2P + Q + R obeys the law of
   mass action. Let y(t) be the number of moles per liter of reactant B
   which h ave reacted at time t, if CA(t) and CB(t) are the concentrations
   of substances A and B at time t , if CA (0) and CB (0) a re the initial