Two chemical species and in a well–stirred reactor can reversibly transform into one…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Two chemical species

X

and

Y

in a well–stirred reactor can reversibly transform into one another. Experimental chemists report the following verbal observations. • “The instantaneous rate at which

X

is consumed is directly proportional to the current amount of

Y

present.” • “The instantaneous rate at which

Y

is consumed is directly proportional to the current amount of

X

present.” Let

x(t)

and

y(t)

denote the masses (in grams) of

X

and

Y

at time

t

minutes, respectively. The proportionality constants are denoted

k_1>0

and

k_2>0

. Answer the following questions in the context of this reaction. (a) Translate the chemists’ verbal observations into a system of two first–order differential equations involving

x(t)

and

y(t)

. (b) Give a physical interpretation of the signs of

k_1

and

k_2

in the system you wrote in part (a). (c) Suppose

k_1>k_2

. Assume

x(0)=x_0>0

and

y(0)=y_0>0

. Without explicitly solving the system, explain which species you expect to dominate (have the larger mass) after a long time and justify your reasoning using the structure of the equations.

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