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CalculusCalculus: Early TranscendentalsIn an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: d [ C ] d t = k [ A ] [ B ] (See Example 3.7.4.) Thus, if the initial concentrations are [A] = a moles/L and [B] = b moles/L and we write x = [C], then we have d x d t = k ( a − x ) ( b − x ) (a) Assuming that a ≠ b , find x as a function of t . Use the fact that the initial concentration of C is 0. (b) Find x ( t ) assuming that a = b. How does this expression for x ( t ) simplify if it is known that [ C ] = 1 2 a after 20 seconds?BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 9.3, Problem 40E

Textbook Problem

In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B:

(See Example 3.7.4.) Thus, if the initial concentrations are [A] = *a* moles/L and [B] = *b* moles/L and we write *x =* [C], then we have

(a) Assuming that *a* ≠ *b*, find *x* as a function of *t*. Use the fact that the initial concentration of C is 0.

(b) Find *x*(*t*) assuming that *a = b.* How does this expression for *x*(*t*) simplify if it is known that

Expert Solution

(a)

To determine

**To find:** The function *x*(*t*) by solving the initial value problem:

**Given:**

In a chemical reaction the molecules A and B are reactants. The product of reactant is *C*.

The concentration of A and B is given below,

The initial concentration is given below,

[A] = *a* moles/L and [B] = *b* moles/L then *x =* [C]

**Formula used:**

Partial fraction of

**Calculation:**

Consider the differential equation,

Simplify the above equation as follows.

Here,

Take integral on both sides,

By using partial fraction of

Substitute

Substitute

Substitute *A* and *B* value in the obtained solution,

Substitute

Expert Solution

(b)

To determine

**To find:** The function *x*(*t*) by solving the initial value problem:

Calculus: Early Transcendentals

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