   Chapter 12.5, Problem 38E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Bimolecular chemical reactions A bimolecular chemical reaction is one in which two chemicals react to form another substance. Suppose that one molecule of each of the two chemicals reacts to form two molecules of a new substance. If x represents the number of molecules of the new substance at time t, then the rate of change of x is proportional to the product of the numbers of molecules of the original chemicals available to be converted. That is, if each of the chemicals initially contained A molecules, then d x d t = k ( A − x ) 2 where k is a constant. If 40% of the initial amount A is converted after 1 hour, how long will it be before 90 % is converted?

To determine

To calculate: The time that will it be before 90% is converted, if 40% of the initially amount A is converted after 1 hours.

Explanation

Given Information:

A biomolecular chemical reaction is one in which two chemical reacts to form another substance.

The one molecule of each of the two chemical reacts to form two molecule of a new substance.

If x represent the number of molecule of the new substance at time t, then the rate of change of x is proportional to the product of the number of molecule of the original chemicals available to be converted.

If the each of chemically initially contained A molecules, then

dxdt=k(Ax)2

Where k is constant, if 40% of the initially amount A is converted after 1 hours.

Formula used:

The differential equation can be equivalently expressed in the form

g(y)dy=f(x)dx

Then the equation is separable.

The solution of a separable differential is obtained by integrating both sides of the equation after the variable have been separated.

Calculation:

Consider a biomolecular chemical reaction is one in which two chemical reacts to form another substance.

The one molecule of each of the two chemical reacts to form two molecule of a new substance.

If x represent the number of molecule of the new substance at time t, then the rate of change of x is proportional to the product of the number of molecule of the original chemicals available to be converted.

If the each of chemically initially contained A molecules, then

dxdt=k(Ax)2

Where k is constant, if 40% of the initially amount A is converted after 1 hours.

Firstly, separate the above differential equation as:

dx(Ax)2=kdt

Integrate the provided function as:

dx(Ax)2=kdt(Ax)11=kt+C1(Ax)1=kt+C

At the given condition t=0

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