   Chapter 10.4, Problem 4E ### 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

# Reaction rates The velocity v of an autocatalytic reaction can be represented by the equation v = x ( a   —   x ) where a is the amount of material originally present and X is the amount that has been decomposed at any given time. Find the maximum velocity of the reaction.

To determine

To calculate: The maximum velocity of the reaction.

Explanation

Given Information:

Velocity of a reaction is represented as v=x(ax)

Where, a is the original amount of material and x is the amount of decomposed material at any time.

Formula used:

To find the maximum value, first calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is less than zero, then substituting the value of the independent variable will give the maximum value of the equation.

The product rule is used when the function is a product of two differentiable terms. If p(x)=f(x)g(x), then the derivative is

p'(x)=f'(x)g(x)+f(x)g'(x)

Calculation:

As given, velocity of a reaction is represented as v=x(ax)

Where, a is the original amount of material and x is the amount of decomposed material at any time

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