# The rate of reaction at time t . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.8, Problem 22E

(a)

To determine

## To find: The rate of reaction at time t.

Expert Solution

The rate of reaction with respect to time d[C]dt is a2k(akt+1)2.

### Explanation of Solution

Given:

The expression is given as below.

[C]=a2ktakt+1 (1)

Here, the variable [C] is the product, and a is the initial concentration of [A]=[B] in moles/L.

Calculation:

Differentiate equation (1) with respect t.

d[C]dt=[(akt+1)(a2k)](a2kt)(ak)(akt+1)2=(a3k2t+a2k)(a3k2t)(akt+1)2

d[C]dt=a2k(akt+1)2 (2)

Hence, the rate of reaction with respect to time d[C]dt is a2k(akt+1)2.

(b)

To determine

Expert Solution

### Explanation of Solution

Proof:

The equation is given as below.

dxdt=k(ax)2 (3)

Here the condition x=[C] applies.

Substitute the value of a2ktakt+1 for x from equation (1) in (3).

dxdt=k(aa2ktakt+1)2=k(a(akt+1)(a2kt)akt+1)2=k((a2kt+a)(a2kt)akt+1)2

dxdt=k(aakt+1)2

dxdt=ka2(akt+1)2 (4)

Compare equation (2) and (4).

Thus, the condition dxdt=k(ax)2 is true.

(c)

To determine

### To show: The concentration if time t tends to ∞.

Expert Solution

The concentration as t tends to is a.

### Explanation of Solution

Determine the concentration as t tends to .

Substitute t tends to (t) in equation (1).

limt[C]=limta2ktakt+1=a2k()ak()+=a2kak×11=a

Therefore, the concentration value as t tends to is a.

(d)

To determine

### To find: The rate of reaction if time t tends to ∞.

Expert Solution

The rate of reaction as t tends to is 0.

### Explanation of Solution

Determine the rate of reaction as t tends to .

Substitute t tends to (t) in equation (2).

limtd[C]dt=limta2k(akt+1)2=a2k(ak()+)2=a2k=0

Therefore, the rate of reaction as t tends to is 0.

(e)

To determine

Expert Solution