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

To compare: The practical means of part (c) and (d).

Expert Solution

Compared the practical means of part (c) and (d).

Explanation of Solution

The result from part (c) suggests that the concentration value as t tends to reduces to zero.

The result from part (d) suggests that the rate of reaction value as t tends to attains zero and no further reaction takes place.

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