#### To determine

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

#### Answer

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

#### Explanation

**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.

#### To determine

**To show:** The equation
dxdt=k(a−x)2, where
[C]=x.

#### Explanation

**Proof:**

The equation is given as below.

dxdt=k(a−x)2 (3)

Here the condition
x=[C] applies.

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

dxdt=k(a−a2ktakt+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(a−x)2 is true.