Introduction:
First-order reactions are those in which the reaction rate depends on the concentration of one reactant raised to the first power. These reactions are widely used in many industries, including the pharmaceutical and chemical industries. In this response, we will show that the time taken for the completion of 99.9% of a first-order reaction is ten times the time required for half change of the reaction.

Explanation:
The rate equation for a first-order reaction is given by:

Rate = k[A]

Where k is the rate constant and [A] is the concentration of the reactant.

Half-life of first-order reaction:
The half-life of a first-order reaction is defined as the time taken for the concentration of the reactant to reduce to half its initial value. The half-life of a first-order reaction is given by:

t1/2 = ln(2) / k

Where ln is the natural logarithm.

Time taken for completion of 99.9%:
The time taken for the completion of 99.9% of a first-order reaction can be calculated as follows:

ln([A]t/[A]0) = -kt

Where [A]t is the concentration of the reactant at time t, [A]0 is the initial concentration of the reactant, and k is the rate constant.

Solving for t, we get:

t = ln(0.001) / -k

Relationship between time taken for completion of 99.9% and half-life:
Substituting the expression for k from the rate equation into the expression for the time taken for completion of 99.9%, we get:

t = ln(0.001) / (-ln(2) / t1/2)

Simplifying this expression, we get:

t = 10t1/2 ln(1000)

Therefore, the time taken for the completion of 99.9% of a first-order reaction is ten times the time required for half change of the reaction.

Conclusion:
In this response, we have shown that the time taken for the completion of 99.9% of a first-order reaction is ten times the time required for half change of the reaction. This relationship is important in many industries, where the rate of reactions is a crucial parameter for the production of various products.