Introduction:
In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The equation for a first-order reaction is given by:

Rate = k[A]

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

Given:
Time required for 75% completion (t) = 9 hours

Calculating the Half-Life Period:
The half-life period is the time taken for the reactant concentration to reduce to half its initial value.

Step 1: Calculate the Reaction Constant (k)
First, we need to find the reaction constant (k) using the given information.

We know that the reaction is first order, so the integrated rate equation for a first-order reaction is:

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

Where [A]t is the concentration at time t and [A]0 is the initial concentration.

Given that the reaction is 75% complete, [A]t/[A]0 = 0.75 and t = 9 hours.

ln(0.75) = -9k

Solving for k:

k = -ln(0.75)/9

Step 2: Calculate the Half-Life Period (t1/2)
The half-life period (t1/2) can be calculated using the equation:

t1/2 = ln(2)/k

Substituting the value of k we calculated earlier:

t1/2 = ln(2)/(-ln(0.75)/9)

Simplifying the expression:

t1/2 = ln(2) * 9 / ln(0.75)

Result:
The half-life period (t1/2) for the given first-order reaction is the calculated value obtained from the equation above.

Conclusion:
In this response, we have calculated the half-life period for a first-order reaction when the time required for 75% completion is given. The half-life period is an important characteristic of a reaction and represents the time taken for the reactant concentration to reduce to half its initial value.

4.5 hour