Rate Constant of Decomposition:
To calculate the rate constant of decomposition, we can use the equation for the first-order reaction:

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, k is the rate constant, and t is the time.

Using the given data, we can calculate the rate constant as follows:

ln(69.13/100) = -k * 60

Simplifying the equation, we have:

ln(0.6913) = -k * 60

Taking the natural logarithm of both sides, we get:

-0.371 = -k * 60

Dividing both sides by -60, we find:

k = 0.00618 min^-1

Therefore, the rate constant of decomposition is 0.00618 min^-1.

Time for Half Decomposition:
To determine the time required for half of the ester to decompose, we can use the first-order integrated rate equation:

[A]t = [A]0 * e^(-kt)

where [A]t is the concentration of the reactant at time t, [A]0 is the initial concentration of the reactant, k is the rate constant, and e is the base of the natural logarithm.

We can rearrange the equation to solve for t:

[A]t = [A]0/2

[A]0 * e^(-kt) = [A]0/2

e^(-kt) = 1/2

Taking the natural logarithm of both sides, we have:

-kt = ln(1/2)

Simplifying the equation, we find:

t = ln(2)/k

Substituting the value of k we calculated earlier (0.00618 min^-1), we can determine the time required for half of the ester to decompose:

t = ln(2)/0.00618

t ≈ 112.16 minutes

Therefore, it would take approximately 112.16 minutes for half of the ester to decompose.

Explanation:
The hydrolysis of propyl acetate in the presence of dilute HCl in aqueous solution is a first-order reaction. The rate of decomposition of the ester depends only on the concentration of the ester itself. The given data allows us to determine the rate constant of decomposition and the time required for half of the ester to decompose.

By using the first-order rate equation and the given data, we can calculate the rate constant by comparing the concentration of the ester at a specific time to its initial concentration. Taking the natural logarithm of the ratio of concentrations allows us to solve for the rate constant.

Similarly, to determine the time required for half of the ester to decompose, we use the integrated rate equation for a first-order reaction. By setting the concentration of the ester at a specific time equal to half of its initial concentration and rearranging the equation, we can solve for the time needed for half decomposition.

These calculations provide valuable information about the reaction rate and the time required for a significant amount of the ester to decompose.