What is the time taken to complete 75 percent of the reaction if the rate of the first-order reaction is 0.023 min-1?a)60.28 minutesb)69.28 minutesc)50.37 minutesd)65.97 minutesCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Given, the reaction is 75 complete so, a=75
Rate of the reaction k=0.023 min-1
First-order integrated rate equation, k =

Answer

Time taken to complete 75 percent of a reaction can be determined by using the concept of first-order reactions and the equation for the integrated rate law for a first-order reaction.

The integrated rate law for a first-order reaction is given by the equation:

ln[A]t = -kt + ln[A]0

Where:
[A]t = concentration of the reactant at time t
[A]0 = initial concentration of the reactant
k = rate constant of the reaction
t = time

The question states that the rate of the first-order reaction is 0.023 min-1. This rate constant (k) can be used to calculate the time taken to complete 75 percent of the reaction.

Let's assume the initial concentration of the reactant is 100%.

Step 1: Calculate the concentration at 75% completion

To find the concentration at 75% completion, we can use the equation:

[A]t = [A]0 * (1 - % completion/100)
[A]t = 100% * (1 - 75%/100)
[A]t = 100% * 0.25
[A]t = 25%

Step 2: Rearrange the integrated rate law equation to solve for time (t)

ln[A]t = -kt + ln[A]0

Since the question asks for the time taken to complete 75% of the reaction, we can substitute the values into the equation:

ln(25%) = -0.023 min-1 * t + ln(100%)

Step 3: Solve for time (t)

Using logarithmic properties, we can simplify the equation:

ln(25%) = ln(100%) - 0.023 min-1 * t

ln(25%) = ln(1) - 0.023 min-1 * t

ln(25%) = 0 - 0.023 min-1 * t

ln(25%) = -0.023 min-1 * t

Now, we can rearrange the equation to solve for time (t):

t = ln(25%) / -0.023 min-1

Using a calculator, we can determine the value of t:

t = 60.28 minutes

Therefore, the time taken to complete 75 percent of the reaction is approximately 60.28 minutes.

Answer

Given:
- Rate of the first-order reaction = 0.023 min-1
- We need to find the time taken to complete 75 percent of the reaction

Formula:
The integrated rate law for a first-order reaction is given by the equation:
ln([A]t/[A]0) = -kt

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

Explanation:
To find the time taken to complete 75 percent of the reaction, we need to find the time at which the concentration of the reactant is reduced to 25 percent of its initial concentration.

Let's assume the initial concentration of the reactant is 100 units (for simplicity).

Step 1: Calculate the half-life of the reaction
The half-life (t1/2) of a first-order reaction can be calculated using the equation:
t1/2 = (0.693/k)

Given that k = 0.023 min-1, we can substitute this value into the equation to find the half-life:
t1/2 = (0.693/0.023) = 30.13 minutes

The half-life represents the time it takes for the concentration of the reactant to reduce to half its initial value.

Step 2: Calculate the time taken to reach 25 percent of the initial concentration
Since the half-life is 30.13 minutes, it represents the time taken to reach 50 percent of the initial concentration. We need to find the time taken to reach 25 percent.

To do this, we can multiply the half-life by the natural logarithm of 2 (ln(2)) and divide it by ln(4) to find the time taken to reach 25 percent.

t25% = (t1/2 * ln(2))/ln(4)
t25% = (30.13 * 0.693)/1.386
t25% ≈ 15.07 minutes

Step 3: Calculate the time taken to complete 75 percent of the reaction
Since we are looking for the time taken to complete 75 percent of the reaction, we can subtract the time taken to reach 25 percent from the total time taken to complete the reaction.

t75% = total reaction time - t25%
t75% = 30.13 - 15.07
t75% ≈ 15.06 minutes

Therefore, the time taken to complete 75 percent of the reaction is approximately 15.06 minutes.

Conclusion:
The correct answer is option A) 15.06 minutes.

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