Answer :
Final answer:
To calculate the time for a first-order reaction to be 60% completed, the rate constant k is determined from the given 15% completion time. Using this rate constant, one can find the time for 60% completion, which will be approximately 100 minutes.
Explanation:
A first-order reaction's completion time for a certain percentage can be determined using the formula t = -(1/k) ln(1 - fraction completed), where k is the rate constant of the reaction and the fraction completed is the percentage of reaction in its decimal form (e.g., 15% is 0.15). To find the time when 60% of the reaction is completed (0.60 in decimal form), we need to first find the rate constant k using the given 15% completion time.
Using the 15% data, we solve for k:
t = -(1/k) ln(1 - 0.15), and with t = 20 minutes, we get:
20 = -(1/k) ln(0.85)
k = -1 / (20 * ln(0.85))
Now that we have k, we can find the time when 60% of the reaction is complete by plugging in the values:
t = -(1/k) ln(0.40)
After calculating t using the above steps with the determined value of k, you'll find the answer closest to the options provided (a) 60 min, (b) 80 min, (c) 100 min, (d) 120 min. The calculation will show that the correct answer is option (c) 100 minutes for the reaction to be 60% completed.
