A radioactive substance has a half life of 4 months. After how many months will three-fourths of the original sample have decayed?

Introduction / Context:
This numerical chemistry question checks your understanding of the basic idea of half life in radioactive decay. Half life problems are very common in nuclear chemistry, physics and environmental science because they help estimate how long a radioactive material remains significantly active.

Given Data / Assumptions:

• Half life of the radioactive substance, t(1/2) = 4 months.
• We want the time required for three-fourths (3/4) of the original sample to decay.
• This means only one-fourth (1/4) of the original sample will remain undecayed.
• Radioactive decay follows first order kinetics and the definition of half life applies repeatedly.

Concept / Approach:
Half life is defined as the time required for half of the radioactive nuclei in a sample to decay. After one half life, the activity or amount becomes 1/2 of the initial value. After two half lives, it becomes 1/2 * 1/2 = 1/4 of the original amount. By expressing the desired remaining fraction in terms of powers of 1/2, we can easily count how many half lives have passed and then multiply by the given half life time to get the total elapsed time.

Step-by-Step Solution:
1) Let the initial amount of the radioactive substance be N0.2) After one half life (4 months), the amount left is N0 / 2.3) After two half lives, the remaining amount is (N0 / 2) / 2 = N0 / 4.4) N0 / 4 corresponds to one-fourth of the original amount, which means three-fourths have decayed.5) Two half lives correspond to 2 * 4 months = 8 months.

Verification / Alternative check:
We can verify with the general formula for first order decay: N = N0 * (1/2)^(t / t(1/2)). We want N = N0 / 4, so N0 / 4 = N0 * (1/2)^(t / 4). Cancelling N0 and writing 1/4 as (1/2)^2 gives (1/2)^2 = (1/2)^(t / 4). This implies t / 4 = 2, so t = 8 months, which matches the earlier reasoning from repeated half lives.

Why Other Options Are Wrong:
3 months: This is less than one half life, so more than half of the substance would still remain, not just one fourth.
4 months: After one half life only 1/2 of the substance has decayed, not 3/4, so this is insufficient time.
12 months: This corresponds to three half lives (12 / 4 = 3), at which point the remaining amount would be N0 / 8, meaning 7/8 decayed, more than required.
16 months: This is four half lives, leaving N0 / 16 remaining, so far more than three-fourths would have decayed.

Common Pitfalls:
A frequent mistake is to think that three-fourths decayed means three half lives, because of the number three in the fraction, but this is incorrect. Students must focus on the remaining fraction, not the decayed fraction. Another error is trying to subtract half life linearly from 100 percent, which ignores the exponential nature of radioactive decay. Always express remaining fraction as (1/2)^n to find the correct number of half lives that have passed.

Final Answer:
Three-fourths of the radioactive substance will have decayed after 8 months.