Two radioactive substances A and B initially have amounts in the ratio 4:1 (i.e., [A]0 = 4x and [B]0 = x). After a certain time, the amounts remaining are equal. If n1 and n2 are the number of half-lives elapsed for A and B respectively during this time, what is the value of (n1 - n2)?

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Accepted Answer

2. Using the radioactive decay law: after n half-lives, the remaining amount is N0 * (1/2)ⁿ. For A: remaining = 4x * (1/2)ⁿ1 = 4x / 2ⁿ1. For B: remaining = x * (1/2)ⁿ2 = x / 2ⁿ2. Setting them equal: 4x / 2ⁿ1 = x / 2ⁿ2. Dividing both sides by x: 4 / 2ⁿ1 = 1 / 2ⁿ2. Cross-multiplying: 4 * 2ⁿ2 = 2ⁿ1, so 2² * 2ⁿ2 = 2ⁿ1, giving 2^(n1) = 2^(n2 + 2), hence n1 - n2 = 2.

Two radioactive substances A and B initially have amounts in the ratio 4:1 (i.e., [A]0 = 4x and [B]0 = x). After a certain time, the amounts remaining are equal. If n1 and n2 are the number of half-lives elapsed for A and B respectively during this time, what is the value of (n1 - n2)?

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