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The half-life of a radioactive substance is T0. At t = 0, the number of active nuclei is N0. Which of the following statements is correct?

1. The number of nuclei that have decayed in the time interval 0 to t is N0 * e^(-lambda*t)
2. The number of nuclei that have decayed in the time interval 0 to t is N0 * (1 - e^(-lambda*t))
3. The probability that a given radioactive nucleus does not decay in the interval 0 to t is e^(-lambda*t)
4. The probability that a given radioactive nucleus does not decay in the interval 0 to t is 1 - e^(-lambda*t)

Correct answer: The number of nuclei that have decayed in the time interval 0 to t is N0 * (1 - e^(-lambda*t))

Solution

At time t, the number of active nuclei is N(t) = N0 * e^(-lambda*t). Nuclei decayed = N0 - N(t) = N0*(1 - e^(-lambda*t)), confirming option B. For a single nucleus, probability of surviving (not decaying) = e^(-lambda*t), making option C also correct. However, option B is unambiguously correct, and option C is also correct. In standard MCQ context (single correct), both B and C are right. The question likely expects both B and C as correct (multi-correct type).

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