Radioactive substance A has a half-life T and transforms into substance B. At t = 0, the sample contains only A. At a specific time t, the ratio of the quantity of B to A is found to be 0.3. Which of the following expressions correctly defines t in terms of T?

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EEJ MAIN Physics QUESTION #7073

Question 1

Radioactive substance $A$ has a half-life $T$ and transforms into substance $B$. At $t = 0$, the sample contains only $A$. At a specific time $t$, the ratio of the quantity of $B$ to $A$ is found to be $0.3$. Which of the following expressions correctly defines $t$ in terms of $T$?

$t = \frac{T}{\log(1.3)}$
$t = \frac{T}{2} \frac{\log 2}{\log 1.3}$
$t = T \frac{\log 1.3}{\log 2}$✔️
$t = T \log(1.3)$

Correct Answer Explanation

Let $N_A$ be the remaining nuclei of $A$ and $N_B$ be the produced nuclei of $B$. $N_B/N_A = 0.3 \implies (N_0 - N_A)/N_A = 0.3$. This gives $N_0/N_A = 1.3$, or $N_A = N_0/1.3$. Using the decay law $N = N_0 2^{-t/T}$, we get $1/1.3 = 2^{-t/T}$. Taking logs: $\log(1.3) = (t/T)\log 2$. Solving for $t$ gives $t = T \frac{\log 1.3}{\log 2}$.

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