Question

Is the following relation true or false: $t_{\frac{1}{2}}=2\times t_{\frac{1}{4}}$.

• A. True
• B. False

Answer 1

The correct option is B False

Say I have a sample of 100 nuclei, whose half-life is 10 days$(t_{\frac{1}{2}}=10days)$. Which means, after 10 days, I will have 50 nuclei left, which is $\frac{1}{2}$ the original.
$t_{1}4$ by definition is the time after which I will have $\frac{1}{4}th$ the initial number of nuclei left, or 25 nuclei left in my sample.
If at the end of day 10, I have 50 nuclei left, I will have 25 left after another half-life, i.e. at the end of day 20.
$\therefore t_{\frac{1}{4}}=2\times t_{\frac{1}{2}}$.
Thus the relation given in the question is false.
P.S : In general, you can say $t_{{\frac{1}{2}}^n}=n\times t_{\frac{1}{2}}$. Convince yourself.