The number of nuclei left after time $t$ in radioactive decay is given by:
$$N(t) = N_0 e^{-t/ \tau}$$
Now if we put $N(t)$ as $\dfrac{N_0}2$, we get half-life. But, if we had put $\dfrac{N_0}4$, we would have quarter-life, which is also independent of $N_0$.
Is there anything special about half-life as opposed to quarter-life
Best Answer
The decay time $t_{1/2}$ of half the given number $N_0$ of atoms atoms is just convenient and visually appealing. Of the unit fractions it is also nearest to the decay time constant (mean lifetime) $\tau$ $t_{1/2}=0.6931 \tau$. The decay time to a unit fraction $1/n$ given by the positive integer $n$ is $$t_{1/n}=\tau \cdot ln(n)$$