# Why not quarter-life?

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

• Not really, $1/2$ is just a nice number. Commented Feb 26, 2018 at 10:51
• Note also that if you go for $\frac{N_0}{e}$ then the time $t=\tau$ is the time it takes to become one-eeeth of its original value. Commented Feb 26, 2018 at 11:48
• There is the very practical reason that if it were "quarter" you wouldn't know whether it referred to a quarter gone or a quarter left. Commented Feb 26, 2018 at 13:15

## 2 Answers

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 $$t_{1/n}=\tau \cdot ln(n)$$

Radioactive decay is an example of an exponential random process. Two key statistics for any exponential random process are the median and the mean. (The standard deviation is equal to the mean for an exponential random process.)

The half life $t_{1/2}$ is the median. The time constant $\tau$ is the mean. There's nothing particularly meaningful about the quarter life.

• Well, it's the first quartile, so there's that. Commented Feb 26, 2018 at 13:14