My friend told me about this idea.

Let there be $N_0$ atoms of a particular radioactive material. Then the number of atoms at time $t$, $N_t$, we already know, is given by:

$$N_t = N_0 e^{-t/\tau},\; where\; \tau = \frac{t_{1/2}}{\ln(2)}$$

where Half-life of the material: $t_{1/2}$.

Is the following quantity useful in any physics or nuclear engineering or something:

$$T_{decay} = \tau(1 + \ln{N_0}) = \dfrac{t_{1/2}}{\ln{2}} (1 + \ln{N_0})$$

This is basically the sum of the time it takes for the number of atoms/nuclei to reduce to $1$ and the mean lifetime of a single atom/nucleus.

My friend told me about this idea.

Let there be $N_0$ atoms of a particular radioactive material. Then the number of atoms at time $t$, $N_t$, we already know, is given by:

$$N_t = N_0 e^{-t/\tau},\; where\; \tau = \frac{t_{1/2}}{\ln(2)}$$

where Half-life of the material: $t_{1/2}$.

Is the following quantity useful in any physics or nuclear engineering or something:

$$T_{decay} = \tau(1 + \ln{N_0}) = \dfrac{t_{1/2}}{\ln{2}} (1 + \ln{N_0})$$

This is basically the sum of the time it takes for the number of atoms/nuclei to reduce to $1$ and the mean lifetime of a single atom/nucleus.

Isn't $T_{decay}$ the time it takes for all nuclei to have decayed/changed?