Question

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And How to derive it?

Answer 1

The correct answer is $\Delta N$ is proportional to $N\Delta t$, or the amount of decayed particles is proportional to the initial number number of particles and time of decay. Mathematically, it can be written as

$$\Delta N\propto N\Delta t$$

(do not confuse the proportionality symbol $\propto$ with a Greek letter $\alpha$, or alpha).

Indeed, imagine that we have N particles in a closed system. Imagine that we started the observation at t₁ and finished at t₂. Assume that k particles decayed. It means that during time $\Delta t=t_2-t_1$ the number of particles that decayed is proportional to $\Delta N=-kN\Delta t,$ the negative sign shows that the number of particles left decreases. Mathematically, we have

$$\Delta N=-kN\Delta t\leftrightarrow\Delta N\propto N\Delta t.$$