Question
Derive the equation $\lambda=\frac{\left(-\frac{ dN }{ dt }\right)}{ N }$ and write what does $\lambda$ denotes.

Answer

The rate of decay of a radioelement at any instant is proportional to the number of nuclei (atoms) present at that instant. It can be represented as, $-\frac{ dN }{ dt } \propto N \quad \text { or }-\frac{ dN }{ dt }=\lambda N \ldots \ldots . \text {.(i) }$ Where, $-\frac{ dN }{ dt }=$ Rate of decay at any time, $t$
$\lambda=$ Decay constant
$N =$ Number of nuclei (atoms) present at time, $t$
From equation (i),
$\lambda=\frac{\left(-\frac{ dN }{ dt }\right)}{ N } \text { or } \lambda=-\frac{ dN }{ dt } \times \frac{1}{ N }$ Decay constant $(\lambda)$ is the fraction of nuclei decaying in unit time. ###It is the ratio of the amount of substance disintegrated per unit time to the amount of substance present at that time.

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