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Radioactive decay formulae

$$A$$ and $$N$$ are related by:$$$A=\lambda N$$$$$\lambda$$ is the constant of proportionality, called the decay constant. While the activity depends on the amount of material present, the decay constant does not depend on any quantity of matter but is a characteristic for every radioactive isotope.

The activity $$A$$, count rate $$C$$ and the number of nuclei $$N$$ of a material after a certain time $$t$$ are given by: $$$\begin{align*}A(t)=A_{0}e^{-\lambda t}\\C(t)=C_{0}e^{-\lambda t}\\N(t)=N_{0}e^{-\lambda t}\end{align*}$$$

The constants $$A_{0}$$, $$C_{0}$$ and $$N_{0}$$ are the activity, count rate and number of nuclei of a material at $$t=0$$. $$e\approx 2.718$$ is the mathematical constant called Euler's number. It is the base of the natural logarithm $$(\ln)$$.

The graphs of $$N$$, $$A$$ and $$C$$ vs $$t$$ all show an exponential decreasing trend.

The number of radioactive nuclei $$N$$in a material can be derived from the activity $$A$$. This is given by:$$$\begin{align*}&A=-\frac{dN}{dt}\\\Rightarrow&\lambda N=-\frac{dN}{dt}\\\Rightarrow&\lambda=-\frac{1}{N}\frac{dN}{dt}\\\Rightarrow&\lambda\text{ }dt=-\frac{1}{N}\text{ }dN\end{align*}$$$

Taking the integral of both sides, we have:$$$\begin{align*}\int^{t}_{0}\lambda\text{ }dt&=-\int^{N}_{\text{N}_{0}}\frac{1}{N}\text{ }dN\\\Rightarrow\lambda t&=-\ln{(\frac{N}{N_{0}})}\\\Rightarrow N&=N_{0}e^{-\lambda t}\end{align*}$$$