Derive the law of radioactive decay on the basis of the fact that the decay probability for a nucleus is independent of the number of nuclei and is proportional to the period of observation.

Derive the law of radioactive decay on the basis of the fact that the

1.	Derive the law of radioactive decay on the basis of the fact that the decay probability for a nucleus is independent of the number of nuclei and is proportional to the period of observation.
Answer» Solution :The decay probability is the ratio of the number of the NUCLEI that have EXPERIENCED decay to the TOTAL number of nuclei: `omega=-(dN)/N`. It is proportional to the TIME of observation, `omega=lamdadt,i.e.(dN)/N=-lamdadt`. We have `int(dN)/N=-lamdaintdt,"so "lnN+C=-lamdat` At the initial point of time t = 0, N = `N_(0)`, so `lnN_(0)+C=0andC=-lnN_(0)`. Substituting this into the above formula, we obtain `lnN-lnN_(0)=-lamdat,orN=N_(0)e^(-lamdat)`

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