Physics · Ch 15 — Structure of Atoms and Nuclei

Average Life of a Radioactive Species

15.10.2

Average Life of a Radioactive Species

Different individual nuclei of the very same radioactive species do not all decay at the same moment -- some decay almost immediately, others survive for a very long time -- so it is natural to ask for the AVERAGE (mean) lifetime of a nucleus of that species, denoted $\tau$ . This average is computed by weighting each possible decay time t by how many nuclei actually decay AT that time: in the short interval between t and t + dt, the number of nuclei decaying is $\lambda N_0e^{-\lambda t}\,dt$ (from the decay law), so the average lifetime is $\tau=\frac{1}{N_0}\int_0^\infty t\,\lambda N_0e^{-\lambda t}\,dt$ . Carrying out this integral gives the strikingly simple result $\tau=\frac{1}{\lambda}$ -- the average life is just the reciprocal of the decay constant. …

Misc Ex.15.9Decay constant, average life and activity of a 1.5 mg sample from its half-life

Worked out. For a nuclear species X with half-life 3.2 days, the decay constant follows directly as $\lambda=0.693/T_{1/2}=0.2166$ per day, which converts to $2.507\times10^{-6}$ s $^{-1}$ ; the average life is then $\tau=1/\lambda=4.617$ days. For the activity of a 1.5 mg sample, the number of nuclei present is first found from Avogadro's number and the (unspecified, symbolic) molar mass Y of species X, $N=6.02\times10^{23}\times1.5\times10^{-3}/Y$ , and the activity $A=\lambda N$ is then expressed in curie by dividing by $3.7\times10^{10}$ , giving a final symbolic result of about $6.1\times10^4/Y$ Ci -- demonstrating that activity for a fixed mass of material depends on the species' molar mass, since a fixed mass of a lighter isotope contains more …

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