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Question 11.12: The half-life of (t1/2) an excited state is the time it woul...

The half-life of \left(t_{1 / 2}\right) an excited state is the time it would take for half the atoms in a large sample to make a transition. Find the relation between t_{1 / 2} and τ (the “lifetime” of the state).

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N(t)=e^{-t / \tau} N(0) .  (Eqs. 11.65 and 11.66). After one half-life,  <br /> N(t)=\frac{1}{2} N(0), \quad \text { so } \quad \frac{1}{2}=e^{-t / \tau}, \text { or } \quad 2= e^{t / \tau}, \text { so } t / \tau=\ln 2, \quad \text { or } \quad t_{1 / 2}=\tau \ln 2 .

N_{b}(t)=N_{b}(0) e^{-A t}                  (11.65).

\tau=\frac{1}{A}                  (11.66).

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