29. (a) The decay rate is given by R = N, where is the disintegration constant and N is the number
of undecayed nuclei. Initially, R = R0 = N0, where N0 is the number of undecayed nuclei
at that time. One must find values for both N0 and . The disintegration constant is related
to the half-life T1/2 by = (ln 2)/T1/2 = (ln 2)/(78 h) = 8.89 × 10-3 h-1. If M is the mass
of the sample and m is the mass of a single atom of gallium, then N0 = M/m. Now, m =
(67 u)(1.661 × 10-24 g/u) = 1.113 × 10-22 g and N0 =(3.4g)/(1.113 × 10-22 g) = 3.05 × 1022. Thus
R0 =(8.89 × 10-3 h-1)(3.05 × 1022) =2.71 × 1020 h-1 =7.53 × 1016 s-1.
(b) The decay rate at any time t is given by
R = R0 e-t
where R0 is the decay rate at t =0. At t =48 h, t =(8.89 × 10-3 h-1)(48 h) = 0.427 and
R =(7.53 × 1016 s-1) e-0.427 =4.91 × 1016 s-1 .
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