47. The fraction of undecayed nuclei remaining after time t is given by
N
= e-t = e-(ln 2)t/T1/2
N0
where  is the disintegration constant and T1/2 (= (ln 2)/) is the half-life. The time for half the original
238 244
U nuclei to decay is 4.5 × 109 y. For Pu at that time,
(ln 2)t (ln 2)(4.5 × 109 y)
= =38.0
T1/2 8.2 × 107 y
and
N
= e-38.0 =3.1 × 10-17 .
N0
248
For Cm at that time,
(ln 2)t (ln 2)(4.5 × 109 y)
= = 9170
T1/2 3.4 × 105 y
and
N
= e-9170 =3.31 × 10-3983 .
N0
For any reasonably sized sample this is less than one nucleus and may be taken to be zero. A standard
calculator probably cannot evaluate e-9170 directly. Our recommendation is to treat it as (e-91.70)100.