An unknown radioactive element decays into nonradioactive su

An unknown radioactive element decays into non-radioactive substances. In 340 days the radioactivity of a sample decreases by 29 percent. (a) What is the half-life of the element? half-life: (days) (b) How long will it take for a sample of 100 mg to decay to 58 mg? time needed: (days)

Solution

rate of decay is proportional to present sample (N)
(- dN/dt) = k N
dN/N = - k dt
log N = - kt + C
N = No e^-kt -------------- (1)
where No = original sample at t=0
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half life that time (T) in which N = No/2
No/2 = No e^-kT
(2)^-1 = e^-kT
e^kt = 2
kT = log(e) [2] ------------------- (2)
================
given, decay by 29% >> mean N = present value = 100 - 29 = 71%
N/No = 0.71, for t = 340 days >>>>>>> use (1) again
e^-340k = 0.71 = 71/100
e^340 k = 100/71 = 1.408
340 k = log(e) [1.408] = 0.3422 --------- (3)
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divide (2) & (3)
T/340 = log(e)[2]/log(e)[1.408] = 0.6931/0.3422
T = 668.63 days
==========================
b) No=100 mg, N = 58 mg
e^-kt = 0.58
kt = log(e) [100/58] = 0.5447 --------- (4)
>>> t/340 = 0.5447/ 0.3422
t = 541.198 days