An unknown radioactive element decays into nonradioactive su

An unknown radioactive element decays into non-radioactive substances. In 600 days the radioactivity of a sample decreases by 73 percent.

A) What is the half life of the element?

B) How long will it take for a sample of 100 mg to decay to 73 mg?

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 73% >> mean N = present value = 100 - 73 = 27%
N/No = 0.27, for t = 600 days >>>>>>> use (1) again
e^-600k = 0.27 = 27/100
e^600 k = 100/27 = 3.703
600 k = log(e) [3.703] = 0.006 --------- (3)
-----------------------
divide (2) & (3)
T/600 = log(e)[2]/log(e)[3.703] = 0.54
T = 317 days
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b) No=100 mg, N = 73 mg
e^-kt = 0.73
kt = log(e) [100/73] = 0.314--------- (4)
>>> t/600 = 0.314/0.54
t = 348 days