Overall Scenario Question One a What is the halflife of the

Overall Scenario:

Question One:

(a) What is the half-life of the element? --(answer in Days)

Solution

(- 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)
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given, decay by 43% >> mean N = present value = 100 - 43 = 57%
N/No = 0.57, for t = 220 days >>>>>>> use (1) again
e^-220k = 0.57 = 57/100
e^220 k = 100/57 = 1.754
220k = log(e) [1.754] = 0.76175--------- (3)
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divide (2) & (3)
T/220= log(e)[2]/log(e)[1.754] = 0.6931/ 0.76175 =0.90987
T = 200.17 days=200 days(approx.)
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b) No=100 mg, N = 47 mg
e^-kt = 0.47
kt = log(e) [100/47] = 0.9240--------- (4)
>>> t/220 =0.9240/0.76175==1.21299=1.213
t = 266.86 days=267 days(approx.)