The radioactive isotope 210Bi is the daughter product of 210Pb and decays by -emission to 210Po, wh — Physical Chemistry Chemistry Question
Radioactive decay
The radioactive isotope 210Bi is the daughter product of 210Pb and decays by -emission to 210Po, which is also radioactive. 210Po decays by -emission to the stable 206Pb .
210Pb (T1/2 = 22.3 y) 210Bi (T1/2 = 5.01 d) 210Po (T1/2 = 138.4 d) 206Pb
A sample of radiochemically pure 210Bi was freshly isolated from 210Pb and was allowed to stand for the growth of 210Po. The radioactivity of the freshly purified 210Bi sample was 100 Ci. (1 Ci = 3.7×10^10 disintegration per second.)
What is the initial mass of the sample (210Bi)?
Model Answer
1 Ci = 3.7×10^4 disintegrations per second (dps).
Initial -activity = 3.7×10^6 dps
-dN1/dt = 1N1 = 3.7×10^6 dps
where N1 is the number of atoms of 210Bi at t = 0 and 1 is its decay constant.
N1 = 2.31×10^12
Initial mass of 210Bi = (2.31×10^12 / 6.02×10^23) × 210 g = 8.06×10^-10 g
Calculate the time it takes for the amount of 210Po in the sample to grow to its maximum value. How much is the maximum amount of 210Po?
Model Answer
Number of atoms of 210Bi at time t is given by
N1 = N1,0 e^{-1t}
The number of atoms of 210Po, N2, is given by equation
dN2/dt = 1N1 - 2N2
where 2 is the decay constant of 210Po.
Using the integrating factor e^{2t} and integrating, with condition N2=0 at t=0:
N2 = 1 N1,0 / (2 - 1) * (e^{-1t} - e^{-2t})
The time t = T when N2 is maximum is given by the condition dN2/dt = 0
which gives
T = ln(2/1) / (2 - 1) = 24.9 d
At t = T, N2 can be calculated from above.
N2 = 2.04×10^12
Mass of 210Po at t = T, m(210Po) = 7.11×10^-10 g
Determine the -disintegration rate of 210Po and -disintegration rate of 210Bi at that time.
Model Answer
At t = T
-disintegration rate of Po = 1.18×10^5 dps
-disintegration rate of 210Bi = -disintegration rate of 210Po = 1.18×10^5 dps