Many radioisotopes have important industrial, medical, and research applications

Question

Many radioisotopes have important industrial, medical, and research applications. One of these is 60Co, which has a half-life of 5.20 years and decays by the emission of a beta particle (energy 0.310 MeV) and two gamma photons (energies 1.17 MeV and 1.33 MeV). A scientist wishes to prepare a 60Co sealed source that will have an activity of at least 11.4 Ci after 42.0 months of use.
If the activity is 11.4 Ci, how many 60Co atoms are in the source?

What is the minimum number of nuclei in the source at the time of creation?

What is the minimum initial mass of 60Co required?

please show work!

Explanation / Answer

Some of the data given in the question are irrelevant to answer the particular question, hence it's misleading.

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part 1,
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You need to find the number of atoms when the activity is 11.5Ci, so you have to use the equation,

R = L* N ------------------(1) ; L - decay constant,

Since you are not given the value of L, you have to find it using,

L = ln(2) / Half Life -------------------(2)

(1) and (2) gives,

R = ln(2) * N / half_life

=>

N = R * half_life / ln(2)

now substitute values. Here R is given in Ci, so you have to convert it to Bq by multiplying it by 3.7*10^(10), because => 1 Ci = 3.7*10^(10)Bq

N = (11.5 * 3.7*10^(10)) * (5.2*365*24*60*60) / (ln(2))
N = 1.007 * 10^20


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part 2,
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Now use the equation,

N = No * exp (-Lt)

N is the number we just found, t is the time in seconds (41 months or 41/12 years)

No = N * exp (+Lt)
No = N * exp ( ln(2) * t / half_life )
No = 1.007 * 10^20 * exp (ln (2) * (41/12)/ 5.2)
No = 1.588 * 10^20

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