Please make sure to show step by step work including the equations used and thei

Question

Please make sure to show step by step work including the equations used and their derivations. Will be sure to rate. (H8-16) Many radioisotopes have important industrial, medical, and research applications. One such radioisotope is ouCo, which has a half-life of 5.2 yr and decays by the emission of a beta particle (energy 0.31 MeV) and two gamma photons (energies 1.17 MeV and 1.33 MeV). A scientist wishes to prepare a Co sealed source that will have an activi of at least 9.4 Ci after 26 months of use. What is the minimum initial mass o required?

Explanation / Answer

Let initial activity of the isotope be A0. Final isotope after 26 months is A = 9.4 Ci.

Then, we have,

= ln2/t1/2 = 0.693/5.2yr, where is the decay constant.

Initial and final activity is related as,

A = A0e-t, putting  A = 9.4 Ci, = 0.693/5.2yr and t = 26 months = 26/12 yr = 2.166 yr we get,

9.4 Ci = A0e-(0.693/5.2yr)(2.167 yr)

or, 9.4 Ci = A0e-0.288

or,  9.4 Ci = A0(0.75)

or, A0 = 12.55 Ci

Now, we know that A0 = N, where N is the number of atoms of isotope A0 is the activity when the number of atom is N.

So, A0 = N, given

12.55 Ci = (0.693/5.2yr)N, we know that 1 Ci = 3.7X1010 decays/s, so

12.55(3.7X1010 decays/s) = [0.693/(5.2X365X24X3600 s)]N, (as 1 yr = 365X24X3600 s)

or, N = 1.098X1020

Co-60 has atomic mass of 60. So 1 mole or 6.022X1023 of Co-60 will have 60 g of Co

So, 1.098X1020 of Co will have,

m = [60g/( 6.022X1023)](1.098X1020)

m = 10.945X10-3g

or, m = 10.945 mg of initial mass is required.

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