Calculate the number of atoms of strontium-90?

Question

According to current regulations, the maximum permissible dose of strontium-90 in the body of an adult is 1 x 10^6 Ci.

A) Using the relationship, rate = kN, calculate the number of atoms of strontium-90 to which this dose corresponds (half-life for strontium-90 is 28.8 years).

B) To what mass of strontium-90 does this correspond ?

Answer 1

I think you meant to type 1 x 10^-6 Ci, as 10^6 Ci in your body would kill you outright.  So I will do the math that way, but if not only the numbers change and nothing else.

A)

First I'd convert Curies into Bequerels:

1 Ci = 3.7*10^10 Bq        - >   1 x 10^-6 Ci x 3.7 x 10^10 Bq/Ci = 3.7 x 10^4 Bq

The rate constant k is related to the half-life t_1/2 by:

k = ln 2 / t_1/2

We're given a half-life in terms of years, and Bequerels are in units of disintigrations per second, so we'll need to convert our time units in order to be consistant.

1 yr = 3.15 x 10^7 sec       - >          28.8 yr x 3.15 x 10^7 sec/yr  - >  t_1/2 = 9.08 x 10^8 sec

We can then solve for the rate constant k:

k = ln 2 / (9.08 x10^8 sec)     - >    k = 7.63 x 10^-10 /sec

Now we have a rate and the constant k so we can solve for the number of atoms N using rate = kN

3.7 x 10^4 atom/sec = 7.63 x 10^-10/sec x N - > N = 4.85 x 10^13 atoms

B)

The molar mass of Sr-90 is about 89.9 g/mol.  We know how many atoms we have so we can easily determine how many moles there are.

1 mol = 6.022 x 10^23 atoms - > 4.85 x 10^13 atoms / (6.022 x 10 ^23 atoms/mol) = 8.05 x 10^-11 mol

8.05 x 10^-11 mol x 89.9 g/mol = 7.24 x 10^-9 g