Question

A particular smoke detector contains 1.95 μCi of ²⁴¹Am, with a half-life of 458 years. The isotope is encased in a thin aluminum container. Calculate the mass of ²⁴¹Am in grams in the detector.

Express your answer numerically in grams.

Answer 1

This involves calculating the specific activity of Am-241.

A half-life of 458 years works out to an average lifetime of 661 years. (Divide by the natural logarithm of 2 to get this.) The average lifetime is the length of time it would take for the entire sample to decay to zero, if it continued to decay at the current rate. (Of course, it doesn't. Rate is proportional to quantity.)

Since uCi is in units of disintegrations per second (dps), convert years to seconds. One year is 31.56 million seconds, so 661 * 31.56 * 10^6 = 20861 * 10^6 seconds, or 2.086 * 10^10 seconds.

The decay constant is the reciprocal of the average lifetime, and in this case, 1/2.086 * 10^(-10) = 0.4794 * 10^(-10)

Each atom of the Am-241 sample has a probability of 4.794 * 10^(-11) of decaying in the next second. In one mole, we expect 6.02 * 10^23 * 4.794 * 10^(-11) = 28.86 * 10^14 atoms of Am-241 to decay every second.

Specific activity is often specified per gram. Since the atomic weight of Am-241 is right around 241 grams/mole, we divide our decay rate by 241 to get
28.86 * 10^12 / 241 = 0.1198 * 10^12 = 1.198 * 10^11 dps/g

One Ci = 3.7 * 10^10 dps
One gram of Am-241 contains 1.198 * 10^11 / 3.7 * 10^10 = 3.238 Ci = 3.238 * 10^6 uCi.

1.95 uCi * 1 g / (3.238 * 10^6 uCi) = 5.94 * 10^(-7) g = 0.0.594 ug Ans

now your formula becomes:

SA = (NA * MW) / (T * 3.7 * 10^10)
SA = Specific Activity
NA = Avogadro's Number
AW = Atomic Weight
T = average lifetime

A good reasonableness check is to recall that the Curie is defined as the activity present in one gram of radium-226.
The half-life of radium-226 is 1620 years.
The atomic weight is 226 g/mole.

1620/458 = 3.54 increase in dps/mole
226/241 = 0.938 decrease in moles/gram
3.54*0.938 = 3.32.

This is not very far from our calculated specific activity, so we can assume there are no major errors in the calculation