The decay of americium-241 follows an exponential decay model, so we can use the formula:
A = A0*e^(-kt)
where:
A0 = initial amount of americium-241 (20 g in this case)
A = amount of americium-241 after time t
k = decay constant (we can find it from the given decay rate)
t = time (in years)
The decay rate is 0.1604%, which is equivalent to 0.001604 as a decimal.
We can use the fact that when the mass decays to 7.2 g, the remaining mass is 7.2 g and the initial mass was 20 g. So we can write:
7.2 = 20*e^(-0.001604t)
Divide both sides by 20:
0.36 = e^(-0.001604t)
Take the natural logarithm of both sides:
ln(0.36) = -0.001604t
Solve for t:
t = -ln(0.36)/0.001604
t ≈ 114.5
Rounding to the nearest year, it will take about 115 years for a 20 g mass of americium-241 to decay to 7.2 g.