Final answer:
The half-life of Americium-241 is 432.7 years. The rate of decay per year, k, is approximately 0.0016. After 700 years, approximately 0.192 mg of Americium-241 will remain. It takes approximately 682.4 years for 70% of the material to decay.
Step-by-step explanation:
To model the decay of Americium-241, we can use the formula A(t) = A₀e^(-kt), where A(t) represents the amount of Americium-241 remaining after time t, A₀ is the initial amount of Americium-241, and k is the rate of decay per year. To find k, we can use the half-life equation: half-life = ln(2) / k. Plugging in the given half-life of 432.7 years, we can solve for k:
half-life = ln(2) / k
432.7 = ln(2) / k
k = ln(2) / 432.7
Using a calculator, we find that k ≈ 0.0016.
(b) To find the amount of Americium-241 remaining after 700 years, we can use the formula A(t) = A₀e^(-kt) and plug in the values:
A(700) = 0.61e^(-0.0016*700)
Solving this equation, we find that approximately 0.192 mg of Americium-241 will remain after 700 years.
(c) To find the time it takes for 70% of the material to decay, we can set up the equation:
0.3A₀ = A₀e^(-kt)
Simplifying, we have:
e^(-kt) = 0.3
Taking the natural log of both sides:
-kt = ln(0.3)
Solving for t, we find that approximately 682.4 years are needed for 70% of the material to decay.