Radioactive Isotopes, exponential decay

Question

The radioactive isotope 226Ra has a half-life of approximately 1599 years. There are 25g of 226Ra now.

(1) How much of it remains after 1200 years? ____ g

(2) How much of it remains after 12000 years? ____ g

Answer 1

For a quantity y that decays exponentially, the decay constant is k and the half-life t is the amount of time after which half of the original quantity remains.

Write the equation y = y₀e^kt (where y₀ is the original quantity and e is the base of the natural logarithm and equal to 2.718281828. Then 0.5y₀ = y₀e^kt which gives kt = ln 0.5 or (-ln 2).

For a half-life of 1599 years, write 0.5y₀ = y₀e^k(1599) or 1599k = ln 0.5 and k equals ln 0.5/1599.

The equation y = 25e^{(ln 0.5/1599)(1200)} gives y after 1200 years as 14.86025685 grams equivalent to 14.86 grams.

The equation y = 25e^{(ln 0.5/1599)(12000)} gives y after 12000 years as 0.1376585151 grams equivalent to 0.138 grams.