Radioactive Isotopes, Exponential Decay

Question

The radioactive isotope 226Ra has a half-life of approximately 1599 years. After 1800 years, there are 2.5g of 226Ra.

(1) What was the initial quantity? ____ g

(2) How much of it remains after 18000 years? _____ g

Answer 1

From y = y₀e^kt, obtain 2.5 = y₀ times (e=2.718281828)^k(1800). It remains to pinpoint the value of k in order to determine y₀.

To find k, write 0.5y₀ (half the original quantity) = y₀e^k(1599) (the original quantity times the base of the natural logarithm e raised to the product of the decay constant sought and the given half-life time), which gives 1599k = ln 0.5 or k equals ln 0.5/1599.

Then obtain the original amount y₀ as 2.5 = y₀e^{(1800ln 0.5/1599)} which simplifies to y₀ = 5.455198371 grams, equivalent to 5.455 grams.

For the amount remaining in 18000 years, write y = (5.455198371)e^{(18000ln 0.5/1599)}which reduces to

y = 0.00228974785 grams equivalent to 0.002 grams.