Question

A sample of a certain radioactive material decays to 89.36% of its mass after 2 years.

a. What is the half-life of the material? Show your calculations and keep four significant figures of accuracy.

b. How long would it take for the sample to decay to 10% of its original mass? How much longer after that would it take to decay to 1% of its original mass?

Answer 1

(a): Given time, t = 2 years

Initial amount, Ao = 100.00 %

Amount left after 2 year, At = 89.36 %

For 1st order radioactive decay,

kt = ln(Ao / At)

=> k x 2 year = ln(100 / 89.36)

=> k = 0.05625 year-1

Hence half-life, t1/2 = 0.693 / k = 0.693 / 0.05625 year-1 = 12.32 year (answer)

(b): At = 10 %

=> kt = ln(100% / 10%) = ln10

=> t = ln10 / k = ln10 / 0.05625 year-1 = 40.93 year (answer)

At = 1 %

=> kt = ln(100% / 1%) = ln100

=> t = ln100 / k = ln100 / 0.05625 year-1 = 81.87 year

Hence total time required = 81.87 - 40.93 = 40.94 year (answer)