Rates of Decay Reactions

Question

Thallium-201 is used medically to diagnose heart problems. The half-life of thallium-201 is 72.9 hours. If you begin with 39.8 mg of this isotope, what mass remains after 200 hours have passed?

Answer 1

You can set up an equation for this that allows you to plug in t = 200.

Let A(t) be the amount of material remaining after "t" hours. Let A₀ be the amount of material at time t = 0 (the initial amount of material). And let "h" be the half life.

Then A(t) = A₀(1/2)^t/h so in this case:

A(t) = 39.8(1/2)^t/72.9

For 200 hours: A(200) = 39.8(1/2)^200/72.9 = 39.8(1/2)^2.743484 = 39.8(0.1493238) = 5.94 mg

Answer 2

1/2= e^72.9r

72.9r = ln.5

r = ln.5/72.9 = -.0095=-0.95% decay rate per hour

A = 39.8e^200(-.0095)

= 39.8e^-1.9

= 39.8/e^1.9

=about 5.95 mg after 200 hours, starting with 39.8 mg