rates of decay reactions: half life & #3 is carbon dating.

Question

1)

Thallium-201 is used medically to diagnose heart problems. The half-life of thallium-201 is 72.9 hours. How much time is required for the activity of a sample of thallium-201 to fall to 23.1 percent of its original value?

answer in hrs

2)

What is the half-life of boron-8 if 2.14 seconds are required for the activity of a sample of boron-8 to fall to 14.6 percent of its original value?

Answer in sec

3)

An artifact classified as charcoal, found in a site at Wadi el Jilat, Jordan, is found to have a ¹⁴C radioactivity of 3.77×10^-2 counts per second per gram of carbon. If living carbon-containing objects have an activity of 0.255 counts per second per gram of carbon, estimate the age of the artifact?

The half-life of ¹⁴C is 5730 years.

 y

Answer 1

There are various ways to approach these problems, but for first order decay, one easy way is to use the following relationship:

FR = 0.5ⁿ

where FR = fraction remaining

n = number of half lives that have elapsed

(1). half life = 72.9 hours; fraction remaining (FR) = 23.1% = 0.231

FR = 05ⁿ and we will solve for n (number of half lives that have elapsed)

0.231 = 0.5ⁿ

log 0.231 = n log 0.5

-0.636 = -0.301 n

n = 2.11 half lives have elapsed, and since each half life is 72.9 hours, we can find the time it took...

2.11 half lives x 72.9 hours / half life = 154 hours

(2). Fraction remaining (FR) = 14.6% = 0.146

FR = 0.5ⁿ

0.146 = 0.5ⁿ

log 0.146 = n log 0.5

-0.836 = -0.301 n

n = 2.78 half lives have elapsed and this is equal to the 2.14 seconds that it took, so we can find the 1/2 life..

2.14 sec / 2.78 half lives = 0.770 seconds / half life

So the half life of boron-8 would be ~ 0.770 seconds

See if you can do #3. Hint: FR = 3.77x10^-2 / 0.255. Answer should be in the neighborhood of 15,800 years