Question

Radioactive isotope X has a half-life of 500,000 years. This isotope may be found in some types of volcanic rocks. A particular sample of volcanic rock taken from a layer that covered up some of the earliest known human-like footprints contains 0.125 mg of isotope X. The volcanic rock sample originally contained 8.00 mg of isotope X. How long ago were these footprints made?

Answer 1

Radioactive decay is characterized by a continuous exponential decay curve given by:

N(t)=Ae−kt

N(t)=Ae−kt

where N(t) is the number of grams of radium present at time t measured in years and A and k are constants. After one half-life, half of the radium remains that was present at t=0. We get:

N(500000)A=12=e−k(500000)

N(1599)A=12=e−k(50000)

ln(12)=−50000

kln⁡(12)=−500000k

k=−1

500000ln(12)

k=−500000ln⁡(12)

k=0.00043349

k=0.00043349

Given at time t=2,000 years there were 5 grams of radium, we can substitute to find A:

5=Ae−0.00043349(2000)

5=Ae−0.00043349(2000)

A=5e0.00043349

(2000)=11.8985

A=5e0.00043349(2000)=11.8985

Now that we have determined the number of grams of radium present at any time t, N

(t)=11.8985e−0.00043349t

N(t)=11.8985e−0.00043349t

, we can calculate the number of grams present at t=5,000 years as follows:

N(500000)=11.8985e−0.00043349(500000)=1.3620

N(5000)=11.8985e−0.00043349(5000)=1.3620

Therefore, 1.36 g of radium remain after 5,000 years rounded to two decimal places.