Answer to Question #138470 in General Chemistry for ava

Question

The volcanic rock sample originally contained 8.00 mg of isotope X. How long ago were these footprints made?

Answer 1

Complete problem:

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?

The following equation gives the quantitative relationship between the original number of nuclei present at time zero (N₀) and the number (N) at a later time t:

$N = N_0e^{−λt}$

λ is the decay constant

The relationship between the decay constant λ and the half-life t_1/2 is

$λ = \frac{ln(2)}{t_{1/2}} ≈ \frac{0.693}{t_{1/2}}$

$λ = \frac{0.693}{500000} = 1.386\times 10^{-6}$

$\frac{N}{N_0} = e^{−λt}$

$\frac{0.125}{8.0} = e^{−1.386\times 10^{-6}t}$

$ln(0.0156) = −1.386\times 10^{-6}t$

$-4.1605 = −1.386\times 10^{-6}t$

$t = 3\times 10^{6} \;years$

Answer: 3 million years old footprints