Question 24.6: Applying Radiocarbon Dating Problem The charred bones of a s......
Applying Radiocarbon Dating
Problem The charred bones of a sloth in a cave in Chile represent the earliest evidence of human presence at the southern tip of South America. A sample of the bone has a specific activity of 5.22 disintegrations per minute per gram of carbon (d/min·g). If the ^{12}C/^{14}C ratio for living organisms results in a specific activity of 15.3 d/min·g, how old are the bones (t_{1/2}\text{ of }^{14}C = 5730 yr)?
Plan We calculate k from the given t_{1/2} (5730 yr). Then we apply Equation 24.5 to find the age (t) of the bones, using the given activities of the bones (𝒜_t = 5.22 d/min·g) and of a living organism (𝒜_0 = 15.3 d/min·g).
t = \frac{1}{k} \ln \frac{𝒜_0}{𝒜_t} (24.5)
Solution Calculating k for ^{14}C decay:
k = \frac{\ln 2}{t_{1/2}} = \frac{0.693 }{5730 yr}
= 1.21×10^{−4} yr^{−1}
Calculating the age (t) of the bones:
t = \frac{1}{k} \ln \frac{𝒜_0}{𝒜_t} = \frac{1}{1.21×10^{−4} yr^{−1}} \ln \left( \frac{15.3 d/\min ·g}{ 5.22 d/\min·g}\right)
= 8.89×10^3 yr
The bones are about 8890 years old.
Check The activity of the bones is between \frac{1}{2} and \frac{1}{4} of the activity of a living organism, so the age of the bones should be between one and two half-lives of ^{14}C (from 5730 to 11,460 yr).