If tissue samples from the skeleton contain about 95.17% of the inital carbon 14 expexted in living tissue. what year did they die? the half life for carbon 14 is 5730 years. (intial concentration of carbon 14 is 1.00)
The correct answer and explanation is:
The problem gives the following information:
- Remaining carbon-14 fraction = 95.17% = 0.9517
- Initial carbon-14 fraction = 1.00
- Half-life of carbon-14 = 5730 years
We need to find the number of years since the organism died.
Step 1: Understand the decay formula
The decay of carbon-14 follows first-order kinetics and is given by the formula: N(t)=N0×(12)tT1/2N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
Where:
- N(t)N(t) = amount of carbon-14 remaining after time tt
- N0N_0 = initial amount of carbon-14
- T1/2T_{1/2} = half-life of carbon-14
- tt = time elapsed since death
Step 2: Set up the equation using the given values
0.9517=1.00×(12)t57300.9517 = 1.00 \times \left(\frac{1}{2}\right)^{\frac{t}{5730}}
Step 3: Solve for tt
Take the natural logarithm (ln) on both sides: ln(0.9517)=ln[(12)t5730]\ln(0.9517) = \ln\left[\left(\frac{1}{2}\right)^{\frac{t}{5730}}\right]
Using log properties: ln(0.9517)=t5730×ln(12)\ln(0.9517) = \frac{t}{5730} \times \ln\left(\frac{1}{2}\right)
Step 4: Calculate values
ln(0.9517)≈−0.0494\ln(0.9517) \approx -0.0494 ln(12)=ln(0.5)≈−0.6931\ln\left(\frac{1}{2}\right) = \ln(0.5) \approx -0.6931
Substitute: −0.0494=t5730×(−0.6931)-0.0494 = \frac{t}{5730} \times (-0.6931)
Divide both sides by −0.6931-0.6931: −0.0494−0.6931=t5730\frac{-0.0494}{-0.6931} = \frac{t}{5730} 0.0713=t57300.0713 = \frac{t}{5730}
Multiply both sides by 5730: t=0.0713×5730≈408.45 yearst = 0.0713 \times 5730 \approx 408.45 \text{ years}
Final answer:
The skeleton died approximately 408 years ago.
Explanation
Carbon-14 dating is based on the principle that living organisms constantly take in carbon, including the radioactive isotope carbon-14. When the organism dies, it stops absorbing carbon-14, and the isotope begins to decay at a known rate measured by its half-life (5730 years for carbon-14). By measuring the remaining percentage of carbon-14 in a tissue sample, scientists can estimate how long it has been since the organism died.
In this case, the tissue contains 95.17% of the original carbon-14, meaning only a small amount has decayed. The decay follows an exponential function related to the half-life. Using logarithms and the decay formula, the calculation shows that the skeleton died roughly 408 years ago. This relatively short time frame compared to the half-life indicates the sample is quite recent in archaeological terms.
This method is widely used in archaeology, geology, and other fields to date formerly living materials up to around 50,000 years old, beyond which the carbon-14 amount becomes too small to measure accurately.