133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?

Verified step by step guidance

1. Understand that the problem involves radioactive decay, where the amount of carbon-14 decreases over time according to an exponential decay model.
2. Recall the decay formula: \(N(t) = N_0 \times e^{-\lambda t}\), where \(N(t)\) is the amount of carbon-14 remaining at time \(t\), \(N_0\) is the original amount, and \(\lambda\) is the decay constant.
3. Since 90% of the carbon-14 has decayed, only 10% remains, so set \(\frac{N(t)}{N_0} = 0.10\) and write the equation \(0.10 = e^{-\lambda t}\).
4. Take the natural logarithm of both sides to solve for \(t\): \(\ln(0.10) = -\lambda t\), which gives \(t = -\frac{\ln(0.10)}{\lambda}\).
5. Use the known half-life of carbon-14 (about 5730 years) to find \(\lambda\) using \(\lambda = \frac{\ln(2)}{\text{half-life}}\), then substitute \(\lambda\) back into the equation to calculate \(t\).

Key Concepts

Radioactive Decay and Half-Life

Radioactive decay is the process by which unstable nuclei lose particles over time, decreasing the amount of a radioactive isotope. The half-life is the time required for half of the original radioactive atoms to decay. Understanding half-life allows us to relate the remaining amount of a substance to the elapsed time.

Carbon-14 Dating Method

Carbon-14 dating estimates the age of organic materials by measuring the remaining carbon-14 isotope, which decays at a known rate. Since living organisms constantly replenish carbon-14, the decrease after death indicates elapsed time. This method is widely used in archaeology and geology.

Exponential Decay Formula

The exponential decay formula, N(t) = N_0 * (1/2)^(t/T), relates the remaining quantity N(t) to the initial amount N_0, elapsed time t, and half-life T. This formula helps calculate the age of a sample when the fraction of remaining radioactive material is known.