What is the age in years of a bone in which the ¹⁴C/¹²C ratio is measured to be 1.65 x 10⁻¹³?

검증된 단계별 안내

Determine the initial ratio of ¹⁴C/¹²C in a living organism. This is typically known to be approximately 1.3 × 10⁻¹². This value will serve as the initial concentration (N₀) of ¹⁴C relative to ¹²C.

Use the radioactive decay formula:

N = N

₀e⁻λt, where N is the current ratio (1.65 × 10⁻¹³), N₀ is the initial ratio (1.3 × 10⁻¹²), λ is the decay constant, and t is the time (age of the bone).

Calculate the decay constant λ using the half-life of ¹⁴C, which is approximately 5730 years. The relationship is

λ = \frac{ln (2)}{T}

_1/2, where T₁/₂ is the half-life.

Rearrange the decay formula to solve for time t:

t = ln (\frac{N}{N}

₀)⁻λ. Substitute the known values of N, N₀, and λ into this equation.

Simplify the expression to calculate the age of the bone in years. Ensure that the units are consistent throughout the calculation.

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주요 개념

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Radiocarbon Dating

Radiocarbon dating is a method used to determine the age of organic materials by measuring the amount of carbon-14 (¹⁴C) remaining in a sample. Since ¹⁴C is radioactive and decays over time, scientists can calculate the time elapsed since the death of the organism by comparing the ratio of ¹⁴C to stable carbon-12 (¹²C) isotopes.

Half-Life of Carbon-14

The half-life of carbon-14 is approximately 5,730 years, which is the time it takes for half of the original amount of ¹⁴C in a sample to decay into nitrogen-14. This property is crucial for radiocarbon dating, as it allows researchers to estimate the age of a sample based on the remaining ¹⁴C content.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of radiocarbon dating, the amount of ¹⁴C in a sample decreases exponentially over time, which can be mathematically modeled to determine the age of the sample based on the measured ¹⁴C/¹²C ratio.