The half-life for radioactive decay of 14C is 5730 years. An archaeological artifact containing wood had only 80% of the 14C found in a living tree. Estimate the age of the sample.

To estimate the age of the sample, we use the radioactive decay formula:

N = N₀ * e^(-λt)

where:
N = remaining quantity of substance
N₀ = initial quantity of substance
λ = decay constant
t = time elapsed

Step 1: Expressing Decay in Terms of Half-life
The decay constant λ is related to the half-life (T₁/₂) by the equation:

λ = ln(2) / T₁/₂

Given that the half-life of carbon-14 (T₁/₂) is 5730 years:

λ = ln(2) / 5730
λ ≈ 0.000121

Step 2: Setting Up the Equation
We are given that the remaining percentage of carbon-14 is 80% of the original, so:

N / N₀ = 0.80

Substituting into the decay formula:

0.80 = e^(-0.000121 * t)

Step 3: Solving for t
Taking the natural logarithm on both sides:

ln(0.80) = -0.000121 * t

-0.2231 = -0.000121 * t

Dividing both sides by -0.000121:

t = 1844 years

Final Answer:
The estimated age of the sample is 1844 years.