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

Answer

To estimate the age of the archaeological artifact containing wood with only 80% of the carbon-14 found in a living tree, we can follow these steps:

### Step 1: Understand the Given Information
- The half-life of carbon-14 (\(^{14}C\)) is 5730 years.
- The artifact contains 80% of the original carbon-14 found in a living tree.

### Step 2: Calculate the Decay Constant (\(k\))
The decay constant can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
where \(t_{1/2}\) is the half-life.

Substituting the given half-life:
\[ k = \frac{0.693}{5730 \text{ years}} \approx 1.209 \times 10^{-4} \text{ years}^{-1} \]

### Step 3: Set Up the Equation for Age Calculation
The age of the sample can be calculated using the formula:
\[ t = \frac{2.303}{k} \log\left(\frac{N_0}{N}\right) \]
where:
- \(N_0\) = initial amount of carbon-14 (100%),
- \(N\) = current amount of carbon-14 (80%).

### Step 4: Substitute Values into the Equation
Substituting the values into the equation:
\[ t = \frac{2.303}{1.209 \times 10^{-4}} \log\left(\frac{100}{80}\right) \]

### Step 5: Calculate the Logarithm
Calculate \(\log\left(\frac{100}{80}\right)\):
\[ \frac{100}{80} = 1.25 \quad \Rightarrow \quad \log(1.25) \approx 0.09691 \]

### Step 6: Complete the Calculation
Now substitute this value back into the equation:
\[ t = \frac{2.303}{1.209 \times 10^{-4}} \times 0.09691 \]
Calculating this gives:
\[ t \approx \frac{2.303 \times 0.09691}{1.209 \times 10^{-4}} \approx 1846 \text{ years} \]

### Conclusion
The estimated age of the archaeological artifact is approximately **1846 years**.

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