A sample of wood contains carbon-14 to carbon-12 atoms in the ratio 1:4 x 10^12. Calculate how many years ago the tree died.

Steps to Solve

1. Write the formula for carbon dating

The formula used in carbon dating is:

$N(t) = N_0 e^{-\lambda t}$

Where: $N(t)$ is the amount of carbon-14 at time $t$ $N_0$ is the initial amount of carbon-14 $\lambda$ is the decay constant $t$ is the time elapsed (age of the sample)

2. Determine the ratio $N(t)/N_0$

The given ratio of carbon-14 to carbon-12 in the sample is $1:4 \times 10^{12}$. We assume that the ratio of carbon-14 to carbon-12 in the living tree (initial ratio) was $1:1$. Therefore, the ratio $N(t)/N_0$ is:

$\frac{N(t)}{N_0} = \frac{1}{4 \times 10^{12}}$

3. Find the decay constant $\lambda$

The decay constant $\lambda$ is related to the half-life ($T_{1/2}$) of carbon-14, which is approximately 5730 years via: $\lambda = \frac{ln(2)}{T_{1/2}} = \frac{ln(2)}{5730}$

4. Solve for $t$ (time elapsed)

Using the carbon dating formula: $\frac{N(t)}{N_0} = e^{-\lambda t}$ $\frac{1}{4 \times 10^{12}} = e^{-\lambda t}$

Take the natural logarithm (ln) of both sides: $ln(\frac{1}{4 \times 10^{12}}) = -\lambda t$ $ln(1) - ln(4 \times 10^{12}) = -\lambda t$ $-ln(4 \times 10^{12}) = -\lambda t$ $t = \frac{ln(4 \times 10^{12})}{\lambda}$

Substitute the value of $\lambda$

$t = \frac{ln(4 \times 10^{12})}{\frac{ln(2)}{5730}}$ $t = \frac{ln(4 \times 10^{12}) \times 5730}{ln(2)}$

5. Calculate the value of $t$

$t = \frac{ln(4 \times 10^{12}) \times 5730}{ln(2)}$ $t = \frac{29.4234 \times 5730}{0.6931}$ $t = \frac{168585.18}{0.6931}$ $t \approx 243232.3$ years