Vid analysen fann man att halten kol-14 var 52% av halten i levande material.För hur många år sedan dog Ismannen?

1. Break Down the Problem

To determine how many years ago Ötzi the Iceman (Ismannen) died, we will use the concept of radiocarbon dating, which is based on the decay of carbon-14 ($^{14}C$). We know that the current ratio of $^{14}C$ in the sample is 52% of that in living material.

2. Relevant Concepts

The decay of carbon-14 can be modeled with the half-life formula, which states: $N(t) = N_0 \left( \frac{1}{2} \right)^{t/T_{1/2}}$ where:

• $N(t)$ = the amount of carbon-14 remaining at time $t$,
• $N_0$ = the original amount of carbon-14,
• $T_{1/2}$ = the half-life of carbon-14 (approximately 5730 years),
• $t$ = time that has elapsed.

Given that $N(t) = 0.52 N_0$, we can rearrange the equation to solve for $t$.

3. Analysis and Detail

Rearranging the decay formula for our specific case: $0.52 N_0 = N_0 \left( \frac{1}{2} \right)^{t/5730}$ Dividing both sides by $N_0$: $0.52 = \left( \frac{1}{2} \right)^{t/5730}$ Taking the logarithm of both sides: $\log(0.52) = \frac{t}{5730} \log\left(\frac{1}{2}\right)$ Solving for $t$: $t = 5730 \times \frac{\log(0.52)}{\log(0.5)}$

4. Verify and Summarize

Now we substitute the logarithmic values: $\log(0.52) \approx -0.2816\\ \log(0.5) \approx -0.3010$

Calculating $t$: $t \approx 5730 \times \frac{-0.2816}{-0.3010} \approx 5730 \times 0.9355 \approx 5356.5 \text{ years}$

Final Answer

Ötzi the Iceman died approximately 5357 years ago.