In 1946, Willard Libby first proposed the metnod of radiocar bon olating. The radioactive isotope carbon-14 is Known to have a half-life of 5730 ± 40 years. King Tuts date of death is a pproximated as 1325 BC. If tested today, what proportion of C-14 remains in the mummy? lo (7)
In 1946, Willard Libby first proposed the metnod of radiocar bon olating. The radioactive isotope carbon-14 is Known to have a half-life of 5730 ± 40 years. King Tuts date of death is a pproximated as 1325 BC. If tested today, what proportion of C-14 remains in the mummy? lo (7)
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![### Understanding Radiocarbon Dating
**Overview:**
In 1946, Willard Libby introduced the method of radiocarbon dating. This process uses the radioactive isotope Carbon-14, which is known to have a half-life of approximately 5730 ± 40 years. For example, if King Tutankhamun's date of death is approximated as 1325 B.C., we can calculate the proportion of Carbon-14 remaining in the mummy if tested today.
#### Key Calculation Steps:
1. **Find the Decay Constant (K):**
- The formula begins with the relationship:
\[
\frac{1}{2} = 1 \cdot e^{K \times 5730}
\]
- Taking the natural logarithm of both sides:
\[
\ln\left(\frac{1}{2}\right) = \ln\left(e^{K \times 5730}\right)
\]
- Which simplifies to:
\[
\ln\left(0.5\right) = K \times 5730
\]
- Solve for \(K\):
\[
K = \frac{\ln\left(0.5\right)}{5730} \approx -0.000121
\]
2. **Exponential Decay Function:**
- The decay of Carbon-14 over time is given by the function:
\[
C(t) = C_0 \cdot e^{-0.000121 \times t}
\]
### Explanation of Diagram:
- **Equation Derivations:**
- The board illustrates the derivation of the decay constant \(K\), emphasizing the natural logarithm operations needed to rearrange the half-life formula into a usable form for further calculations.
- **Calculation Flow:**
- The flow from determining \(\ln\left(\frac{1}{2}\right)\) through to solving for \(K\) is mapped with lines, indicating logical progression.
Understanding how to derive constant values and use exponential functions is crucial to applying radiocarbon dating methods accurately in archaeology and other fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F113b0878-5af8-4998-be4b-de5d66497a2a%2F82602aad-ea88-4cef-b823-4c3a508d5167%2Fa74c26m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Radiocarbon Dating
**Overview:**
In 1946, Willard Libby introduced the method of radiocarbon dating. This process uses the radioactive isotope Carbon-14, which is known to have a half-life of approximately 5730 ± 40 years. For example, if King Tutankhamun's date of death is approximated as 1325 B.C., we can calculate the proportion of Carbon-14 remaining in the mummy if tested today.
#### Key Calculation Steps:
1. **Find the Decay Constant (K):**
- The formula begins with the relationship:
\[
\frac{1}{2} = 1 \cdot e^{K \times 5730}
\]
- Taking the natural logarithm of both sides:
\[
\ln\left(\frac{1}{2}\right) = \ln\left(e^{K \times 5730}\right)
\]
- Which simplifies to:
\[
\ln\left(0.5\right) = K \times 5730
\]
- Solve for \(K\):
\[
K = \frac{\ln\left(0.5\right)}{5730} \approx -0.000121
\]
2. **Exponential Decay Function:**
- The decay of Carbon-14 over time is given by the function:
\[
C(t) = C_0 \cdot e^{-0.000121 \times t}
\]
### Explanation of Diagram:
- **Equation Derivations:**
- The board illustrates the derivation of the decay constant \(K\), emphasizing the natural logarithm operations needed to rearrange the half-life formula into a usable form for further calculations.
- **Calculation Flow:**
- The flow from determining \(\ln\left(\frac{1}{2}\right)\) through to solving for \(K\) is mapped with lines, indicating logical progression.
Understanding how to derive constant values and use exponential functions is crucial to applying radiocarbon dating methods accurately in archaeology and other fields.
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