Problem 65 - Chemistry and Chemical Reactivity

Question

A piece of charred bone found in the ruins of a Nativee \(e\) American village has a \(^{14} \mathrm{C} /^{12} \mathrm{C}\) ratio that is \(72 \%\) of the ratio found in living organisms. Calculate the age of the bone fragment. $$\left(t_{1/2} \text { for }^{14} \mathrm{C} \text { is } 5.73 \times 10^{3}\text { years.) }\right.$$

Step-by-Step Solution

Answer

The bone fragment is approximately 2715 years old.

Step 1: Understand the Relationship

The relationship between the age of a sample and the ratio of carbon isotopes is governed by exponential decay. The formula to find the age \(t\) is given by \[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] where \(N/N_0\) is the remaining ratio, and \(t_{1/2}\) is the half-life of the isotope.

Step 2: Apply the Given Values

You're given that the ratio \(N/N_0 = 0.72\) and the half-life \(t_{1/2} = 5.73 \times 10^3\) years. Substitute these into the equation: \[ 0.72 = \left(\frac{1}{2}\right)^{\frac{t}{5730}} \]

Step 3: Solve for Time \(t\)

To solve for \(t\), take the natural logarithm (ln) of both sides. This gives: \[ \ln(0.72) = \frac{t}{5730} \ln\left(\frac{1}{2}\right) \] Then rearrange to solve for \(t\): \[ t = \frac{\ln(0.72)}{\ln(0.5)} \times 5730 \]

Step 4: Calculate the Value

Calculate the numerical values: \[ \ln(0.72) \approx -0.3285 \] \[ \ln(0.5) \approx -0.6931 \] Now substitute these into the rearranged equation: \[ t = \frac{-0.3285}{-0.6931} \times 5730 \] \[ t \approx 0.4742 \times 5730 \approx 2715 \]

Step 5: Conclusion

The age of the bone fragment is approximately 2715 years.

Key Concepts

Radioactive Decay

Radioactive decay is a naturally occurring process wherein an unstable atomic nucleus loses energy by emitting radiation. This process results in the transformation of elements over time. For carbon dating, we particularly focus on the radioactive isotope of carbon, known as carbon-14 (\(^{14}C\)).
Carbon-14 is constantly formed in the atmosphere and absorbed by living organisms. Once the organism dies, it no longer absorbs carbon-14, and the existing \(^{14}C\) begins to decay.
Understanding this decay process is crucial for calculating the age of organic materials. The decay follows an exponential pattern, meaning it decreases at a rate proportional to its current value. This allows us to estimate ages of artifacts in a very precise manner using known decay rates.

Half-life Calculation

The concept of half-life is key in radioactive decay. Half-life is the time required for half of the radioactive nuclei in a sample to decay. For \(^{14}C\), this is approximately 5730 years.
Calculating the half-life allows us to determine how long it takes for a given amount of substance to diminish.To calculate the age of an artifact using half-life, we compare the current isotopic ratio of \(^{14}C\) in the sample to that of a living organism. In the exercise, the sample has 72% of the \(^{14}C\) compared to living organisms.
The half-life formula used is \[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] where \(N/N_0\) is the proportion of remaining \(^{14}C\) and \(t_{1/2}\) is the half-life. By solving the equation, we determine the age of samples based on the remaining carbon-14.

Isotopic Ratio

Isotopes are different forms of an element with the same number of protons but a different number of neutrons. For carbon, \(^{14}C\) and \(^{12}C\) are two common isotopes.
When we talk about isotopic ratios in carbon dating, we specifically refer to the ratio of \(^{14}C\) to \(^{12}C\). This ratio decreases over time as \(^{14}C\) decays after the organism dies.
The isotopic ratio is crucial for dating because it provides a measurable indication of time elapsed. A decrease in the \(^{14}C/^{12}C\) ratio from the reference ratio in a living organism allows us to calculate the time that has passed since death. Understanding isotopic ratios is hence vital to the accuracy of carbon dating.

Archaeological Dating

Archaeological dating employs methods to determine the age of artifacts, sites, or historical works. Carbon dating is one of the most popular techniques, especially useful for dating organic materials like bones, wood, and charcoal.
The age determination based on carbon dating contributes tremendously to archaeology by providing a timeline for discoveries. This helps assess cultural and historical contexts.
By calculating the age of organic remains, archaeologists can effectively piece together ancient human activities and environmental conditions.

• Enhances historical knowledge by providing dates directly linked to artifacts.
• Utilizes scientific methods to offer objective data.
• Correlates archaeological findings with historical records.

By understanding archaeological dating, we can better understand past human civilizations and their evolution over time.