Problem 40 - Chemistry The Central Science

Question

A wooden artifact from a Chinese temple has a \({ }^{14} \mathrm{C}\) activity of 38.0 counts per minute as compared with an activity of 58.2 counts per minute for a standard of zero age. From the halflife for \({ }^{14} \mathrm{C}\) decay, \(5715 \mathrm{yr}\), determine the age of the artifact.

Step-by-Step Solution

Answer

The age of the artifact is approximately \(2797\) years.

Step 1: Calculate the decay constant (\(\lambda\))

We know the halflife of \({}^{14}\mathrm{C}\), which is \(5715 \mathrm{yr}\). We can use the formula: \(\lambda = \frac{ln(2)}{T_{1/2}}\) Where: - \(\lambda\) is the decay constant - \(T_{1/2}\) is the halflife Plugging in the values, \(\lambda = \frac{ln(2)}{5715 \mathrm{yr}} = 1.21 \times 10^{-4} \mathrm{yr}^{-1}\)

Step 2: Apply the decay equation to find the age of the artifact

We need to find the time \(t\), using the decay equation: \(N = N_0 \times e^{-\lambda t}\) Rearrange the equation to solve for \(t\): \(t = \frac{ln(\frac{N}{N_0})}{-\lambda}\) Plugging in the values, where \(N = 38.0\) counts/min and \(N_0 = 58.2\) counts/min, \(t = \frac{ln(\frac{38.0}{58.2})}{-1.21 \times 10^{-4} \mathrm{yr}^{-1}} = 2797.49 \mathrm{yr}\)

Step 3: Final Answer

The age of the artifact is approximately \(2797\) years.

Key Concepts

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This process occurs naturally in various isotopes, such as carbon-14, which is widely used in carbon dating. This decay transforms the element over time and involves the emission of particles or electromagnetic waves, leading to a different, more stable, nucleus.
The rate at which radioactive decay occurs is characterized by its decay constant, denoted as \( \lambda \). This constant is essential for understanding how different isotopes behave over time. Each isotope, like carbon-14, has its decay constant, reflecting the speed of its decay process.
Some key notes about radioactive decay:

• It is a random process, and while individual atoms cannot be predicted, large numbers follow predictable patterns.
• Different isotopes have unique decay rates, identified by their decay constants.
• It forms the basis for dating ancient objects, like artifacts, by measuring their remaining radioactive isotopes.

The decay constant is crucial in calculations for determining the age of artifacts by interpreting radioactive decay activity compared to fresh samples.

Half-life

Half-life is the time it takes for half of a given amount of a radioactive isotope to decay. For carbon-14, this half-life is approximately 5715 years, which means after this period, only half of the original carbon-14 remains in a sample.
This concept of half-life allows scientists to estimate the ages of ancient objects by measuring the remaining radioactivity and comparing it to the known initial quantity.
Key aspects of half-life include:

• It is unique to every isotope; for example, carbon-14 has a half-life of 5715 years, quite different from other isotopes.
• Half-life provides a "clock" to measure the age since radioactive decay operates predictably over time.
• It helps in situations where the decay constant \( \lambda \) is used, utilizing the formula \( \lambda = \frac{\ln(2)}{T_{1/2}} \) to relate the half-life with the decay rate.

Understanding half-life is crucial when using carbon-14 dating, as it's the reference point to measure changes in radioactivity within ancient artifacts, hence determining their age.

Exponential Decay Formula

The exponential decay formula captures the process by which radioactive substances reduce over time. This mathematical expression describes how the quantity of a radioactive isotope decreases at a rate proportional to its current amount. For carbon dating, this is particularly useful.
The decay formula is written as \( N = N_0 \times e^{-\lambda t} \). Here, \( N_0 \) represents the initial quantity of the isotope, and \( N \) is the remaining quantity after time \( t \). The decay constant \( \lambda \) is used to influence the rate of decay, reflecting how quickly the substance transforms.
Important points about the exponential decay formula:

• It is derived from the need to understand changes in quantity over time, central to processes like carbon dating.
• Rearranging the formula to solve for time \( t \) provides a way to calculate the age of samples when original and remaining quantities are known.
• It relies on understanding the natural logarithm \( \ln \), which is crucial for calculating the decay constant and time correctly.

The exponential decay formula is invaluable in carbon dating, offering a method to calculate the age of artifacts by understanding how much carbon-14 is left compared to when it was 'freshly' formed.