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 half-life for \({ }^{14} \mathrm{C}\) decay, 5715 yr, determine the age of the artifact.

Short answer

The age of the wooden artifact can be determined using the radioactive decay formula, given the initial and final activities and the half-life of carbon-14. We first calculate the decay constant, plug in the known values into the decay formula, and then solve for time. The result is that the artifact is approximately 3789 years old.

Step-by-step solution

Understand the radioactive decay formula

The radioactive decay formula is given by: \[A = A_0e^{-kt}\] Where: - \(A\) is the final activity in counts/min - \(A_0\) is the initial activity in counts/min - \(k\) is the decay constant (in \(yr^{-1}\)) - \(t\) is the time (in years)

Calculate the decay constant

The decay constant \(k\) is related to the half-life \(t_{1/2}\) by the following formula: \[k = \frac{\ln 2}{t_{1/2}}\] Using the given half-life of 5715 years, we can calculate the decay constant: \[k = \frac{\ln 2}{5715} \approx 0.000121\, yr^{-1}\]

Plug in the known values into the decay formula

Now, we can plug in the given values for the initial activity (\(A_0 = 58.2\), counts/min), final activity (\(A = 38.0\), counts/min), and decay constant (\(k = 0.000121\, yr^{-1}\)) into the decay formula: \[38.0 = 58.2e^{-0.000121t}\]

Solve for time (age of the artifact)

Our goal is to find the value of \(t\), which represents the age of the artifact in years: \[\frac{38}{58.2} = e^{-0.000121t}\] We need to isolate \(t\), so we'll take the natural logarithm of both sides: \[\ln\left(\frac{38}{58.2}\right) = -0.000121t\] Now, we can divide by \(-0.000121\) to solve for \(t\): \[t = \frac{\ln\left(\frac{38}{58.2}\right)}{-0.000121} \approx 3789 \, years\] The artifact is approximately 3789 years old.

Key concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This process transforms the original nucleus into a different element or a different state. Radioactive decay occurs at a characteristic rate for each radioactive isotope, which is unaffected by external conditions such as temperature and pressure. This constancy of rate enables scientists to use radioactive decay as a reliable clock for dating artifacts and fossils.One of the key principles of radioactive decay is that it follows an exponential decay law, which can be expressed with the formula:

• The remaining activity, \(A\), over time is given by \(A = A_0e^{-kt}\)
• Where \(A_0\) is the initial activity, \(k\) is the decay constant, and \(t\) is the elapsed time.

This formula allows scientists to measure the current activity of a sample and determine its age by comparing it with its original activity.

Half-Life

Half-life is a crucial concept in understanding radioactive decay. It refers to the amount of time it takes for half of the original amount of a radioactive substance to decay. The half-life of an isotope is a specific constant and remains unchanged under typical environmental conditions. This predictability makes using half-life a reliable method for dating ancient artifacts.For carbon-14, an isotope used in archaeological dating, the half-life is 5715 years. During this time, half of the original \(^{14}C\) atoms in a sample will decay. This gradual reduction leads us to a very important realizations:

• It provides a chronological framework for determining the age of organic artifacts by comparing their current \(^{14}C\) activity against that expected in a living organism.
• The decay formula and half-life together allow the calculation of the age of artifacts by applying known rates of decay.

Understanding the half-life helps in decoding the timelines involved in the natural processes of radioactive decay.

Decay Constant

The decay constant, denoted \(k\), is a parameter that quantifies the rate of decay of a radioactive isotope. It is directly linked to the half-life and provides an alternative method for expressing how quickly a substance undergoes radioactive decay.The relationship between the decay constant and half-life is mathematically given by:

• \(k = \frac{\ln 2}{t_{1/2}}\)
• Where \(t_{1/2}\) is the half-life of the isotope.

In the case of carbon-14, with a half-life of 5715 years, the decay constant comes out to be approximately \(0.000121\, \text{yr}^{-1}\).This constant is crucial because it allows you to plug into the decay formula and estimate the time elapsed since the 'activity' of the sample has decreased. Effectively, it forms a bridge between the theoretical aspect of radioactive decay and its practical application in dating techniques.

Artifact Age Calculation

Artifact age calculation using carbon-14 dating involves a logical application of the decay formula and understanding of the decay constant. This application allows archaeologists to ascertain the age of ancient relics with remarkable accuracy.The process begins with measuring the current \(^{14}C\) activity in the sample (e.g., 38 counts per minute for a wooden artifact), and comparing it with the activity of a contemporary, or zero-age, sample of the same type (e.g., 58.2 counts per minute).Using the decay formula:

• \(A = A_0e^{-kt}\)
• We rearrange the formula to solve for the time of decay \(t\) as shown: \(t = \frac{\ln\left(\frac{A}{A_0}\right)}{-k}\)

For the given problem, this results in:

• \(\frac{38}{58.2} = e^{-0.000121t}\)
• Applying logarithms we get: \(t = \frac{\ln\left(\frac{38}{58.2}\right)}{-0.000121} \approx 3789 \text{ years}\)

This detailed method provides a scientific basis for calculating the age of ancient artifacts, offering insights into historical timelines and the age of cultural heritage.