Question

Crater Lake in Oregon was formed by volcanic explosion. If a tree charred by the explosion gave an activity of \(\sim 7.7 \mathrm{dpm}\) and the half-life of \(\mathrm{C}-14\) is 5730 years, what is the approximate age of Crater Lake?

Short answer

The approximate age of Crater Lake is about 5730 years.

Step-by-step solution

Understand the Given Data

You are given that the current activity of the tree is \(7.7\) dpm (disintegrations per minute). You also know that the half-life of \(^{14}\!C\) is \(5730\) years. Your task is to use this data to estimate the age of Crater Lake.

Determine Initial Activity

The initial activity of carbon in a living tree would have been \(15.3\) dpm, as this is the expected activity for \(^{14}\!C\) in living organisms.

Use the Decay Formula

Use the decay formula: \[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \(A\) is the current activity (\(7.7\) dpm), \(A_0\) is the initial activity (\(15.3\) dpm), \(t\) is the time elapsed, and \(T_{1/2}\) is the half-life (5730 years).

Solve for Time Elapsed (t)

Rearrange the formula to solve for \(t\): \[ 7.7 = 15.3 \left( \frac{1}{2} \right)^{\frac{t}{5730}} \] Take natural logarithms on both sides and solve for \(t\): \[ \ln(7.7) = \ln(15.3) + \frac{t}{5730} \ln\left( \frac{1}{2} \right) \] \[ t = \frac{5730 \times (\ln(7.7) - \ln(15.3))}{\ln(\frac{1}{2})} \]

Calculate t

Evaluate the expression: calculate \(\ln(7.7) - \ln(15.3)\), then divide by \(\ln(0.5)\), and multiply by 5730 to find \(t\). You should find: \[ t \approx 5730 \times \frac{\ln(7.7) - \ln(15.3)}{\ln(0.5)} \approx 5730 \times \frac{-0.693}{-0.693} \approx 5730 \] This results in \(t \approx 5730\) years (approximately).

Conclude the Estimate

Based on the calculation, the approximate age of Crater Lake (since the volcanic explosion) is about 5730 years.

Key concepts

Carbon-14 Decay

Carbon-14 is a radioactive isotope of carbon that is present in all living organisms. It is continuously being formed in the atmosphere through the interaction of cosmic rays with nitrogen, and then gets incorporated into plants through photosynthesis. Animals and humans consume these plants, maintaining a constant level of carbon-14 in their bodies during their lifetimes.

When an organism dies, it no longer takes in carbon-14, and the existing carbon-14 atoms begin to decay into nitrogen at a constant rate. This decay process follows a predictable pattern known as radioactive decay. Specifically, carbon-14 decays by beta emission, releasing a small amount of energy.

• This natural decay process is constant and unaffected by environmental factors.
• It is characterized by a specific decay, or disintegration, rate known as disintegrations per minute (dpm).
• The rate of decay is used to determine the age of ancient biological materials through radioactive dating techniques.

Half-Life Calculation

The term "half-life" refers to the time required for half of the radioactive atoms in a sample to decay. For carbon-14, this time period is 5730 years, which means that in 5730 years, half of the carbon-14 atoms in a sample will have decayed and turned into nitrogen.

The concept of half-life is crucial in radioactive dating because it allows scientists to estimate the age of an object by understanding how much of the radioactive material remains. In calculations, the half-life is denoted as \( T_{1/2} \), and knowing this value, scientists can determine the elapsed time since the death of a living organism, by comparing the remaining activity with the initial activity when the organism was alive.

• This is done by using the exponential decay formula to find the time period that has passed.
• Half-life calculations are essential in applications like archaeology and geology where understanding the age of artifacts or geological formations is required.

Exponential Decay Formula

The exponential decay formula is a mathematical expression that describes the process of radioactive decay, allowing us to calculate the age of an object based on remaining radioactivity. The formula is expressed as follows:
\[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
In this formula:

• \( A \) is the current activity of the sample, measured in disintegrations per minute (dpm).
• \( A_0 \) is the initial activity of the sample, or the activity the sample would have had when the organism was still alive.
• \( t \) is the time that has passed since the death of the organism.
• \( T_{1/2} \) is the known half-life of the radioactive isotope, for carbon-14, this is 5730 years.

Solving this equation for \( t \) involves using logarithms, as it allows the unknown \( t \) to be isolated on one side of the equation. This process helps in deriving the age of archaeological finds, geological samples, as well as enabling the precise dating of historical events.