Question

Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of \({ }^{14} \mathrm{C}\) is 5715 years.)

Short answer

To calculate the time elapsed, calculate the right hand side of the final equation. The result will be the time elapsed since the tree was burned to make the ancient charcoal in years.

Step-by-step solution

Understanding the half-life formula

The formula for radioactive decay is \[N = N_0 \cdot (0.5)^{t/T}\]. Where:\nN is the final amount of the substance\nN_0 is the initial amount of the substance\nT is the half-life period of the substance (in this case, 5715 years for Carbon-14)\nt is the time elapsed.

Plugging in the given values

In our case, \(N = 0.15N_0\) of modern charcoal is left. Thus, we can substitute these values into the formula to get \(0.15 = (0.5)^{t/5715}\).

Solving for time elapsed (t)

Next, we solve for t by taking the natural log on both sides, we get: \(ln(0.15) = \frac{t}{5715} \cdot ln(0.5)\). Solving for t gives: \(t = \frac{5715 \cdot ln(0.15)}{ln(0.5)}\). Calculate the right-hand-side to find the time elapsed.

Key concepts

Radioactive Decay Formula

The radioactive decay formula is pivotal in understanding the behavior of isotopes over time. It establishes how the quantity of a radioactive element decreases over time. This process is described by the equation:
\[N = N_0 \cdot (0.5)^{t/T}\]
where:

• \(N\) represents the amount of the substance that remains after time \(t\).
• \(N_0\) is the original amount of the substance.
• \(T\) is the half-life of the substance, which is the time required for half of the original amount to decay.
• \(t\) is the elapsed time since the beginning of the decay process.

To put this into context, imagine you have a certain number of radioactive atoms. After each half-life period, half of the remaining atoms will have decayed, and half will remain. This decay process continues over successive half-lives until the amount of the substance approaches insignificance.
This formula is integral for Carbon-14 dating, where researchers determine the age of archaeological finds by measuring the remaining Carbon-14.

Half-Life Calculation

Calculating a half-life is crucial for interpreting the age of objects using Carbon-14 dating. The half-life of a radioactive element is constant and does not depend on the initial amount of material. As seen in the exercise, the half-life of Carbon-14 (\(^{14}C\)) is 5715 years. Knowing this, we can determine how many half-lives have passed since a sample stopped absorbing Carbon-14, which is typically when the organism died.
To calculate the time passed using a half-life, we need to manipulate the radioactive decay formula to solve for \(t\):
\[t = \frac{ln(N/N_0)}{ln(0.5)} \cdot T\]
With this equation, the time that has elapsed since the organism, such as a tree, died can be calculated. In the context of the example with the ancient charcoal, by solving the half-life formula with the given percentage of Carbon-14 remaining, we can determine the tree's burn date, thus revealing the charcoal's age.

Radioactive Carbon

Radioactive Carbon, or Carbon-14 (\(^{14}C\)), is a naturally occurring isotope of carbon that is an essential tool in archaeological and geological dating. Unlike stable isotopes, such as Carbon-12, Carbon-14 is unstable and decays over time. It is continually formed in the upper atmosphere as cosmic rays interact with nitrogen atoms, and it is assimilated into living organisms during their lifetime.
Once the organism dies, it stops exchanging Carbon-14 with the environment, and the radioactive carbon within begins to decay at a known rate, making it an excellent marker for determining the age of organic remains. This decay makes it possible to measure the proportion of radioactive carbon in a sample and, using the decay formula, estimate when the organism stopped taking in \(^{14}C\), which essentially marks the time of its death. Carbon-14 dating can provide estimates for organic material up to about 50,000 years old, beyond which there's typically not enough \(^{14}C\) remaining to measure reliably.