Question

Charcoal found in the Lascaux cave in France, site of many prehistoric cave paintings, was observed in 1950 to decay at a rate of \(2.4\) disintegrations/min per gram of carbon. What is the age of the charcoal \(\mathrm{BP}\) if currently living organisms decay at the rate of \(15.3\) disintegrations/min per gram of carbon? The halflife of \({ }^{14} \mathrm{C}\) is 5715 years.

Short answer

Age of charcoal: Approx. 17,300 years BP.

Step-by-step solution

Understand the Problem

We need to determine the age of the charcoal in years Before Present (BP) using the decay rate of carbon. The decay rate of the charcoal is given as \(2.4\) disintegrations/min per gram of carbon, and the decay rate of currently living organisms is \(15.3\) disintegrations/min. The halflife of \(^{14}C\) is \(5715\) years. We will use the formula that relates the current decay rate, initial decay rate, and the half-life to calculate the age.

Write the Formula for Decay

The formula for radioactive decay is \( N(t) = N_0 \cdot (\frac{1}{2})^{\frac{t}{T_{1/2}}} \), where \( N(t) \) is the current decay rate, \( N_0 \) is the initial decay rate, \( T_{1/2} \) is the half-life of the substance, and \( t \) is the age. In our case, \( N(t) = 2.4 \) and \( N_0 = 15.3 \).

Key concepts

Half-Life

The concept of half-life is essential in understanding radioactive decay, especially for elements like carbon-14. A half-life, denoted as \(T_{1/2}\), is the time required for half of the radioactive atoms in a sample to decay into a more stable form. For carbon-14, this amount of time is 5,715 years.
This means that if we start with a given quantity of carbon-14, after 5,715 years, only half of that amount will remain; the rest would have decayed into nitrogen-14.

• Understanding half-life allows us to estimate the age of organic materials by calculating how much carbon-14 has decayed over time.
• This concept is crucial in radiocarbon dating because it provides a consistent measure to compare the decay of carbon-14 regardless of initial amounts or external conditions.

Imagine a stopwatch that keeps resetting every 5,715 years. Each reset tells us half of the original substance remains, providing a reliable way to track the decay progress through ages.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. For carbon-14, which is a radioactive isotope of carbon, this decay occurs when it transforms into a more stable element, nitrogen-14, over time.
The decay process is random, yet it occurs at a predictable rate, which is why it's incredibly valuable for dating ancient objects.

• The decay of carbon-14 follows an exponential pattern, represented mathematically by the formula \( N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \).
• Here \(N(t)\) represents the amount of carbon-14 remaining at time \(t\), \(N_0\) is the initial amount, and \(T_{1/2}\) is the half-life of carbon-14.

This predictable nature of radioactive decay forms the backbone of radiocarbon dating, as scientists can measure how much carbon-14 remains in a sample and use the exponential decay formula to calculate the sample's age.

Carbon-14 Decay Rate

The decay rate of carbon-14 is a crucial factor in determining the age of archeological samples. For living organisms, the decay rate is measured in disintegrations per minute (dpm) per gram of carbon. In the given exercise, it states currently living organisms decay at a rate of 15.3 dpm, while the charcoal sample from Lascaux cave decays at 2.4 dpm.
To find the age of the charcoal, we observe how this decay rate has changed over time. The lower the decay rate from the current standard, the older the sample is, as it signifies more carbon-14 has decayed.

• By comparing the present decay rate \(N(t) = 2.4\) dpm with the initial (or living) decay rate \(N_0 = 15.3\) dpm while factoring in the known half-life, the sample's age can be calculated.
• This is achieved using the decay formula mentioned earlier, where the age \(t\) can be solved to show how many half-lives have passed.

This decay rate analysis allows researchers to paint a more accurate picture of historical timelines, understanding more about when specific artifacts or remains originated.