Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

(Carbon Dating) All living things contain carbon 12 , which is stable, and carbon 14, which is radioactive. While a plant or animal is alive, the ratio of these two isotopes of carbon remains unchanged, since the carbon 14 is constantly renewed: after death, no more carbon 14 is absorbed. The half-life of carbon 14 is 5730 years. If charred logs of an old fort show only \(70 \%\) of the carbon 14 expected in living matter, when did the fort burn down? Assume that the fort burned soon after it was built of freshly cut logs.

Step-by-Step Solution

Answer

The fort burned down approximately 3017 years ago.

Step 1: Understanding Carbon Dating

Carbon dating is based on the decay of carbon-14, a radioactive isotope of carbon. The ratio of carbon-12 to carbon-14 remains constant while the organism is alive, but after death, carbon-14 decays with a half-life of 5730 years, meaning every 5730 years, half of the carbon-14 present decays.

Step 2: Understanding the Problem

The problem states that the charred logs contain only 70% of the carbon-14 expected in a living organism. This means 30% of the original carbon-14 has decayed.

Step 3: Write Down the Decay Formula

The formula for radioactive decay is given by \[N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where \(N(t)\) is the amount of substance that still remains after time \(t\), \(N_0\) is the initial amount of substance, and \(T_{1/2}\) is the half-life of the substance. In this problem, \(T_{1/2}\) is 5730 years.

Step 4: Set Up the Initial Equation

We want to find out when only 70% of the initial carbon-14 remains. Thus, we set up the equation:\[0.7 N_0 = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{5730}}\]

Step 5: Simplify and Solve the Equation

Divide both sides by \(N_0\) to get:\[0.7 = \left(\frac{1}{2}\right)^{\frac{t}{5730}}\]Take the natural logarithm of both sides:\[\ln(0.7) = \frac{t}{5730} \times \ln\left(\frac{1}{2}\right)\]Solve for \(t\):\[t = \frac{\ln(0.7)}{\ln\left(\frac{1}{2}\right)} \times 5730\]

Step 6: Calculate the Result

Now we just compute the value:\[\ln(0.7) \approx -0.3567, \quad \ln\left(\frac{1}{2}\right) \approx -0.6931\]\[t \approx \frac{-0.3567}{-0.6931} \times 5730 \approx 3016.52\]Rounding the result gives \(t \approx 3017\) years.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. In the case of carbon dating, the focus is specifically on the decay of carbon-14, a radioactive isotope of carbon. This process is essential for determining the age of once-living materials. When living organisms die, they stop absorbing carbon-14, and the existing carbon-14 begins to decay.
The rate of decay is predictable and is used to estimate the time that has elapsed since the organism died.

• The decay process results in the conversion of carbon-14 into nitrogen-14.
• This gradual transformation involves the release of beta particles.
• The rate of decay is characterized by the half-life, which is 5730 years for carbon-14.

Using these principles, scientists can determine the age of archaeological finds and geological samples by measuring the remaining carbon-14 content relative to its initial amount.

Half-life

The half-life is a crucial concept when it comes to understanding radioactive decay and carbon dating. It represents the time required for half of a particular radioactive isotope to decay.
For carbon-14, the half-life is precisely 5730 years.
This means that if you start with 100 atoms of carbon-14, only 50 would remain after 5730 years.

• The half-life remains constant, unaffected by external factors like temperature or pressure.
• It is a statistical measure that describes how long it takes for half of the isotope in a sample to decay.
• Knowing the half-life allows us to calculate how long it has been since the death of an organism by analyzing how much carbon-14 remains.

In the given exercise, because the amount of carbon-14 is 70% of what was originally present, the half-life concept helps us determine that approximately 3017 years have passed since the fort burned down.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons. In the context of carbon dating, we deal with two isotopes of carbon: carbon-12 and carbon-14.
Carbon-12 is stable and does not undergo radioactive decay.
Carbon-14, however, is unstable and radioactive.

• Both isotopes have six protons, but carbon-14 has two more neutrons than carbon-12, giving it a total of eight neutrons.
• The presence of more neutrons in carbon-14 makes it heavier and radioactive.
• Isotopes play a critical role in carbon dating because the decay of carbon-14 allows scientists to determine how long it has been since an organism died.

Understanding isotopes helps explain why living organisms have a constant ratio of carbon-12 to carbon-14, which changes after death as carbon-14 decays.

Carbon-14

Carbon-14 is a naturally occurring isotope of carbon that is radioactive. Its presence and eventual decay are central to the process of carbon dating.
Carbon-14 is continuously formed in the atmosphere through the interaction of cosmic rays with nitrogen.
Living organisms incorporate carbon-14 along with carbon-12 from the carbon dioxide they ingest.

• Once an organism dies, it no longer absorbs carbon-14 from the atmosphere.
• The carbon-14 present in the organism begins to decay at a known rate, defined by its half-life of 5730 years.
• By measuring the remaining carbon-14 in a sample and comparing it to the expected level in living organisms, scientists estimate the elapsed time since the organism’s death.

The understanding of carbon-14's formation and decay process is essential to implementing carbon dating techniques correctly and interpreting results accurately in archaeological and geological studies.