Question

Suppose that you measure the intensity of radiation from carbon-14 in an ancient piece of wood to be 6 percent of what it would be in a freshly cut piece of wood. How old is this artifact?

Short answer

The artifact is approximately 24,710 years old.

Step-by-step solution

Understand the Problem

We have two pieces of wood: one freshly cut with 100% intensity of radiation and one ancient with only 6% intensity. The radioactive decay of carbon-14 is used to determine the age of the ancient piece based on its reduced radiation intensity.

Use the Half-Life Concept

The half-life of carbon-14 is approximately 5730 years. This means that every 5730 years, the amount of carbon-14 will be reduced by half.

Set Up the Exponential Decay Formula

The formula to calculate the remaining amount of a substance due to radioactive decay is:\[ N(t) = N_0 imes (0.5)^{t/T} \]where \( N(t) \) is the remaining amount, \( N_0 \) is the initial amount, \( t \) is the time in years, and \( T \) is the half-life of carbon-14 (5730 years).

Substitute Known Values

Since the ancient artifact has 6% of its original carbon-14, we set \( N(t)/N_0 = 0.06 \). The formula becomes:\[ 0.06 = (0.5)^{t/5730} \]

Solve for the Time Variable

To solve for \( t \), we use logarithms. Take the logarithm of both sides:\[ \log(0.06) = \log((0.5)^{t/5730}) \]Apply the logarithm power rule:\[ \log(0.06) = \frac{t}{5730} \times \log(0.5) \]

Calculate the Time

Rearrange the equation to solve for \( t \):\[ t = \frac{\log(0.06)}{\log(0.5)} \times 5730 \]Calculate the value:\[ t \approx \frac{-1.2218}{-0.301} \times 5730 \approx 24710 \text{ years} \]

Conclude the Calculation

The ancient piece of wood is approximately 24,710 years old based on the decay of carbon-14.

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. It's essential to understand that not all isotopes are stable; some, like carbon-14, undergo this decay over time. During radioactive decay, the parent isotope transforms into a different element or a different isotope of the same element.
Key points about radioactive decay include:

• The process is spontaneous and random. We can't predict exactly when a particular atom will decay, but we can predict the average time for a large number of atoms to decay.
• Radioactive decay is characterized by its half-life, a significant concept we'll explore next, which helps understand how quickly a material loses its radioactivity over time.

Carbon-14 undergoes beta decay, where it turns into nitrogen-14, emitting a beta particle. This property of carbon-14 is what allows scientists to estimate the age of previously living things, like wood in our exercise.

Half-Life Calculation

The half-life is the time required for half of the radioactive isotopes in a sample to decay. In the example of carbon-14, its half-life is approximately 5730 years.
Understanding half-life helps in determining how old an object could be based on the remaining radioactive isotopes it contains. If you start with a certain amount of a radioactive substance, after one half-life, only half of the original quantity remains.
For the problem at hand:

• The freshly cut wood has 100% of its carbon-14 content.
• The ancient wood containing 6% of its original carbon-14 amount means it has undergone several half-lives.

Calculating the number of half-lives that have passed helps in finding out the age of a particular object.

Carbon-14

Carbon-14 is a radioactive isotope of carbon, used widely in dating archaeological findings. Every living organism contains carbon-14 due to its interaction with the atmosphere.
When organisms perish, they cease taking in new carbon-14, and the existing isotopes start to decay at a known rate, governed by their half-life.
Carbon-14 dating, known as radiocarbon dating, enables scientists to determine how long ago an organism died by measuring the remaining carbon-14. This technique is remarkably useful in archaeology, providing insights into the timeline of human activities.

• It’s important because it's present in organic materials, enabling the dating of artifacts like wood and cloth once alive.
• Carbon-14 concentration reduction from its original state helps estimate the age of a sample, as we observed in the solution to the given exercise.

Exponential Decay Formula

The exponential decay formula, vital for understanding radioactive decay, expresses how the quantity of a radioactive substance decreases over time.
This mathematical representation helps scientists predict how long it will take for a certain percentage of the isotope to remain.
The formula is given by:\[ N(t) = N_0 \times (0.5)^{t/T} \]where:

• \( N(t) \) represents the remaining amount of the substance after time \( t \).
• \( N_0 \) is the initial quantity.
• \( T \) stands for the half-life.

In our exercise, we know the current amount, the initial amount, and the half-life of carbon-14, allowing us to find the time since the artifact was last alive by solving the formula for \( t \). The solution included using logarithms to re-arrange the equation and solve for the elapsed time. This approach underlines its importance in actual applications of carbon dating.