Question

The \({ }^{14} \mathrm{C}\) content of an ancient piece of wood was found to be one-tenth of that in living trees. How many years old is this piece of wood? \(\left(t_{1 / 2}=5730\right.\) years for \({ }^{14} \mathrm{C}\).)

Short answer

The piece of wood is approximately 18998 years old.

Step-by-step solution

Understand the Half-Life Concept

The half-life of a radioactive isotope, in this case, Carbon-14 (C^{14}), is the amount of time it takes for half of the radioactive atoms in a sample to decay. The given half-life of Carbon-14 is 5730 years.

Determine the Number of Half-Lives Passed

Since the C^{14} content of the wood is one-tenth that of living trees, we need to find out how many half-lives have passed to reach this fraction. This can be expressed as (1/2)^n = 1/10, where n is the number of half-lives.

Solve for the Number of Half-Lives (n)

Taking the log base 2 of both sides of the equation gives log_2(1/2)^n = log_2(1/10), which simplifies to n = log_2(1/10). Solving for n gives n approx 3.322.

Calculate the Age of the Wood

Multiply the number of half-lives by the duration of one half-life to find the age of the wood: Age = n times t_{1/2} = 3.322 times 5730 years.

Key concepts

Carbon-14 Dating

Carbon-14 dating is an invaluable tool for archaeologists and geologists, allowing them to determine the age of ancient organic materials. This technique hinges on the presence of carbon-14, a radioactive isotope that decays over time. Living organisms constantly exchange carbon with their surroundings, so the ratio of carbon-14 to carbon-12 remains fairly constant. However, when the organism dies, it no longer absorbs carbon, and the carbon-14 begins to decay at a predictable rate, known as the half-life.

The task of dating entails measuring the remaining carbon-14 in a sample and comparing it to the expected amount in a living organism. Through mathematical calculations using the half-life, scientists can estimate the time that has passed since the death of the organism.

For effective use of carbon-14 dating, understanding its principles—including the concept of half-life and the logarithmic nature of radioactive decay—is key. When the ratio deviates from the living standard, such as the case in our exercise where the wood contains one-tenth the carbon-14, it indicates that multiple half-lives have passed since the wood was part of a living tree.

Half-Life of Isotopes

The half-life of an isotope is a fundamental concept in radioactive dating. It is defined as the time required for half the atoms of a radioactive sample to decay. Half-life values vary dramatically among different isotopes, from fractions of a second to billions of years.

In our example of Carbon-14, with a half-life of 5730 years, it means that every 5730 years, half of the carbon-14 atoms in a given sample will have decayed to form nitrogen-14. After one half-life, the remaining amount of carbon-14 would be half of the original. After two half-lives, one quarter would remain, and so on. This predictable pattern allows scientists to retroactively calculate how many half-lives have passed since the organism stopped absorbing carbon-14, thereby revealing its age.

Understanding the half-life concept helps in grasping the bigger picture of isotopes and their decay over time, and it paves the way towards understanding more complex decay patterns seen in different elements.

Radioactive Decay

Radioactive decay is a spontaneous process where unstable atomic nuclei lose energy by emitting radiation. The nature of the emitted radiation can be alpha particles, beta particles, or gamma rays. This process is random for individual atoms, yet for large samples, it occurs at a statistically consistent rate described by the half-life.

In relation to our exercise, carbon-14 undergoes beta decay, where a neutron in the nucleus transforms into a proton, emitting an electron (beta particle) and an anti-neutrino in the process. The result is an atom of nitrogen-14. The carbon-14 decay rate is crucial for dating samples, as it is constant and dependable over extended periods, allowing for accurate age determinations. The decay of carbon-14 to nitrogen-14 at a consistent rate is an exemplary model of this process, demonstrating the steady clockwork that makes such dating methods possible.

Logarithmic Equations

Logarithmic equations are vital in solving problems involving exponential relationships, such as the decay of radioactive isotopes. Logarithms are the inverse operations of exponentiation. The logarithm of a number is the power to which a given base must be raised to obtain that number.

In our example, to find the number of half-lives (n) that have passed, we employed a logarithmic equation using the base 2 because the half-life concept deals with a quantity being reduced by half each time. The equation \( (1/2)^n = 1/10 \) is an exponential equation where \( n \) is the exponent we want to find. By taking the logarithm based on 2 of both sides (to 'undo' the exponentiation), we can solve for \( n \) and find that approximately 3.322 half-lives have passed. When working with logarithms, it is crucial to understand properties like the product rule, quotient rule, and power rule, to effectively manipulate and solve equations. Logarithms simplify the process of working with exponential data, making them indispensable in fields like radiometric dating, sound intensity, pH chemistry, and more.