Question

Anthropologists can estimate the age of a bone or other sample of or- ganic matter by its carbon-14 4 content. The carbon-14 in a living organ- ism is constant until the organism dies, after which carbon-14 decays with first-order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains 19.5\(\%\) of the \(\mathrm{C}-14\) found in living or- ganisms. How old is the bone?

Short answer

The bone is approximately 11789 years old.

Step-by-step solution

- Understanding the half-life formula

The decay of carbon-14 follows first-order kinetics, which can be described with the formula for half-life decay: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the initial amount of substance, and \( t_{1/2} \) is the half-life of the substance which is given as 5730 years for carbon-14.

- Setting up the equation

For the bone, we have that it contains 19.5\% of the original carbon-14. We can set up the equation: \[ 0.195N_0 = N_0 \left(\frac{1}{2}\right)^{\frac{t}{5730}} \] where \( t \) is the age of the bone we want to find.

- Solving for t

Divide both sides of the equation by \( N_0 \) to get \[ 0.195 = \left(\frac{1}{2}\right)^{\frac{t}{5730}} \]. Take the natural logarithm of both sides to solve for \( t \), \[ \ln(0.195)= \ln\left(\left(\frac{1}{2}\right)^{\frac{t}{5730}}\right) = \frac{t}{5730} \ln\left(\frac{1}{2}\right) \]. We can then calculate \( t \): \[ t = \frac{\ln(0.195)}{\ln(1/2)} \cdot 5730 \].

- Calculate the age of the bone

Use a calculator to compute the value of \( t \): \[ t = \frac{\ln(0.195)}{\ln(0.5)} \cdot 5730 \approx 11789 \] years. Therefore, the bone is approximately 11789 years old.

Key concepts

Half-life Decay

Half-life decay is a fundamental concept in understanding radioactive decay processes, including the decay of carbon-14 which is used in radiocarbon dating. It refers to the time required for half of a given amount of a radioactive isotope to decay into a stable form. For instance, if you start with a certain amount of a radioactive element, the half-life is the time it takes for half of those atoms to transform.

In mathematical terms, this process can be expressed with the equation:
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]
where:

• \( N(t) \) is the quantity of the substance remaining after time \( t \).
• \( N_0 \) is the original quantity of the substance.
• \( t_{1/2} \) is the half-life of the substance.

For carbon-14, this half-life is approximately 5730 years, which means that every 5730 years, the amount of carbon-14 in a sample will be reduced to half of its initial quantity.

First-order Kinetics

First-order kinetics is a principle that describes how the rate of a reaction depends on the concentration of one reactant. Specifically, in first-order reactions, the rate at which a substance decays is directly proportional to its concentration at any given moment. This type of reaction is described by an exponential decay model.

In the context of carbon-14 decay, this implies that the decay rate of carbon-14 is dependent solely on its current amount, rather than being influenced by other factors. Mathematically, first-order kinetics can be correlated with the half-life equation provided for carbon-14 decay:
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]
This equation represents how carbon-14 atoms decrease exponentially over time, distinct from zero-order reactions where the rate is constant or second-order reactions where the rate depends on the concentration of two reactants.

Radiocarbon Dating

Radiocarbon dating, also known as carbon-14 dating, is a technique used to estimate the age of organic materials by measuring the amount of carbon-14 they contain. Carbon-14 is a naturally occurring radioactive isotope of carbon that is absorbed by living organisms during their life.

When an organism dies, it stops absorbing carbon-14, and the isotope begins to decay at a predictable rate governed by its half-life. By comparing the remaining carbon-14 in a sample with the expected amount in a living organism, scientists can calculate the amount of time that has passed since the death of the organism.

The key to this method is understanding that while alive, the carbon-14 concentration remains relatively constant due to the organism's consumption and exchange with the environment. Once dead, the absence of exchange allows the carbon-14 to decay undisturbed, functioning effectively as a molecular clock.

Archaeological Age Estimation

Archaeological age estimation using radiocarbon dating is an invaluable tool for historians and archaeologists. It allows them to determine the age of artifacts and sites with a previously unknown history. By obtaining organic samples from these sites – such as wood, bone, or textile fibers – and measuring their carbon-14 content, researchers can estimate when the organisms from which the samples were derived lived, and in turn, when the artifacts or structures were likely used by past civilizations.

However, it's important to remember that this method is not without limitations. For example, it assumes the amount of atmospheric carbon-14 has remained constant over the ages, which is not entirely accurate due to fluctuations caused by solar radiation and industrial activities. Despite these variations, [calibration|calibrated] radiocarbon dating remains a cornerstone technique for age estimation in archaeology, offering the chance to peer into historical timelines and stitch together narratives of human and environmental history.