Problem 16 - Campbell Biology Concepts & Connections

Question

Measurements indicate that a fossilized skull you unearthed has a carbon-14: carbon- 12 ratio about \(1 / 16\) th that of the skulls of present-day animals. What is the approximate age of the fossil? (The half-life of carbon-14 is 5,730 years.)

Step-by-Step Solution

Answer

The approximate age of the fossil is 22,920 years.

Step 1 Title - Understand the Concept of Half-Life

The half-life of a substance is the time it takes for half of it to decay. Carbon-14 has a half-life of 5,730 years, meaning that every 5,730 years, the amount of Carbon-14 in a sample is reduced by half.

Step 2 Title - Calculate Number of Half-Lives

We know that the fossilized skull has a carbon-14 to carbon-12 ratio that is 1/16th of a present-day skull. Since each half-life reduces the amount of carbon-14 by half, we need to determine how many half-lives result in a reduction to 1/16th.

Step 3 Title - Determine the Reduction Factor in Exponential Terms

The relationship can be expressed as \(\left(\frac{1}{2}\right)^n = \frac{1}{16}\), where \(n\) is the number of half-lives. Solving for \(n\): \(\left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^4\). Hence, \(n = 4\). This shows that the fossilized skull has undergone 4 half-lives.

Step 4 Title - Calculate the Age of the Fossil

Multiply the number of half-lives by the duration of one half-life to find the total age. \(\text{Total Age} = 4 \text{ half-lives} \times 5730 \text{ years per half-life} = 22920\text{ years} \)

Key Concepts

Half-Life

The half-life of an element is the time it takes for half of a given amount of the substance to decay. For Carbon-14, this duration is 5,730 years. Understanding half-life is essential because it allows us to date ancient objects. For example, if you start with a certain amount of Carbon-14, after one half-life (5,730 years), only half of it will remain. After another 5,730 years, only a quarter (half of the remaining half) will be left. This decay continues in a predictable pattern, which is crucial for calculating the age of fossils.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. The process follows a specific pattern over time. With Carbon-14 dating, the main idea is that this isotope of carbon undergoes radioactive decay to Nitrogen-14 at a constant rate. This makes it a reliable tool to determine the age of carbon-containing materials. As the Carbon-14 isotopes decay over millions of years, counting back from the present ratio of Carbon-14 to Carbon-12 gives us an age estimate for the sample.

Fossil Age Calculation

To calculate the age of a fossil, we need to understand its Carbon-14 to Carbon-12 ratio compared to a reference value, usually of a present-day sample. For instance, if a fossil has \({1/16}\) the Carbon-14 of a living organism, we need to calculate how many half-lives it took to reach that ratio. Since \({1/16} = (1/2)^4\), this tells us that the sample has gone through four half-lives. Each half-life is 5,730 years, so the fossil is \[4 \times 5730 = 22920\] years old. This step-by-step degradation helps us pinpoint time points in ancient history.

Exponential Decay

Exponential decay describes a process that decreases at a rate proportional to its current value. In terms of Carbon-14 dating, exponential decay means that the rate at which Carbon-14 reduces resembles an exponential function. Mathematically, this follows the equation \(\frac{1}{2^n}\), where \({n}\) represents the number of half-lives. So each half-life divides the remaining amount by two, creating a smooth, predictable curve of decline over time. This exponential nature of radioactive decay is what allows us to confidently date ancient carbon-based materials, such as fossils.