Question

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.27 \times 10^{-12} $$

Short answer

The estimated age of the fossil is approximately 8430 years.

Step-by-step solution

Rearrange the Formula

First, rearrange the decay formula to solve for \(t\). The formula is \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\). We can divide both sides by \(10^{-12}\) and take the logarithm of base \(0.5\) to get \(t\). This leaves us with the formula \(t= 5715 \times \log_{0.5}\left(\frac{R}{10^{-12}}\right)\)

Substitute the Given Value of \(R\)

Now, substitute the given value of \(R=0.27 \times 10^{-12}\) into the rearranged formula to find \(t\).\nThis gives \(t= 5715 \times \log_{0.5}\left(\frac{0.27 \times 10^{-12}}{10^{-12}}\right)\)

Calculate the Value of \(t\)

Now calculate the value of \(t\) by finding the logarithm and multiplying it by 5715. After doing the calculations, we find that \(t\approx 8430\) years.

Key concepts

Understanding Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics and chemistry, which describes the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs naturally and spontaneously in many elements, including carbon-14, which is commonly used for radiocarbon dating. The decay continues until the unstable nucleus has transformed into a more stable form.

In the context of carbon dating, it's crucial to understand that once organic material dies, its carbon-14 isotopes (which are radioactive) begin to decay at a predictable rate. This rate is measured in terms of the half-life of the isotope, which is the time taken for half of the initial number of radioactive atoms to decay.

To enhance student comprehension and relate to everyday experiences, one can compare the radioactive decay process to the melting of ice in a glass on a hot day. Just as the rate at which the ice melts can be predicted under consistent conditions, so can the rate of radioactive decay for carbon-14 isotopes.

The Significance of Half-Life

The half-life of a radioactive substance is a fundamental property that is key to understanding nuclear decay. It is defined as the amount of time required for half of the radioactive atoms in a sample to decay. For carbon-14, this period is approximately 5,715 years.

In practical terms, the concept of half-life allows scientists to determine the age of ancient organic materials. By measuring the remaining concentration of carbon-14 in a fossil and comparing it to the expected concentration in living organisms, the time that has elapsed since the organism died can be estimated. It is crucial for educational content to stress that the half-life concept does not change with time—it's a constant property of the substance.

Educators might further demonstrate the concept using visuals or animations that depict the decay of a fixed number of atoms over successive half-lives, showing clearly how the number decreases by about half each time.

Using Logarithms in Carbon Dating

Logarithms play a critical role in carbon dating, as they allow us to solve equations involving exponential decay functions. A logarithm is the inverse operation to exponentiation and tells us the power to which we must raise a base number to get another number. In carbon dating, logarithms help us work backwards from observed data (the current proportion of carbon-14) to find the time that has elapsed since the organism died.

To calculate the age of a fossil, we need to invert the exponential decay formula, which can be done using logarithms. This mathematical step is crucial for determining the time variable in our carbon dating equation. For students to fully grasp logarithms, it's beneficial to use concrete examples and emphasize the logarithms' property of turning multiplication into addition and exponentiation into multiplication, greatly simplifying complex calculations.

For a rich learning experience, engaging in hands-on activities like calculating the age of a hypothetical fossil using logarithms might solidify students' understanding of this abstract concept.