A layer of peat beneath the glacial sediments of the last ice age has a carbon-14 to carbon-12 ratio that is 22.8% of that found in living organisms. How long ago was this ice age?

Verified step by step guidance

insert step 1> Determine the half-life of carbon-14, which is approximately 5730 years.

insert step 2> Use the formula for radioactive decay: \( N = N_0 \times (0.5)^{t/t_{1/2}} \), where \( N \) is the remaining amount of carbon-14, \( N_0 \) is the initial amount, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life.

insert step 3> Set up the equation using the given ratio: \( 0.228 = (0.5)^{t/5730} \).

insert step 4> Solve for \( t \) by taking the natural logarithm of both sides: \( \ln(0.228) = \frac{t}{5730} \times \ln(0.5) \).

insert step 5> Rearrange the equation to solve for \( t \): \( t = \frac{\ln(0.228)}{\ln(0.5)} \times 5730 \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Carbon Dating

Carbon dating is a method used to determine the age of organic materials by measuring the ratio of carbon-14 to carbon-12 isotopes. Living organisms maintain a consistent ratio of these isotopes while alive, but upon death, carbon-14 decays at a known rate (its half-life is about 5730 years). By comparing the current ratio of carbon-14 to carbon-12 in a sample to that of living organisms, scientists can estimate the time since the organism's death.

Half-Life

The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay into a different element or isotope. For carbon-14, this half-life is approximately 5730 years. Understanding half-life is crucial for calculating the age of a sample based on the remaining amount of carbon-14, as it allows for the determination of how many half-lives have passed since the organism died.

Isotope Ratios

Isotope ratios refer to the relative abundance of different isotopes of an element in a sample. In the context of carbon dating, the ratio of carbon-14 to carbon-12 is used to assess the age of organic materials. A decrease in the carbon-14 to carbon-12 ratio over time indicates the passage of years since the organism's death, allowing for age estimation based on the remaining isotopes.