Archaeological samples are often dated by radiocarbon dating. The half-life of carbon-14 is 5,700 years. a. After how many half-lives will the sample have only \(1 / 64\) as much carbon-14 as it originally contained? b. How much time will have passed? c. If the daughter product of carbon- 14 is present in the sample when it forms (even before any radioactive decay happens) you cannot assume that every daughter you see is the result of carbon-14 decay. If you did make this assumption, would you overestimate or underestimate the age of a sample?

Radioactive decay is a cornerstone of radiocarbon dating. One crucial concept is the half-life of carbon-14. The half-life of carbon-14 is 5,700 years. This means it takes 5,700 years for half of the carbon-14 atoms in a sample to decay into nitrogen-14. The half-life is useful because it remains constant regardless of the material conditions.
Understanding the half-life allows us to calculate how old a sample is based on how much carbon-14 remains compared to the original amount.
Let's break it down with an analogy. Suppose you have 100 candies, and you eat half every day. After the first day, you'd have 50 candies; after the second day, 25; and so on. Each 'day' you continue to eat half of what's left. Carbon-14 behaves similarly, but its 'days' are measured in thousands of years.

Another important part of radiocarbon dating is how we measure fraction reduction. In the given exercise, we need to find out how many half-lives are required for the sample to reduce to \( \frac{1}{64} \) of its original carbon-14. We can use the equation for fraction reduction: \[ \frac{1}{64} = 2^{-6} \].
This tells us that the sample has gone through 6 half-lives because reducing by \( \frac{1}{64} \) is equivalent to splitting the sample in half 6 times. This understanding helps us calculate the time that has passed.
If the half-life of carbon-14 is 5,700 years, and it has undergone 6 half-lives, we can multiply the two values: \[ 6 \times 5700 = 34200 \] years. Thus, 34,200 years have passed for the carbon-14 to decay to \( \frac{1}{64} \) of its original amount.

Overestimating the age of a sample can be a critical mistake in radiocarbon dating. If a daughter product (the isotopes formed from the decay) is already present in the sample before any decay of carbon-14 has taken place, this can mislead scientists.
Suppose you wrongly assume every daughter product you see is due to carbon-14 decay. In that case, you'll overestimate the sample age. This assumption would mean you think there was more carbon-14 to begin with than there truly was, causing an overestimation.
It's like assuming all the leaves on the ground in the fall came from one tree. If you don't realize other trees also dropped their leaves (the initial presence of daughter product), you'll overestimate how many leaves that one tree lost.

In radiocarbon dating, a 'daughter product' is the result of the decay of a radioactive element, specifically the isotope carbon-14 in this context. As carbon-14 decays, it turns into nitrogen-14, the daughter product.
Measuring the ratios of parent (carbon-14) and daughter (nitrogen-14) products helps determine the age of samples. If nitrogen-14 is present without having been formed by decay, this can create inaccuracies.
It's important to account for the fact that some daughter products might be present due to other processes. Not doing so would be like counting all spilled paint on the floor as being from one can, ignoring possible other sources. Therefore, accurately identifying and accounting for daughter products ensures reliable and precise age estimates.