Accuracy of Carbon 14 Dating I

Task

The half-life of Carbon $14$, that is, the time required for half of the Carbon $14$ in a sample to decay, is variable: not every Carbon $14$ specimen has exactly the same half life. The half-life for Carbon $14$ has a distribution that is approximately normal with a standard deviation of $40$ years. This explains why the Wikipedia article on Carbon $14$ lists the half-life of Carbon 14 as $5730 \pm 40$ years. Other resources report this half-life as the absolute amounts of $5730$ years, or sometimes simply $5700$ years.

1. Explain the meaning of these three quantities ($5730 \pm 40$, $5730$, and $5700$) focusing on how they differ.
2. Can all three of the reported half-lives for Carbon $14$ be correct? Explain.
3. What are some of the benefits and drawbacks for each of the three ways of describing the half-life of Carbon $14$?

IM Commentary

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon $14$ to Carbon $12$. The focus here is on the statistical nature of such dating. The decay of Carbon $14$ into stable Nitrogen $14$ does not take place in a regular, determined fashion: rather it is governed by the laws of probability and statistics formalized in the language of quantum mechanics. As such, the reported half life of $5730 \pm 40$ years means that $40$ years is the standard deviation for the process and so we expect that roughly $68$ percent of the time half of the Carbon $14$ in a given sample will decay within the time span of $5730 \pm 40$ years. If greater likelihood is sought, we could look at the interval $5730 \pm 80$ years, encompassing two standard deviations, and the likelihood that the half-life of a given sample of Carbon $14$ will fall in this range is a little over $95$ percent.

This task addresses a very important issue about precision in reporting and understanding statements in a realistic scientific context. This has implications for the other tasks on Carbon 14 dating which will be addressed in ''Accuracy of Carbon 14 Dating II.''

The statistical nature of radioactive decay means that reporting the half-life as $5730 \pm 40$ is more informative than providing a number such as $5730$ or $5700$. Not only does the $\pm 40$ years provide extra information but it also allows us to assess the reliability of conclusions or predictions based on our calculations.

This task is intended for instructional purposes. Some more information about Carbon $14$ dating along with references is available at the following link: Radiocarbon Dating

Solution

1. Of the three reported half-lives for Carbon $14$, the clearest and most informative is $5730 \pm 40$. Since radioactive decay is an atomic process, it is governed by the probabilistic laws of quantum physics. We are given that $40$ years is the standard deviation for this process so that about $68$ percent of the time, we expect that the half-life of Carbon $14$ will occur within $40$ years of $5730$ years. This range of $40$ years in either direction of $5730$ represents about seven tenths of one percent of $5730$ years.

   The quantity $5730$ is probably the one most commonly used in chemistry text books but it could be interpreted in several ways and it does not communicate the statistical nature of radioactive decay. For one, the level of accuracy being claimed is ambiguous -- it could be being claimed to be exact to the nearest year or, more likely, to the nearest ten years. In fact, neither of these is the case. The reason why $5730$ is convenient is that it is the best known estimate and, for calculation purposes, it avoids working with the $\pm 40$ term.

   The quantity $5700$ suffers from the same drawbacks as $5730$. It again fails to communicate the statistical nature of radioactive decay. The most likely interpretation of $5700$ is that it is the best known estimate to within one hundred years though it could also be exact to the nearest ten or one. One advantage to $5700$, as opposed to $5730$, is that it communicates better our actual knowledge about the decay of Carbon $14$: with a standard deviation of $40$ years, trying to predict when the half-life of a given sample will occur with greater accuracy than $100$ years will be very difficult. Neither quantity, $5730$ or $5700$, carries any information about the statistical nature of radioactive decay and in particular they do not give any indication what the standard deviation for the process is.

2. As was described in part (a) the three reported half-lives, $5730 \pm 40$, $5730$, and $5700$ are all consistent but do not carry the same information. The most informative is $5730 \pm 40$ while the most convenient is $5730$. The number $5700$ has the advantage of communicating the general level of accuracy for when the half-life is likely to occur but it is not as good an estimate as $5730$ and is less informative than $5730 \pm 40$.

3. The advantage to $5730 \pm 40$ is that it communicates both the best known estimate of $5730$ and the fact that radioactive decay is not a deterministic process so some interval around the estimate of $5730$ must be given for when the half-life occurs: here that interval is $40$ years in either direction. Moreover, the quantity $5730 \pm 40$ years also conveys how likely it is that a given sample of Carbon $14$ will have its half-life fall within the specified time range since $40$ years is represents one standard deviation. The disadvantage to this is that for calculation purposes handling the $\pm 40$ is challenging so a specific number would be more convenient.

   The number $5730$ is both the best known estimate and it is a number and so is suitable for calculating how much Carbon $14$ from a given sample is likely to remain as time passes. The disadvantage to $5730$ is that it can mislead if the reader believes that it is always the case that exactly one half of the Carbon $14$ decays after exactly $5730$ years. In other words, the quantity fails to communicate the statistical nature of radioactive decay.

   The number $5700$ is both a good estimate and communicates the rough level of accuracy. Its downside is that $5730$ is a better estimate and, like $5730$, it could be interpreted as meaning that one half of the Carbon $14$ always decays after exactly $5700$ years.