Question

Carbon-14 Dating Measurement Sensitivity To see the effect of a relatively small error in the estimate of the amount of carbon-14 in a sample being dated, answer the following questions about this hypothetical situation. (a) A fossilized bone found in central Illinois in the year A.D. 2000 contains 17\(\%\) of its original carbon-14 content. Estimate the year the animal died. (b) Repeat part (a) assuming 18\(\%\) instead of 17\(\%\) . (c) Repeat part (a) assuming 16\(\%\) instead of 17\(\%\) .

Short answer

The animal is estimated to have died around 15789 years ago for 17% carbon-14 content, 15301 years ago for 18% content and 16312 years ago for 16% content.

Step-by-step solution

Initial calculation for 17%

We start with calculating the year when the animal died assuming the remaining carbon-14 content is 17%. Using the exponential decay formula \(N = N_{0} e^{-kt}\), we can substitute and derive: 0.17 = \(e^{-0.000121t}\). Taking the natural logarithm of both sides we will get: \(\ln(0.17) = -0.000121t\). Solving for \(t\) gives us \(t = \ln(0.17) / -0.000121\), which results in \(t \approx 15789\) years from A.D. 2000.

Calculation for 18%

Now we repeat the calculation with the remaining carbon-14 content as 18%. Following a similar process to that in Step 1, \(\ln(0.18) = -0.000121t\), hence \(t = \ln(0.18) / -0.000121\), which results in \(t \approx 15301\) years from A.D. 2000.

Calculation for 16%

Finally, we repeat the calculation with the remaining carbon-14 content as 16%. Following a similar process to those in Step 1 and 2, \(\ln(0.16) = -0.000121t\), hence \(t = \ln(0.16) / -0.000121\), which results in \(t \approx 16312\) years from A.D. 2000. So we can see that even a 1% difference in the remaining percentage corresponds to an interval of about 500 years.

Key concepts

Exponential Decay

Exponential decay refers to the process by which a quantity decreases at a rate proportional to its current value. This phenomenon is observed in various domains such as physics, chemistry, and biology, with one of the most common applications in determining the age of archaeological finds through radiometric dating.

In mathematics, the exponential decay model is represented by the equation: \(N = N_{0}e^{-kt}\), where:\(N\) is the remaining amount of the substance after time \(t\),\(N_{0}\) is the initial amount of the substance,\(k\) is the decay constant, which is specific to the decaying substance, and \(e\) is the base of the natural logarithms, approximately equal to 2.71828.

This process is particularly relevant to Carbon-14 dating, where the amount of Carbon-14 in a sample decreases over time. As time passes, the ratio of Carbon-14 to other carbon isotopes declines, providing a measure for how long ago an organism containing carbon (for example, a tree or a fossil) ceased exchanging carbon with the environment upon its death.

Radiometric Dating

Radiometric dating is a method used by scientists to date materials, typically based on a comparison between the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates. In the context of Carbon-14 dating, this method revolves around understanding the half-life of Carbon-14, which is about 5,730 years.

When an organism such as a tree or an animal dies, it no longer absorbs Carbon-14, and the isotope begins to decay at a predictable rate. By measuring how much Carbon-14 is left in a sample and comparing it to the known rate of decay, scientists can estimate the time of death. This technique is crucial for archaeologists, geologists, and anyone interested in studying the timing of events in the history of our planet or determining the age of fossils and artifacts.

Natural Logarithms

Natural logarithms are a specific kind of logarithm that are linked intimately with exponential functions involving the number \(e\), which is an irrational number approximately equal to 2.71828. The natural logarithm of a number \(x\) is denoted as \(\ln(x)\) and it answers the question: 'To what power must we raise \(e\) to obtain \(x\)?'

When working with exponential decay in Carbon-14 dating, we frequently need to solve for time \(t\). Doing so often entails taking the natural logarithm of the equation to rearrange it and solve for \(t\). For example, from the exponential decay equation \(N = N_{0}e^{-kt}\), we can derive \(t\) by first dividing both sides by \(N_{0}\), getting the fraction of remaining substance, and then taking the natural logarithm of both sides. This isolates the variable \(t\) on one side of the equation, allowing us to solve for it directly.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in science, engineering, and mathematics to make computations more manageable. In scientific notation, a number is written as the product of two factors: a coefficient and a power of 10.

The general form is: \(a \times 10^{n}\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer. This format allows for the clear display of values such as the decay constant in the Carbon-14 dating equation, which in the given exercises is expressed as \(-0.000121\), a number that would typically be written in scientific notation to improve readability and to reduce the chances of error in calculations or interpretation.