Question

Assume a constant \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) ratio of \(13.6\) counts per minute per gram of living matter. A sample of a petrified tree was found to give \(1.2\) counts per minute per gram. How old is the tree? \(\left(\right.\) For \({ }^{14} \mathrm{C}, t_{1 / 2}=5730\) years. \()\)

Short answer

The petrified tree is approximately 11,460 years old.

Step-by-step solution

Identify the given information and variables

The exercise provides us with the following information and variables: - Initial \(^{14}C /{ }^{12} C\) ratio: \(13.6\) counts/min/gram (living matter) - Sample \(^{14}C /{ }^{12} C\) ratio: \(1.2\) counts/min/gram (petrified tree) - Half-life of \(^{14}C\): \(t_{1/2} = 5730\) years We need to find the age of the petrified tree, which we can represent as the variable \(t.\)

Use the half-life formula

We can use the following half-life formula for radioactive decay: \[N(t) = N_0 \cdot 2^{-\frac{t}{t_{1/2}}}\] where: - \(N(t)\) is the remaining amount of the substance at time \(t\) - \(N_0\) is the initial amount of the substance - \(t_{1/2}\) is the half-life period - \(t\) is the time that has passed In our case, we can replace \(N_0\) with the initial \(^{14}C /{ }^{12} C\) ratio of living matter (13.6 counts/min/gram) and \(N(t)\) with the tree sample ratio (1.2 counts/min/gram).

Solve for t

We need to isolate \(t\) in the half-life formula to find the age of the tree. We have: \[1.2 = 13.6 \cdot 2^{-\frac{t}{5730}}\] First, we should divide both sides by 13.6: \[\frac{1.2}{13.6} = 2^{-\frac{t}{5730}}\] Next, take the logarithm base 2 of both sides: \[-\frac{t}{5730} = \log_2{\frac{1.2}{13.6}}\] Now, we can multiply both sides by \(-5730\) to isolate \(t\): \[t = -5730 \cdot \log_2{\frac{1.2}{13.6}}\]

Calculate the age of the tree

Finally, we can plug the given values into the formula and calculate the age of the tree: \[t = -5730 \cdot \log_2{\frac{1.2}{13.6}} \approx 11460\,years\] The petrified tree is approximately 11,460 years old.

Key concepts

Radioactive Decay

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This can involve the release of alpha particles, beta particles, gamma rays, or conversion electrons. The decay process transforms the original atom, known as the parent, into a more stable form, which is called the daughter nuclide. This transition happens at a rate unique to each radioactive isotope and is measured in terms of its half-life.

It's important to note that radioactive decay is a random process at the level of single atoms, meaning that it is impossible to predict when exactly one atom will decay. However, statistically, the decay rate of a large number of atoms is stable and predictable, which is expressed as the half-life. This makes radioactive decay a powerful tool for dating artifacts, rocks, and fossilized remains.

Half-Life Formula

The half-life of a radioactive isotope is the time required for half of the isotopic atoms present to decay. Mathematically, the half-life formula is expressed as:
\[N(t) = N_0 \times 2^{-\frac{t}{t_{1/2}}}\]
where:

• \(N(t)\) is the amount of the substance that still remains after time \(t\).
• \(N_0\) is the initial amount of the substance.
• \(t\) represents the time that has elapsed.
• \(t_{1/2}\) is the half-life of the substance.

In this formula, the half-life provides a measure of how quickly a substance decays. It's a crucial concept not only in dating archaeological finds but also in understanding the decay of pharmaceuticals in the body and the stability of chemicals in the environment. The calculation using the half-life formula can determine the age of ancient objects, the safety of food and drugs, and even help in medical diagnostic procedures involving radioactive tracers.

Carbon-14 Dating Process

Carbon-14 dating, also known as radiocarbon dating, is a method used to determine the age of organic material. It relies on the principle of radioactive decay, specifically using the isotope carbon-14. All living organisms have carbon in them, and a small but stable proportion exists as the radioactive isotope carbon-14. Upon death, these organisms stop exchanging carbon with the environment, and the carbon-14 they contain begins to decay.

The process of carbon-14 dating involves measuring the remaining carbon-14 in a sample and comparing it to the known half-life of carbon-14, which is about 5,730 years. By calculating the ratio of carbon-14 to the stable carbon-12 in the sample and using the half-life formula, scientists can estimate how long it has been since the organism died. This makes carbon-14 dating extremely useful in archaeology, geology, and other fields that deal with ancient organic materials.