Question

Swedish researchers report that they have discovered the world's oldest living tree. The spruce tree's roots were radiocarbon dated and found to have \(31.5 \%\) of their carbon-14 (C-14) left. The half-life of C-14 is 5730 years. How old was the tree when it was discovered?

Short answer

The tree is approximately 9615 years old.

Step-by-step solution

- Understand the problem

We need to find the age of the tree based on the remaining percentage of carbon-14. The half-life of carbon-14 is 5730 years.

- Use the decay formula

The decay formula for radioactive isotopes is given by \( \frac{N(t)}{N_0} = \frac{1}{2}^{\frac{t}{T}} \), where \( N(t) \) is the remaining quantity of the substance, \( N_0 \) is the original quantity, \( t \) is time, and \( T \) is the half-life.

- Convert to logarithmic form

Rearrange the decay formula to solve for \( t \): \[ \frac{N(t)}{N_0} = \frac{1}{2}^{\frac{t}{T}} \rightarrow \frac{1}{2}^{\frac{t}{T}} = 0.315 \rightarrow \frac{t}{T} = \frac{\text{log}(0.315)}{\text{log}(0.5)} \]

- Plug in the half-life

Substitute the half-life \( T = 5730 \) years into the equation: \[ t = 5730 \times \frac{\text{log}(0.315)}{\text{log}(0.5)} \]

- Calculate the age

Perform the calculation: \[ t = 5730 \times \frac{\text{log}(0.315)}{\text{log}(0.5)} \approx 5730 \times 1.678 \approx 9614.94 \] The tree is approximately 9615 years old.

Key concepts

Carbon-14 Dating

Carbon-14 dating, also known as radiocarbon dating, is a method used by scientists to determine the age of an object containing organic material. This technique is based on the radioactive decay of carbon-14 (C-14), an isotope of carbon that is absorbed by living organisms.
When an organism dies, it stops absorbing carbon-14, which then begins to decay at a known rate. This decay follows a predictable process, making it possible to estimate the time since death. By measuring the remaining amount of C-14 in a sample, scientists can calculate how many half-lives have passed, hence determining the age of the sample.
Radiocarbon dating is especially useful for dating objects up to around 50,000 years old. Beyond this age, the amount of C-14 left in a sample is usually too small to measure accurately. For instance, the world's oldest living tree mentioned in the exercise was dated using this precise method.

Half-Life

The concept of half-life is fundamental in the process of radiocarbon dating. A radioactive half-life is the time required for half of the radioactive atoms in a sample to decay. For Carbon-14, the half-life is 5730 years.
Understanding half-life helps explain why radioactive substances decrease over time. In our exercise, the tree's roots had 31.5% of their original C-14 content left. This percentage indicates how much of the C-14 has decayed and allows us to calculate the tree's age.
We use the decay formula for this purpose: \(\frac{N(t)}{N_0} = \frac{1}{2}^{\frac{t}{T}}\). Here, \(N(t)\) is the remaining quantity of C-14, \(N_0\) is the original quantity, \(\frac{t}{T}\) is the number of half-lives, and \(T\) is the half-life (5730 years for C-14).

Logarithms

Logarithms play a crucial role in solving the decay formula for radiocarbon dating. In our exercise, after establishing the decay relationship \(\frac{N(t)}{N_0} = \frac{1}{2}^{\frac{t}{T}}\), logarithms help us isolate the variable \(t\) (time).
We rearrange the formula to solve for \(t\) using logarithmic principles: \(\frac{t}{T} = \frac{\text{log}(0.315)}{\text{log}(0.5)}\). This transformation simplifies the problem from an exponential equation to a linear one, allowing us to solve it using basic algebra.
By substituting \(\text{log}(0.315) \) and \(\text{log}(0.5)\), we calculate \( t\). This step demonstrates the power of logarithms in dealing with exponential decay models. The final calculation provides the tree's age: approximately 9615 years.