A piece of wood from an archeological source shows a \({ }^{14} \mathrm{C}\) activity which is \(60 \%\) of the activity found in fresh wood today. Calculate the age of the archeological sample. \(\left(t_{1 / 2}{ }^{14} \mathrm{C}=5770\right.\) year \()\)

Carbon-14 dating is a method used by archaeologists and geologists to determine the age of organic material.
It is based on the principle of radioactive decay, which is the process by which an unstable atomic nucleus loses energy by emitting radiation.
When an organism dies, it stops absorbing carbon, including the radioactive isotope carbon-14.

• From this point on, the amount of carbon-14 begins to decrease as it decays into nitrogen
• By comparing the remaining carbon-14 to the expected original amount, scientists can calculate how many half-lives have passed, and thus, the age of the sample

In the problem provided, the sample shows 60% of the carbon-14 activity found in fresh wood, suggesting that some half-lives have passed since the organism that provided the wood died.
This method assumes a constant decay rate and the initial carbon-14 content, which are factors to consider for accurate dating.

The half-life of a radioactive substance is the time required for half of the original amount of radioactive nuclei to decay.
This concept is critical in carbon-14 dating, as it provides a scale to estimate the age of carbon-based materials.

• Knowing the half-life, one can determine the time period over which a certain percentage of the original isotopes has decayed
• The half-life of carbon-14 is approximately 5770 years

The calculations focus on determining the number of half-lives that have elapsed to account for the observed decrease in carbon-14 activity.
With the half-life known, as in the provided example, the age of an archeological sample can be estimated by determining how many half-lives have passed, which is done using exponential decay formulas.

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value.
In the context of radioactive decay, the number of undecayed atoms decreases exponentially over time.

• The general formula for exponential decay is represented as \( N(t) = N_0 \times e^{-kt} \), where \( N_0 \) is the initial quantity, \( N(t) \) is the quantity after time \( t \), and \( k \) is the decay constant
• However, for half-life problems, the formula often simplifies to \( N(t) = N_0 \times \big(\frac{1}{2}\big)^{t/t_{1/2}} \), as seen in the provided exercise

This equation shows that the quantity of a substance decreases by half every \( t_{1/2} \) years, and this pattern continues over each subsequent half-life period.
Understanding exponential decay allows us to solve many real-world problems involving radioactive substances and is essential for carbon dating.

Logarithmic equations are mathematical expressions where the unknown variable appears in the exponent of an equation and logarithms are used to solve for that variable.
They are particularly useful in half-life problems of radioactive decay, where the original exponential decay formula need to be rearranged to isolate the time variable, \( t \).

• To solve for time in exponential decay equations, taking the natural logarithm of both sides, as demonstrated in the solution to the example exercise, often simplifies the calculation
• A logarithm helps to 'bring down' the exponent in the equation, allowing us to solve for the time elapsed since the organism died

The problem provided involves manipulating the decay formula via logarithms and algebraic manipulation to find the age of the wood sample.
In such problems, a clear understanding of logarithmic functions and their properties is crucial for interpreting the decay processes and determining the time that has passed since the death of an organism.