Question

Archaeologists can determine the age of an artifact made of wood or bone by measuring the amount of the radioactive isotope \({ }^{14} \mathrm{C}\) present in the object. The amount of this isotope decreases in a first-order process. If \(15.5 \%\) of the original amount of \({ }^{14} \mathrm{C}\) is present in a wooden tool at the time of analysis, what is the age of the tool? The half- life of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{yr}\).

Short answer

The age of the wooden tool is approximately 15380 years.

Step-by-step solution

- Understand the Problem

You need to calculate the age of a wooden tool by determining the remaining amount of \(^{14}C\) and using its half-life value.

- Establish the Exponential Decay Formula

The decay of \(^{14}C\) follows the first-order kinetics given by the equation: \[ N(t) = N_0 e^{-kt} \] where \(N(t)\) is the amount of \(^{14}C\) at time \(t\), \(N_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time.

- Calculate the Decay Constant (k)

The decay constant \(k\) can be calculated using the half-life formula: \[ k = \frac{\text{ln}(2)}{t_{1/2}} \] Substitute \(t_{1/2} = 5730\text{ yr}\): \[ k = \frac{\text{ln}(2)}{5730} = 0.000121 \text{ yr}^{-1} \]

- Use the Exponential Decay Formula to Find Time

Rearrange the exponential decay equation to solve for \(t\): \[ \frac{N(t)}{N_0} = e^{-kt} \] Given \( N(t) / N_0 = 0.155 \), substitute and solve: \[ 0.155 = e^{-0.000121t} \] Taking the natural logarithm (ln) of both sides: \[ \text{ln}(0.155) = -0.000121t \] \[ t = \frac{\text{ln}(0.155)}{-0.000121} \]

- Calculate the Age (t)

Perform the calculation: \[ t = \frac{\text{ln}(0.155)}{-0.000121} \] \[ t = \frac{-1.861}{-0.000121} \] \[ t = 15380 \text{ yr} \]

Key concepts

first-order kinetics

Radiocarbon dating uses the concept of first-order kinetics to determine the age of an artifact. In first-order kinetics, the rate of decay of a radioactive substance is directly proportional to the amount of substance present. Due to this property, such reactions do not follow a constant rate but instead decay exponentially over time.

The equation used to describe first-order kinetics is: oot _ e^{-kt} where \( N(t) \) is the remaining amount of the radioactive substance at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time.

This exponential decay means that the decrease in the amount of a radioactive substance is rapid at first and slows down over time. This characteristic makes it especially useful in radiocarbon dating, where we measure how much of the initial radioactive isotope \( {}^{14}C \) remains in a sample to estimate its age.

exponential decay

Another crucial concept in radiocarbon dating is exponential decay. This principle explains how the quantity of a radioactive isotope decreases over time. It’s not a simple linear drop but rather a curve that falls off quickly and then levels off.

The formula representing exponential decay is: \[ N(t) = N_0 e^{-kt} \]
The base of the natural logarithm, \( e \approx 2.718 \), plays a significant role in this equation.

This equation tells us that at any given time \( t \), the amount remaining is the initial amount \( N_0 \) multiplied by the exponential factor \( e^{-kt} \). When using this in real-world scenarios, like in our exercise, it's vital to also determine the decay constant \( k \). We then substitute the known values and solve for time \( t \). This allows us to find out how old the sample is.

Exponential decay is seen in many natural processes besides radioactive decay, such as population decline and the cooling of hot objects.

half-life formula

The half-life formula is essential in radiocarbon dating for calculating the decay constant \( k \). The half-life of a radioactive isotope is the time it takes for half of the substance to decay.

Mathematically, half-life is incorporated into first-order kinetics using this formula: \[ k = \frac{{\text{ln}(2)}}{ t_{1/2} } \]
For \( {}^{14}C \), the half-life is \( 5730 \) years. We use the natural logarithm of \( 2 \) (approximately \( 0.693 \)) because it represents the decay to half of the original amount.

Substituting the half-life into the equation helps us find: \[ k = \frac{ 0.693 }{ 5730 \, \text{yr} } = 0.000121 \, \text{yr}^{-1} \]
With \( k \) known, we can return to the exponential decay formula, plug in our values, and solve for \( t \). This is how we determine the age of artifacts by measuring how much of the \( {}^{14}C \) remains.

The concept of half-life is broadly applicable to various fields involving radioactive elements, such as medical treatments, nuclear power, and understanding environmental processes.