Question

Carbon- 14 , which is present in all living tissue, radioactively decays via a first order process. A one-gram sample of wood taken from a living tree gives a rate for carbon- 14 decay of \(13.6\) counts per. If the half-life for carbon-14 is 5720 years. How old is a wood sample that gives a rate for carbon-14 decay of \(3.9\) counts per minute? a. \(1.0 \times 10^{4}\) years b. \(2.1 \times 10^{3}\) years c. \(5.4 \times 10^{3}\) years d. \(3.0 \times 10^{4}\) years

Short answer

The wood sample is approximately 5,400 years old (option c).

Step-by-step solution

Understand the First Order Kinetics

The decay of carbon-14 follows first-order kinetics, which means the rate of decay is proportional to the amount of carbon-14 present. The formula for first-order kinetics is expressed as: \[N = N_0 e^{-kt}\] where \(N\) is the amount of substance remaining, \(N_0\) is the initial amount, \(k\) is the rate constant, and \(t\) is the time elapsed.

Calculate the Rate Constant

Given the half-life (\(t_{1/2}\)) of carbon-14 is 5720 years, we can calculate the rate constant \(k\) using the first-order half-life formula: \[k = \frac{0.693}{t_{1/2}} = \frac{0.693}{5720}\]Calculate \(k\) to use it in the decay equation.

Use the Decay Equation

Using the decay equation \( N = N_0 e^{-kt} \), where \( N_0 = 13.6 \) counts/min and \( N = 3.9 \) counts/min, set up the equation to solve for \( t \): \[3.9 = 13.6 e^{-kt}\] Take the natural logarithm of both sides to solve for \( t \).

Solve for Time 't'

Taking the natural logarithm of the equation \[\ln{(3.9)} = \ln{(13.6)} - kt \] Rearrange to solve for \( t \):\[t = \frac{\ln{(13.6)} - \ln{(3.9)}}{k}\]Insert the value of \( k \) calculated earlier and solve for \( t \).

Compare Result with Options

Calculate the time \( t \) and compare it with the given options: - a. \(1.0 \times 10^{4}\) years - b. \(2.1 \times 10^{3}\) years - c. \(5.4 \times 10^{3}\) years - d. \(3.0 \times 10^{4}\) yearsEnsure that the calculated \( t \) from Step 4 matches one of these options.

Key concepts

First Order Kinetics

In the world of chemistry, understanding reaction rates is crucial. First-order kinetics is a concept that describes how the speed of a reaction is directly proportional to the concentration of one reactant. When something decays, like radioactive carbon-14, it follows this first-order process. With first-order kinetics, the formula \[ N = N_0 e^{-kt} \] is key. Here, \( N \) represents the amount of substance left, \( N_0 \) is the starting amount, \( k \) is the decay constant, and \( t \) is time. What this equation means is that as time progresses, the quantity decreases exponentially. For instance, in carbon-14 decay, each carbon atom will have a different probability of decaying per minute, but on average, they decay at a rate dependent on how many are left. This behavior mimics how we calculate the time it takes for half of a sample to decay, hence the tie-in to half-life calculations. Understanding first-order kinetics makes it possible to predict how much of a substance remains over time. This predictability is especially helpful in processes like archaeological dating.

Carbon-14 Dating

Carbon-14 dating is a fascinating application of radioactive decay, allowing us to determine the age of ancient organic materials. While carbon-14 is a radioactive isotope of carbon, it behaves just like a stable carbon atom in biological reactions. Living things take in carbon dioxide, including a tiny fraction that contains carbon-14. Once an organism dies, it no longer absorbs carbon-14. The isotope begins to decay at a predictable rate, following first-order kinetics. By measuring the remaining carbon-14 in a sample and comparing it to the initial activity, scientists can estimate how long it has been since the organism's death. This method is crucial for archaeologists and paleontologists, providing the tools needed to date artifacts, bones, and even ancient wooden structures. By understanding the principles of carbon-14 decay, we can reconstruct timelines of events stretching thousands of years into the past.

Half-life Calculation

Half-life is a core concept in understanding radioactive decay. It refers to the time required for half of a radioactive substance to decay. For carbon-14, the half-life is 5720 years, which is a bit over five millennia. The beauty of half-life is its predictability. No matter how much substance you start with, the time for half of it to decay remains constant. This constancy makes it a reliable measure for determining the age of materials. To calculate the age of a sample, like in our wood example, we use the formula for first-order kinetics along with the known half-life:\[ k = \frac{0.693}{t_{1/2}} \]where \( k \) is the decay constant. Once we have \( k \), we can rearrange the decay equation and take the natural logarithm to solve for time:\[ t = \frac{\ln{(N_0)} - \ln{(N)}}{k} \]This log transformation helps us find out how long it has been since the carbon-14 began its decay. Whether dating an artifact or solving homework exercises, grasping these concepts allows for deeper insights into the passage of time.