Problem 29 - Chemistry: Principles and Reactions

Question

Carbon from a cypress beam obtained from the tomb of an ancient Egyptian king gave \(9.2\) disintegrations/minute of \(C-14\) per gram of carbon. Carbon from living material gives \(15.3\) disintegrations/min of C-14 per gram of carbon. Carbon-14 has a half-life of 5730 years. How old is the beam?

Step-by-Step Solution

Answer

Question: Calculate the age of a cypress beam taken from an ancient Egyptian king's tomb using the radioactive decay method. The disintegration rate of the beam is 9.2 disintegrations/min per gram of carbon, while the disintegration rate of living material is 15.3 disintegrations/min per gram of carbon. The half-life of Carbon-14 is 5730 years. Answer: The age of the cypress beam is approximately 4,000 years.

Step 1: Understand given information

We are given the following information: - The disintegration rate of the beam is \(9.2\) disintegrations/min per gram of carbon. - The disintegration rate of living material is \(15.3\) disintegrations/min per gram of carbon. - The half-life of Carbon-14 is 5730 years.

Step 2: Identify the decay model

The decay model we will use is exponential decay, which is given by the formula: \(N(t) = N_0 \cdot e^{-kt}\) Where: - \(N(t)\) is the amount of substance at time \(t\). - \(N_0\) is the initial amount of substance. - \(k\) is the decay constant. - \(t\) is the time elapsed.

Step 3: Calculate the decay constant

First, we need to find the decay constant, which is related to the half-life by the formula: \(T_{1/2} = \frac{ln(2)}{k}\) Rearranging the equation and solving for \(k\), we get: \(k = \dfrac{ln(2)}{T_{1/2}}\) Plugging in the values, we get: \(k = \dfrac{ln(2)}{5730} \approx 1.21 \times 10^{-4}\) per year

Step 4: Set up decay equation

Now, the decay equation for our situation can be written as: \(\dfrac{N(t)}{N_0} = e^{-kt}\) We can use the given disintegration rates for living material and the cypress beam to find the fraction \(\frac{N(t)}{N_0}\): \(\dfrac{N(t)}{N_0} = \dfrac{9.2}{15.3}\)

Step 5: Solve for time elapsed (age of the beam)

Substitute the decay constant and the fraction of disintegrations in the decay equation to find \(t\): \(\dfrac{9.2}{15.3} = e^{-\left(1.21 \times 10^{-4}\right)t}\) To isolate the variable \(t\), take the natural logarithm of both sides: \(ln \left( \dfrac{9.2}{15.3} \right) = -\left(1.21 \times 10^{-4}\right)t\) Now, solve for \(t\): \(t = \dfrac{ln\left(\dfrac{9.2}{15.3}\right)}{-1.21 \times 10^{-4}} \approx 4,000\) years The age of the cypress beam is approximately 4,000 years.

Key Concepts

Carbon-14 Half-Life

Understanding the half-life of Carbon-14 (C-14) is crucial when studying radioactive carbon dating. The half-life is the amount of time it takes for half of the radioactive isotopes in a sample to decay. For Carbon-14, this period is approximately 5730 years.

This knowledge allows scientists to calculate the age of ancient organic materials by comparing the remaining amount of C-14 to the expected amount in a current, live sample. If we know the half-life, we can use this as a 'clock' to measure how much time has passed since the death of the organic material. It is important to remember that the half-life remains constant while the proportion of the isotope decreases over time due to its exponential decay nature.

Exponential Decay Model

Radioactive isotopes such as Carbon-14 follow an exponential decay model, described by the formula \(N(t) = N_0 \cdot e^{-kt}\) where \(N(t)\) is the number of remaining radioactive atoms at time \(t\), \(N_0\) is the initial number of radioactive atoms, \(e\) is the base of the natural logarithm, \(k\) is the decay constant, and \(t\) is the time elapsed.

In context, this model shows us that the decay of radioactive isotopes is not linear but decreases faster at the beginning and slows down as time passes. The decay constant \(k\) is unique to each radioactive isotope and strongly connected to its half-life, providing a clear link between the amount of isotope remaining, the time elapsed, and the half-life, all essential factors in radiocarbon dating.

Radiocarbon Dating Calculations

To perform radiocarbon dating calculations, we first establish the decay constant from the known half-life of Carbon-14 using the equation \(k = \dfrac{ln(2)}{T_{1/2}}\).

Then, we compare the disintegration rate of the sample with the disintegration rate of a modern, or living, sample to calculate the fraction representing the amount of C-14 that has decayed. Finally, we use the exponential decay equation to solve for the elapsed time since the death of the sample, which represents its age.

Example Calculation

Given the disintegration rate of an ancient beam and a modern sample, we use these rates in conjunction with the exponential decay formula to determine the age of the beam. By rearranging and solving the equation, the calculated time gives us the approximate age of the ancient material. This process shines light on the history of our planet by allowing us to date artifacts, fossils, and even geologic events.