Problem 36 - Chemistry An Atoms First Approach

Question

Fresh rainwater or surface water contains enough tritium ( \(^{3}_{1} \mathrm{H}\) ) to show 5.5 decay events per minute per \(100 .\) g water. Tritium has a half-life of 12.3 years. You are asked to check a vintage wine that is claimed to have been produced in \(1946 .\) How many decay events per minute should you expect to observe in \(100 .\) g of that wine?

Step-by-Step Solution

Answer

After 75 years, you should expect to observe approximately 0.099 decay events per minute in 100g of the vintage wine produced in 1946.

Step 1: Calculate time elapsed

Given the vintage wine was produced in 1946, we can calculate the time elapsed (in years) by subtracting 1946 from the current year 2021. Time elapsed \(t = 2021 - 1946 = 75\) years Step 2: Calculate the decay factor

Step 2: Calculate decay factor

Using the radioactive decay formula and the given half-life (12.3 years), we can calculate the fraction of tritium that remains in the wine after 75 years. Decay factor \(= (\frac{1}{2})^{\frac{t}{t_{1/2}}} = (\frac{1}{2})^{\frac{75}{12.3}}\) Calculate the decay factor: \( \approx 0.018\) Step 3: Calculate the remaining decay events per minute in the wine

Step 3: Calculate remaining decay events

We are given the initial decay events per minute in fresh rainwater or surface water (5.5 events). We will multiply this initial value by the decay factor we calculated in the previous step to find the remaining decay events per minute in 100g of wine. Remaining decay events \( = N_0 \times\) Decay factor \( = 5.5 \times 0.018\) Calculate the remaining decay events: \( \approx 0.099\) Step 4: Present the result

Step 4: Present the final result

After 75 years (time elapsed from 1946), you should expect to observe approximately 0.099 decay events per minute in 100g of the vintage wine produced in 1946.

Key Concepts

Understanding the Radioactive Decay Formula

Radioactive decay is a fundamental process by which unstable atomic nuclei release energy by emitting radiation. To quantify this process, scientists use the radioactive decay formula. This formula enables us to calculate the remaining quantity of a radioactive substance over time using its half-life, which is the time required for half of the original amount of radioactive material to decay.

The formula expressed mathematically is: \[ N(t) = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{t_{1/2}}} \] Here, \( N(t) \) represents the number of undecayed atoms at time \( t \), \( N_0 \) is the initial number of atoms, \( t \) is the elapsed time, and \( t_{1/2} \) is the half-life of the isotope. Calculating the decay factor, as shown in the provided solution, is a critical step to determine the current activity or concentration of the substance after a certain period has elapsed.

Tritium Half-Life and its Implications

Tritium, or Hydrogen-3, is a radioactive isotope of hydrogen with a half-life of approximately 12.3 years. The half-life of a radioactive element is crucial because it is a stable value that characterizes its rate of decay. Knowing the half-life allows us to predict how long it takes for half of the tritium in a sample to decay.

This knowledge applies widely, from environmental science to vintage wine authentication. In the exercise, the half-life of tritium is used to estimate the number of decay events remaining in a sample of wine that is several decades old. As time passes, the tritium content decreases exponentially, indicating the age and hence the authenticity of the vintage.

Radiometric Dating of Wine

Radiometric dating is a technique used to date materials, like rocks or carbon, usually based on a comparison between the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates. However, this same principle can also apply to wine, using isotopes like tritium for authentication.

By measuring the decay events per minute of tritium in a wine sample, experts can establish if its claimed vintage is accurate. If the tritium content is higher than expected for a given vintage, the wine may not be as old as it is reported to be. In the exercise, the calculation was performed to predict the decay events for a bottle said to be from 1946, providing a scientific method to assess the wine's vintage declaration.