Question

The radioactive isotope tritium, \({ }^{3} \mathrm{H}\), is produced in nature in much the same way as \({ }_{6}^{14} \mathrm{C} .\) Its half-life is 12.3 years. Estimate the \({ }_{1}^{3} \mathrm{H}\) ratio of the tritium of water in the area to the tritium in a bottle of wine claimed to be 25 years old.

Short answer

Answer: The approximate ratio of tritium in water in the area to tritium in the 25-year-old wine is 4.82:1.

Step-by-step solution

Write down the half-life formula for radioactive decay

The formula for radioactive decay is given by: $$N(t) = N_0 \cdot (1/2)^{t / T}$$ where: - \(N(t)\) represents the amount of the radioactive substance remaining after time \(t\) has passed, - \(N_0\) is the initial amount of the substance, - \(t\) is the time in years, and - \(T\) is the half-life of the substance (12.3 years for tritium in this case).

Calculate the decay in tritium for 25 years

We are given that the wine is claimed to be 25 years old. So, we want to determine the amount of tritium remaining in the wine after 25 years. Using the half-life formula and replacing \(t\) with 25 years and \(T\) with 12.3 years, $$N(25) = N_0 \cdot (1/2)^{25 / 12.3}$$

Calculate the tritium ratio in wine

Now that we have the equation for the amount of tritium remaining in the wine, we can write the tritium ratio of wine as: $$\text{Tritium ratio}_{\text{wine}} = \frac{N(25)}{N_0} = (1/2)^{25 / 12.3}$$

Calculate the tritium ratio in water

Since the wine has been stored in a bottle for 25 years, it will not have any more recent tritium added to it. In contrast, the water in the area will have the latest tritium ratios since it is constantly being exposed to new tritium. Hence, water will have a tritium ratio of 1.

Estimate the \({ }_{1}^{3} \mathrm{H}\) ratio of tritium in water to wine

Now we can estimate the \({ }_{1}^{3} \mathrm{H}\) ratio of tritium in water to the tritium in the 25-year-old wine as: $$\frac{\text{Tritium ratio}_{\text{water}}}{\text{Tritium ratio}_{\text{wine}}} = \frac{1}{(1/2)^{25 / 12.3}}$$ Calculate the ratio: $$\frac{1}{(1/2)^{25 / 12.3}} \approx 4.82$$ So, the \({ }_{1}^{3} \mathrm{H}\) ratio of tritium in water in the area to the tritium in the 25-year-old wine is approximately 4.82:1.