Tritium \(\left({ }^{3} \mathrm{H} ; t_{1 / 2}=12.26 \mathrm{yr}\right)\) is continually formed in the upper troposphere by interaction of solar particles with nitrogen. As a result, natural waters contain a small amount of tritium. Two samples of wine are analyzed, one known to be made in 1941 and another made earlier. The water in the 1941 wine has 2.23 times as much tritium as the water in the other. When was the other wine produced?

Tritium, also known as hydrogen-3, is a radioactive isotope of hydrogen. It is continuously formed in the upper atmosphere through the interaction of solar radiation with nitrogen atoms. Over time, tritium decays and its amount decreases. Decay occurs because tritium nuclei are unstable, leading to the radiation of particles. This process, called radioactive decay, changes tritium into a different element, helium-3. Since tritium is found naturally in water, including wine, we can measure how much tritium remains to determine the age of a sample, like an old bottle of wine. This decay follows predictable rates, which makes it excellent for dating purposes.

The half-life of a radioactive substance is the time it takes for half of the initial quantity of the substance to decay. For tritium, the half-life is 12.26 years. This value is crucial because it allows us to determine how much tritium remains over time. To calculate the half-life, we use the formula: \[ t_{1/2} = \frac{\ln 2}{k} \] Here, \( t_{1/2} \) represents the half-life, and \( k \) is the decay constant that indicates the rate of decay. Using the known half-life of 12.26 years, we can derive the decay constant, which will then be used for further calculations. It is through understanding half-life that we can make accurate predictions about the age of tritium-containing samples.

Radioactive decay is described by the exponential decay formula. The equation is: \[ N(t) = N_0 \, e^{-kt} \] Where:

• \(N(t)\) is the amount of substance remaining at time \(t\)
• \(N_0\) is the initial amount of the substance
• \(k\) is the decay constant
• \(t\) is the time elapsed

This formula shows how the amount of radioactive material decreases over time. The term \(e^{-kt}\) represents the fraction of the original amount remaining after time \(t\). Having an accurate exponential decay formula allows scientists to predict how much of a substance will remain after a given time, which is essential for dating ancient objects, like a vintage bottle of wine. By setting up this equation properly, we can solve for unknown ages.

The decay constant \(k\) is a critical value in the study of radioactive decay. It represents how quickly a substance decays, with larger values of \(k\) indicating faster decay rates. The decay constant can be calculated using the half-life formula: \[ k = \frac{\ln 2}{t_{1/2}} \] For tritium, with a half-life of 12.26 years, this becomes: \[ k = \frac{\ln 2}{12.26} \approx 0.0565 \text{ per year} \] Knowing the decay constant allows us to use the exponential decay formula to find out how much of a radioactive substance remains after a certain period. This is invaluable for dating and understanding the timeframes involved in radioactive decay processes.