A pure sample of tritium, 3H, was prepared and sealed in a container for a number of years. Tritium undergoes â decay with a half-life of 12.32 years. How long has the container been sealed if analysis of the contents shows there are 5.25 mol of 3H and 6.35 mol of 3He present?
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The age of wine can be determined by measuring the trace amount of radioactive tritium, 3H, present in a sample. Tritium is formed from hydrogen in water vapor in the upper atmosphere by cosmic bombardment, so all naturally ocurring water contains a small amount of this isotope. Once the water is in a bottle of wine, however, the formation of additional tritium from the water is negligible, so the tritium initially present gradually diminishes by a first-order radioactive decay with a half-life of 12.5 years. If a bottle of wine is found to have a tritium concentration that is 0.116 that of freshly bottled wine (i.e. [3H]t = 0.116 [3H]0), what is the age of the wine?
To solve this problem, we can use the concept of half-life to determine the elapsed time since the tritium was sealed in the container.
Let's first understand the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the original quantity to decay. In this case, the half-life of tritium is given as 12.32 years.
Now, we are given that there are 5.25 mol of tritium (3H) and 6.35 mol of helium-3 (3He) present in the container. Since tritium undergoes β decay and converts into helium-3, the amount of tritium decreases while the amount of helium-3 increases.
The ratio of tritium to helium-3 is 1:1. This means that for every mole of tritium that decays, one mole of helium-3 is formed. Since there are 5.25 mol of tritium and 6.35 mol of helium-3, we can conclude that 5.25 mol of tritium have decayed.
Now, let's calculate the number of half-lives that have elapsed. Since each half-life is 12.32 years, we can divide the total elapsed time by the half-life to find the number of half-lives.
Number of half-lives = elapsed time / half-life
Let's denote the elapsed time as "t." We can rearrange the equation as:
t = (number of half-lives) * (half-life)
We know that 5.25 mol of tritium have decayed, and each mole decays in one half-life. So the number of half-lives is equal to 5.25.
Now, we can substitute the values into the equation:
t = 5.25 * 12.32 years
Calculating the value, we get:
t ≈ 64.68 years
Therefore, the container has been sealed for approximately 64.68 years.
I hope this explanation helps! Let me know if you have any more questions.