College

The half-life of hydrogen-3 is 12.26 years. If 35.8 g of hydrogen-3 are allowed to decay for 36.78 years, what mass of hydrogen-3 will be left?

Answer :

To solve this problem, we need to understand the concept of half-life in radioactive decay.

The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life of hydrogen-3, also known as tritium, is given as 12.26 years.

We are asked to find the remaining mass of hydrogen-3 after 36.78 years, starting with an initial mass of 35.8 grams.

Here's how to solve it step-by-step:

  1. Determine the Number of Half-Lives:

    The number of half-lives that have passed is calculated by dividing the total time by the half-life of the substance:

    [tex]\text{Number of Half-Lives} = \frac{\text{Total Time}}{\text{Half-Life}} = \frac{36.78 \text{ years}}{12.26 \text{ years}} \approx 3[/tex]

  2. Calculate the Remaining Mass:

    Every half-life, the mass of the remaining substance is halved. So, we apply the equation for exponential decay:

    [tex]\text{Remaining Mass} = \text{Initial Mass} \times \left( \frac{1}{2} \right)^n[/tex]

    Where [tex]n[/tex] is the number of half-lives. Plugging in the values we get:

    [tex]\text{Remaining Mass} = 35.8 \text{ g} \times \left( \frac{1}{2} \right)^3[/tex]

    [tex]\text{Remaining Mass} = 35.8 \text{ g} \times \frac{1}{8}[/tex]

    [tex]\text{Remaining Mass} \approx 4.475 \text{ g}[/tex]

So, after 36.78 years, approximately 4.475 grams of hydrogen-3 will remain.

This step-by-step explanation helps illustrate not only how to plug numbers into a formula but also why these steps provide the solution to the problem.