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:
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]
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.
