Question

Tritium has a half-life of 12.3 years. How many years will have elapsed when the radioactivity of a tritium sample has decreased to 10 percent of its original value?

Answer 1

Roughly 49.2 years will have elapsed when the radioactivity of a tritium sample decreases to just below 10 percent of its original value, as it takes more than three half-lives (each of 12.3 years) for the sample to reach this level.

Step-by-step explanation:

Tritium, or hydrogen-3, undergoes radioactive decay, and its half-life is a key factor in determining how much time it will take for a specific amount of tritium to decay to a certain level. The half-life of 12.3 years means that every 12.3 years, the quantity of tritium will be reduced to half of its initial amount. To determine how many years it takes for the radioactivity to decrease to 10 percent of the original value, we need to use the concept of half-lives in sequence.

Each half-life reduces the initial amount by half:

• After 1 half-life (12.3 years): 50% remains.
• After 2 half-lives (24.6 years): 25% remains.
• After 3 half-lives (36.9 years): 12.5% remains.

Since 10% is less than 12.5%, it will take more than three half-lives. After the third half-life, the quantity keeps halving:

After 4 half-lives (49.2 years): 6.25% remains - which is less than 10%.

Thus, the radioactivity of the tritium sample will decrease to just below 10 percent after 4 half-lives, meaning roughly 49.2 years will have elapsed.

Answer 2

Given

Half life of tritium is

`t_{(1)/(2)}=12.3\text{ years}`

The sample is reduced to 10% of its original value.

To find

How many years will have elapsed when the radioactivity of a tritium sample has decreased to 10 percent of its original value?

Explanation

The activity is given by

`\begin{gathered} A=A_oe^(-\lambda t) \\ \Rightarrow0.1A_o=A_oe^{-\frac{0.693}{t_{(1)/(2)}}t} \\ \Rightarrow ln(0.1)=-(0.693)/(1.23)t \\ \Rightarrow-2.302=-(0.693)/(1.23)t \\ \Rightarrow t=4.08 \end{gathered}`

Conclusion

The time taken is 4.08 years