Gold-198 Has A Half-life Of Approximately 3 Days. If A 100 G Sample Of Gold-198 Decays For 9 Days, Approximately How Much Remains?Use The Half-life Formula:1. Number Of Half-lives: $n = \frac{t}{t_{1/2}}$2. $N(t) =

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Introduction

Radioactive decay is a process in which unstable atomic nuclei lose energy through the emission of radiation. This process is characterized by a half-life, which is the time required for half of the initial amount of a radioactive substance to decay. In this article, we will explore the concept of half-life and its application to a specific case study involving Gold-198.

The Half-Life Formula

The half-life formula is a mathematical expression that describes the rate of radioactive decay. It is given by:

  1. Number of half-lives: n=tt1/2n = \frac{t}{t_{1/2}}

where:

  • nn is the number of half-lives
  • tt is the time elapsed
  • t1/2t_{1/2} is the half-life of the radioactive substance
  1. Radioactive decay: N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n

where:

  • N(t)N(t) is the amount of the radioactive substance remaining after time tt
  • N0N_0 is the initial amount of the radioactive substance
  • nn is the number of half-lives

Case Study: Gold-198

Gold-198 is a radioactive isotope of gold with a half-life of approximately 3 days. If a 100 g sample of Gold-198 decays for 9 days, we can use the half-life formula to determine the amount of Gold-198 remaining.

Step 1: Calculate the Number of Half-Lives

Using the half-life formula, we can calculate the number of half-lives as follows:

n=tt1/2=9 days3 days=3n = \frac{t}{t_{1/2}} = \frac{9 \text{ days}}{3 \text{ days}} = 3

Step 2: Calculate the Amount of Gold-198 Remaining

Now that we have calculated the number of half-lives, we can use the radioactive decay formula to determine the amount of Gold-198 remaining:

N(t)=N0(12)n=100 g(12)3=100 g(18)=12.5 gN(t) = N_0 \left(\frac{1}{2}\right)^n = 100 \text{ g} \left(\frac{1}{2}\right)^3 = 100 \text{ g} \left(\frac{1}{8}\right) = 12.5 \text{ g}

Therefore, after 9 days, approximately 12.5 g of Gold-198 remains.

Discussion

The half-life formula is a powerful tool for understanding radioactive decay. By applying this formula to a specific case study, we can gain insight into the rate of decay and the amount of a radioactive substance remaining after a given time.

In this case study, we used the half-life formula to calculate the number of half-lives and the amount of Gold-198 remaining after 9 days. The results show that approximately 12.5 g of Gold-198 remains after 9 days, which is a significant reduction from the initial amount of 100 g.

Conclusion

In conclusion, the half-life formula is a useful tool for understanding radioactive decay. By applying this formula to a specific case study, we can gain insight into the rate of decay and the amount of a radioactive substance remaining after a given time. In this case study, we used the half-life formula to calculate the number of half-lives and the amount of Gold-198 remaining after 9 days.

References

  • Half-life formula: n=tt1/2n = \frac{t}{t_{1/2}} and N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n
  • Gold-198: a radioactive isotope of gold with a half-life of approximately 3 days

Future Work

Future work could involve applying the half-life formula to other radioactive substances and exploring the implications of radioactive decay in various fields, such as medicine, energy, and environmental science.

Limitations

One limitation of this case study is that it assumes a constant rate of decay, which may not be the case in reality. Additionally, the half-life formula is based on a simplified model of radioactive decay and may not account for all the complexities of the process.

Recommendations

Based on this case study, we recommend using the half-life formula to calculate the number of half-lives and the amount of a radioactive substance remaining after a given time. We also recommend exploring the implications of radioactive decay in various fields and considering the limitations of the half-life formula.

Appendix

The following table summarizes the results of this case study:

Time (days) Number of Half-Lives Amount of Gold-198 Remaining (g)
0 0 100
3 1 50
6 2 25
9 3 12.5

Introduction

In our previous article, we explored the concept of radioactive decay and the half-life formula. We used a case study involving Gold-198 to demonstrate how the half-life formula can be applied to calculate the amount of a radioactive substance remaining after a given time. In this article, we will answer some frequently asked questions about radioactive decay and the half-life formula.

Q: What is radioactive decay?

A: Radioactive decay is a process in which unstable atomic nuclei lose energy through the emission of radiation. This process is characterized by a half-life, which is the time required for half of the initial amount of a radioactive substance to decay.

Q: What is the half-life formula?

A: The half-life formula is a mathematical expression that describes the rate of radioactive decay. It is given by:

  1. Number of half-lives: n=tt1/2n = \frac{t}{t_{1/2}}
  2. Radioactive decay: N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n

Q: How do I calculate the number of half-lives?

A: To calculate the number of half-lives, you need to know the time elapsed (t) and the half-life of the radioactive substance (t1/2). You can use the formula:

n=tt1/2n = \frac{t}{t_{1/2}}

Q: How do I calculate the amount of a radioactive substance remaining?

A: To calculate the amount of a radioactive substance remaining, you need to know the initial amount (N0) and the number of half-lives (n). You can use the formula:

N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n

Q: What is the significance of the half-life formula?

A: The half-life formula is a powerful tool for understanding radioactive decay. It allows us to calculate the amount of a radioactive substance remaining after a given time, which is essential in various fields such as medicine, energy, and environmental science.

Q: What are some limitations of the half-life formula?

A: One limitation of the half-life formula is that it assumes a constant rate of decay, which may not be the case in reality. Additionally, the half-life formula is based on a simplified model of radioactive decay and may not account for all the complexities of the process.

Q: Can I use the half-life formula for other types of radioactive decay?

A: Yes, the half-life formula can be used for other types of radioactive decay, such as alpha, beta, and gamma decay. However, the half-life formula is specifically designed for radioactive decay, and you may need to modify it to account for other types of decay.

Q: How do I apply the half-life formula to a real-world problem?

A: To apply the half-life formula to a real-world problem, you need to identify the initial amount of the radioactive substance, the half-life of the substance, and the time elapsed. You can then use the half-life formula to calculate the amount of the substance remaining.

Q: What are some examples of real-world applications of the half-life formula?

A: Some examples of real-world applications of the half-life formula include:

  • Calculating the amount of radioactive waste remaining after a given time
  • Determining the effectiveness of radiation therapy in cancer treatment
  • Understanding the decay of radioactive isotopes in the environment
  • Calculating the amount of radioactive material remaining in a nuclear reactor

Conclusion

In conclusion, the half-life formula is a powerful tool for understanding radioactive decay. By applying this formula to a specific case study, we can gain insight into the rate of decay and the amount of a radioactive substance remaining after a given time. We hope that this Q&A article has provided you with a better understanding of the half-life formula and its applications.

References

  • Half-life formula: n=tt1/2n = \frac{t}{t_{1/2}} and N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n
  • Radioactive decay: a process in which unstable atomic nuclei lose energy through the emission of radiation
  • Half-life: the time required for half of the initial amount of a radioactive substance to decay

Future Work

Future work could involve exploring the implications of radioactive decay in various fields, such as medicine, energy, and environmental science. Additionally, researchers could investigate the limitations of the half-life formula and develop more accurate models of radioactive decay.

Appendix

The following table summarizes the results of this Q&A article:

Question Answer
What is radioactive decay? A process in which unstable atomic nuclei lose energy through the emission of radiation.
What is the half-life formula? A mathematical expression that describes the rate of radioactive decay.
How do I calculate the number of half-lives? Use the formula: n=tt1/2n = \frac{t}{t_{1/2}}
How do I calculate the amount of a radioactive substance remaining? Use the formula: N(t)=N0(12)nN(t) = N_0 \left(\frac{1}{2}\right)^n
What is the significance of the half-life formula? A powerful tool for understanding radioactive decay.
What are some limitations of the half-life formula? Assumes a constant rate of decay and may not account for all the complexities of the process.