If 75% Of \[$^ {238}U\$\] Has Decayed To \[$^ {206}Pb\$\], How Many Half-lives Have Passed?A. One B. Two C. Three D. Four
Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This process is a random and spontaneous event, and it occurs at a constant rate for a given radioactive substance. One of the key concepts in understanding radioactive decay is the concept of half-life, which is the time required for half of the initial amount of a radioactive substance to decay.
What is Half-Life?
Half-life is a measure of the stability of a radioactive substance. It is the time required for half of the initial amount of a substance to decay. The half-life of a substance is a constant value that is independent of the initial amount of the substance. For example, if a substance has a half-life of 10 years, it means that after 10 years, half of the initial amount of the substance will have decayed.
Calculating Half-Lives
To calculate the number of half-lives that have passed, we need to know the initial amount of the substance and the amount that has decayed. In this case, we are given that 75% of the initial amount of {^ {238}U$}$ has decayed to {^ {206}Pb$}$. This means that 25% of the initial amount of {^ {238}U$}$ remains.
Using the Half-Life Formula
The formula for calculating the number of half-lives that have passed is:
n = log(Nf/Ni) / log(2)
where n is the number of half-lives, Nf is the final amount of the substance, Ni is the initial amount of the substance, and log is the logarithm to the base 2.
Applying the Formula
In this case, we are given that 75% of the initial amount of {^ {238}U$}$ has decayed, which means that 25% of the initial amount remains. We can plug in the values into the formula as follows:
n = log(0.25/1) / log(2)
n = log(0.25) / log(2)
n = -0.60206 / 0.30103
n = 2
Conclusion
Therefore, the correct answer is B. Two half-lives have passed.
Understanding the Significance of Half-Lives
Half-lives are an important concept in understanding radioactive decay. They provide a way to measure the stability of a radioactive substance and to calculate the amount of time that has passed since the substance was formed. In this case, we have seen how to calculate the number of half-lives that have passed using the half-life formula.
Real-World Applications of Half-Lives
Half-lives have many real-world applications, including:
- Nuclear Power Plants: Half-lives are used to calculate the amount of time that has passed since a nuclear power plant was built, which is important for determining the safety of the plant.
- Radiation Therapy: Half-lives are used to calculate the amount of time that has passed since a patient received radiation therapy, which is important for determining the effectiveness of the treatment.
- Environmental Monitoring: Half-lives are used to calculate the amount of time that has passed since a radioactive substance was released into the environment, which is important for determining the impact of the release on the environment.
Conclusion
In conclusion, half-lives are an important concept in understanding radioactive decay. They provide a way to measure the stability of a radioactive substance and to calculate the amount of time that has passed since the substance was formed. By understanding half-lives, we can better understand the behavior of radioactive substances and their impact on the environment.
References
- National Institute of Standards and Technology: "Half-Life"
- World Nuclear Association: "Radioactive Decay"
- International Atomic Energy Agency: "Half-Life"
Frequently Asked Questions (FAQs) about Half-Lives =====================================================
Q: What is the difference between half-life and decay rate?
A: Half-life and decay rate are two related but distinct concepts in radioactive decay. Half-life is the time required for half of the initial amount of a substance to decay, while decay rate is the rate at which a substance decays. Decay rate is typically measured in units of time, such as seconds or years, and is usually expressed as a percentage or a fraction of the initial amount of the substance.
Q: How is half-life related to the decay constant?
A: The half-life of a substance is related to the decay constant (λ) by the following equation:
t1/2 = ln(2) / λ
where t1/2 is the half-life, ln(2) is the natural logarithm of 2, and λ is the decay constant.
Q: What is the relationship between half-life and the number of atoms?
A: The half-life of a substance is independent of the number of atoms present. This means that the half-life of a substance will remain the same regardless of the initial amount of the substance.
Q: Can half-life be affected by external factors?
A: No, half-life is a fundamental property of a substance and is not affected by external factors such as temperature, pressure, or radiation. However, the decay rate of a substance can be affected by external factors, such as radiation or chemical reactions.
Q: How is half-life used in real-world applications?
A: Half-life is used in a variety of real-world applications, including:
- Nuclear Power Plants: Half-lives are used to calculate the amount of time that has passed since a nuclear power plant was built, which is important for determining the safety of the plant.
- Radiation Therapy: Half-lives are used to calculate the amount of time that has passed since a patient received radiation therapy, which is important for determining the effectiveness of the treatment.
- Environmental Monitoring: Half-lives are used to calculate the amount of time that has passed since a radioactive substance was released into the environment, which is important for determining the impact of the release on the environment.
Q: Can half-life be used to predict the future behavior of a substance?
A: Yes, half-life can be used to predict the future behavior of a substance. By knowing the half-life of a substance, you can calculate the amount of time that will pass before a certain percentage of the substance has decayed.
Q: What is the significance of half-life in nuclear physics?
A: Half-life is a fundamental concept in nuclear physics and is used to describe the behavior of radioactive substances. It is a measure of the stability of a substance and is used to calculate the amount of time that has passed since a substance was formed.
Q: Can half-life be used to determine the age of a substance?
A: Yes, half-life can be used to determine the age of a substance. By knowing the half-life of a substance and the amount of time that has passed since the substance was formed, you can calculate the age of the substance.
Q: What is the relationship between half-life and the decay curve?
A: The half-life of a substance is related to the decay curve by the following equation:
N(t) = N0 * (1/2)^(t/t1/2)
where N(t) is the amount of the substance at time t, N0 is the initial amount of the substance, t is the time, and t1/2 is the half-life.
Q: Can half-life be used to predict the amount of radiation emitted by a substance?
A: Yes, half-life can be used to predict the amount of radiation emitted by a substance. By knowing the half-life of a substance and the amount of time that has passed since the substance was formed, you can calculate the amount of radiation that will be emitted by the substance.