The Table On The Right Shows The Amount Of A Radioactive Substance In Mg Remaining After { X $}$ Days From An Initial Sample Of 3 Milligrams. Answer Questions (a) Through (c). \[ \begin{tabular}{|c|c|c|c|c|} \hline { X }$ (days) & 0 &

Introduction

Radioactive substances are known to decay over time, releasing energy in the process. This decay is a natural phenomenon that can be modeled using mathematical equations. In this article, we will explore the concept of half-life and how it relates to the decay of radioactive substances. We will use a table to illustrate the amount of a radioactive substance remaining after a certain number of days and answer questions related to the decay process.

The Table

${ x }$ (days)	0	1	2	3	4	5
Amount (mg)	3	2.4	1.8	1.32	0.99	0.74

Question (a)

What is the initial amount of the radioactive substance?

The initial amount of the radioactive substance is 3 milligrams.

Question (b)

What is the decay rate of the radioactive substance?

To determine the decay rate, we need to examine the table and look for a pattern. We can see that the amount of the substance decreases by a factor of 0.8 every day. This means that the decay rate is 0.8 or 80% per day.

Question (c)

What is the half-life of the radioactive substance?

The half-life of a radioactive substance is the time it takes for the amount of the substance to decrease by half. We can see from the table that the amount of the substance decreases by half every 2 days. Therefore, the half-life of this radioactive substance is 2 days.

Understanding Half-Life

Half-life is a fundamental concept in chemistry that describes the rate at which a radioactive substance decays. It is a measure of the time it takes for the amount of the substance to decrease by half. The half-life of a substance is determined by its decay rate, which is a measure of how quickly the substance decays.

Calculating Half-Life

To calculate the half-life of a radioactive substance, we need to know its decay rate. The decay rate is typically expressed as a percentage per unit of time. For example, if a substance decays at a rate of 80% per day, its half-life would be 1 day.

The Formula for Half-Life

The formula for half-life is:

${ \text{Half-life} = \frac{\ln(2)}{\text{Decay rate}} }$

Where:

• Half-life is the time it takes for the amount of the substance to decrease by half
• Decay rate is the rate at which the substance decays, expressed as a percentage per unit of time
• ln(2) is the natural logarithm of 2

Example

Suppose we have a radioactive substance that decays at a rate of 90% per day. We can use the formula to calculate its half-life:

${ \text{Half-life} = \frac{\ln(2)}{0.9} }$

${ \text{Half-life} = \frac{0.693}{0.9} }$

${ \text{Half-life} = 0.77 \text{ days} }$

Conclusion

In conclusion, the half-life of a radioactive substance is a measure of the time it takes for the amount of the substance to decrease by half. It is determined by the decay rate of the substance, which is a measure of how quickly the substance decays. We can use the formula for half-life to calculate the half-life of a substance given its decay rate.

References

• [1] Half-life. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Half-life
• [2] Radioactive decay. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Radioactive_decay

Further Reading

• [1] Radioactivity. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/science/chemistry/radioactivity
• [2] Half-life. (n.d.). In Chemistry LibreTexts. Retrieved from https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_General_Chemistry_(Brown_et_al.)/14%3A_Nuclear_Chemistry/14.2%3A_Half-Life
  Q&A: Radioactive Substances and Half-Life =============================================

Q: What is the difference between half-life and decay rate?

A: Half-life is the time it takes for the amount of a radioactive substance to decrease by half, while decay rate is the rate at which the substance decays, expressed as a percentage per unit of time.

Q: How do I calculate the half-life of a radioactive substance?

A: To calculate the half-life of a radioactive substance, you can use the formula:

${ \text{Half-life} = \frac{\ln(2)}{\text{Decay rate}} }$

Where:

• Half-life is the time it takes for the amount of the substance to decrease by half
• Decay rate is the rate at which the substance decays, expressed as a percentage per unit of time
• ln(2) is the natural logarithm of 2

Q: What is the relationship between half-life and the amount of a radioactive substance?

A: The amount of a radioactive substance decreases exponentially over time, with the half-life being the time it takes for the amount to decrease by half. This means that the amount of the substance will decrease by half every half-life.

Q: Can you give an example of how to calculate the half-life of a radioactive substance?

A: Suppose we have a radioactive substance that decays at a rate of 90% per day. We can use the formula to calculate its half-life:

${ \text{Half-life} = \frac{\ln(2)}{0.9} }$

${ \text{Half-life} = \frac{0.693}{0.9} }$

${ \text{Half-life} = 0.77 \text{ days} }$

Q: What is the significance of half-life in real-world applications?

A: Half-life is an important concept in many real-world applications, including:

• Nuclear power plants: Half-life is used to determine the amount of radioactive waste that will be produced over time.
• Radiation therapy: Half-life is used to determine the amount of radiation that will be delivered to a tumor over time.
• Environmental monitoring: Half-life is used to determine the amount of radioactive substances that will be present in the environment over time.

Q: Can you explain the concept of radioactive decay in simple terms?

A: Radioactive decay is the process by which unstable atoms lose energy and stability by emitting radiation. This process occurs when an atom has too many or too few neutrons, causing it to become unstable and release energy in the form of radiation.

Q: What are some common sources of radioactive substances?

A: Some common sources of radioactive substances include:

• Nuclear power plants
• Radioactive waste
• Medical treatments (such as radiation therapy)
• Environmental contamination (such as nuclear accidents)

Q: How can I protect myself from radioactive substances?

A: To protect yourself from radioactive substances, you can:

• Avoid areas with high levels of radiation
• Wear protective clothing and equipment (such as gloves and masks)
• Follow proper safety procedures when handling radioactive materials
• Stay informed about radiation levels in your area

Q: Can you explain the concept of half-life in terms of a real-world example?

A: Imagine a bottle of radioactive water that contains 100 milliliters of the substance. If the half-life of the substance is 1 day, then after 1 day, the amount of the substance will decrease to 50 milliliters. After another day, the amount will decrease to 25 milliliters, and so on. This is an example of how half-life works in real-world applications.