Finding the Decay Constant

Oct 13, 2025 by wbrum.org 26 views

Introduction

Radioactive substances decay over time, releasing energy in the form of radiation. The half-life of a radioactive substance is a fundamental concept in nuclear physics, representing the time it takes for half of the substance to decay. In this article, we will explore the exponential decay model for a radioactive substance with a half-life of 23.8 years and use it to determine how long it will take for a sample of 400 grams to decay.

Exponential Decay Model

The exponential decay model is given by the equation:

${ A(t) = A_0 e^{kt} }$

where:

${ A(t) }$ is the amount of substance remaining at time ${ t }$
${ A_0 }$ is the initial amount of substance
${ k }$ is the decay constant
${ t }$ is time

The half-life of a radioactive substance is related to the decay constant ${ k }$ by the equation:

${ \frac{1}{2} = e^{k \cdot t_{1/2}} }$

where ${ t_{1/2} }$ is the half-life.

Finding the Decay Constant

Given that the half-life of the substance is 23.8 years, we can use the equation above to find the decay constant ${ k }$ .

${ \frac{1}{2} = e^{k \cdot 23.8} }$

Taking the natural logarithm of both sides:

${ \ln \left( \frac{1}{2} \right) = k \cdot 23.8 }$

Solving for ${ k }$ :

${ k = \frac{\ln \left( \frac{1}{2} \right)}{23.8} }$

Using a calculator to evaluate the expression:

${ k \approx -0.0294 }$

Rounding ${ k }$ to the nearest thousandth:

${ k \approx -0.029 }$

Exponential Decay Model for the Substance

Now that we have found the decay constant ${ k }$ , we can write the exponential decay model for the substance:

${ A(t) = A_0 e^{-0.029t} }$

How Long Will it Take for a Sample of 400 Grams to Decay?

To determine how long it will take for a sample of 400 grams to decay, we need to find the time ${ t }$ when the amount of substance remaining is 200 grams (half of the initial amount).

${ 200 = 400 e^{-0.029t} }$

Dividing both sides by 400:

${ \frac{1}{2} = e^{-0.029t} }$

Taking the natural logarithm of both sides:

${ \ln \left( \frac{1}{2} \right) = -0.029t }$

Solving for ${ t }$ :

${ t = \frac{\ln \left( \frac{1}{2} \right)}{-0.029} }$

Using a calculator to evaluate the expression:

${ t \approx 23.8 }$

Therefore, it will take approximately 23.8 years for a sample of 400 grams to decay.

Conclusion

In this article, we have explored the exponential decay model for a radioactive substance with a half-life of 23.8 years. We have found the decay constant ${ k }$ and used it to write the exponential decay model for the substance. We have also used the model to determine how long it will take for a sample of 400 grams to decay. The results demonstrate the importance of understanding exponential decay in nuclear physics and its applications in various fields.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Discussion

What are some real-world applications of exponential decay in nuclear physics?

How does the half-life of a radioactive substance relate to its decay constant?

What are some common sources of radioactive substances in everyday life?

Introduction

In our previous article, we explored the exponential decay model for a radioactive substance with a half-life of 23.8 years. We also used the model to determine how long it will take for a sample of 400 grams to decay. In this article, we will answer some frequently asked questions about exponential decay and radioactive substances.

Q: What is exponential decay?

A: Exponential decay is a process where the amount of a substance decreases over time at a rate proportional to the amount present. This is often seen in radioactive substances, where the decay rate is constant and the amount of substance remaining decreases exponentially.

Q: What is the half-life of a radioactive substance?

A: The half-life of a radioactive substance is the time it takes for half of the substance to decay. It is a fundamental concept in nuclear physics and is used to describe the decay rate of radioactive substances.

Q: How is the half-life related to the decay constant?

A: The half-life of a radioactive substance is related to the decay constant by the equation:

${ \frac{1}{2} = e^{k \cdot t_{1/2}} }$

where ${ k }$ is the decay constant and ${ t_{1/2} }$ is the half-life.

Q: How do you calculate the decay constant?

A: To calculate the decay constant, you can use the equation:

${ k = \frac{\ln \left( \frac{1}{2} \right)}{t_{1/2}} }$

where ${ t_{1/2} }$ is the half-life.

Q: What is the exponential decay model?

A: The exponential decay model is a mathematical equation that describes the decay of a substance over time. It is given by the equation:

${ A(t) = A_0 e^{kt} }$

where ${ A(t) }$ is the amount of substance remaining at time ${ t }$ , ${ A_0 }$ is the initial amount of substance, ${ k }$ is the decay constant, and ${ t }$ is time.

Q: How do you use the exponential decay model to predict the behavior of a radioactive substance?

A: To use the exponential decay model to predict the behavior of a radioactive substance, you need to know the initial amount of substance, the decay constant, and the time. You can then plug these values into the equation:

${ A(t) = A_0 e^{kt} }$

to find the amount of substance remaining at a given time.

Q: What are some real-world applications of exponential decay in nuclear physics?

A: Exponential decay is used in a variety of real-world applications, including:

Nuclear medicine: Exponential decay is used to determine the half-life of radioactive isotopes used in medical treatments.
Radiation detection: Exponential decay is used to detect and measure the amount of radiation present in a sample.
Nuclear power: Exponential decay is used to predict the behavior of radioactive substances in nuclear reactors.

Q: How can you use the exponential decay model to determine how long it will take for a sample of a radioactive substance to decay?

A: To use the exponential decay model to determine how long it will take for a sample of a radioactive substance to decay, you need to know the initial amount of substance, the decay constant, and the amount of substance remaining. You can then plug these values into the equation:

${ A(t) = A_0 e^{kt} }$

to find the time it will take for the sample to decay.

Conclusion

In this article, we have answered some frequently asked questions about exponential decay and radioactive substances. We have also provided a brief overview of the exponential decay model and its applications in nuclear physics. We hope this article has been helpful in understanding the concept of exponential decay and its importance in nuclear physics.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Discussion

What are some other real-world applications of exponential decay in nuclear physics?

How can you use the exponential decay model to predict the behavior of a radioactive substance in different environments?

What are some common sources of radioactive substances in everyday life?

How can you use the exponential decay model to determine the half-life of a radioactive substance?