The Radioactive Substance Uranium-240 Has A Half-life Of 14 Hours. The Amount $A(t)$ Of A Sample Of Uranium-240 Remaining (in Grams) After $t$ Hours Is Given By The Following Exponential Function:$A(t) =

by ADMIN 209 views

Introduction

Radioactive decay is a fundamental concept in chemistry, where unstable atoms lose energy through radiation. Uranium-240 is a radioactive substance with a half-life of 14 hours, meaning that every 14 hours, the amount of the substance remaining will decrease by half. In this article, we will explore the exponential function that describes the amount of uranium-240 remaining after a certain period of time.

The Exponential Function

The amount $A(t)$ of a sample of uranium-240 remaining (in grams) after $t$ hours is given by the following exponential function:

A(t)=100e−0.0493tA(t) = 100e^{-0.0493t}

where $e$ is the base of the natural logarithm, approximately equal to 2.71828.

Understanding the Exponential Function

The exponential function $A(t) = 100e^{-0.0493t}$ describes the amount of uranium-240 remaining after $t$ hours. The function has two key components:

  • Initial amount: The initial amount of uranium-240 is 100 grams.
  • Decay rate: The decay rate is represented by the coefficient $-0.0493$, which is the natural logarithm of the decay constant.

Half-Life

The half-life of uranium-240 is 14 hours, meaning that every 14 hours, the amount of the substance remaining will decrease by half. To understand this, we can use the exponential function to calculate the amount of uranium-240 remaining after 14 hours.

Calculating the Amount Remaining

To calculate the amount of uranium-240 remaining after 14 hours, we can plug in $t = 14$ into the exponential function:

A(14)=100e−0.0493(14)A(14) = 100e^{-0.0493(14)}

A(14)=100e−0.6922A(14) = 100e^{-0.6922}

A(14)=100(0.4987)A(14) = 100(0.4987)

A(14)=49.87A(14) = 49.87

After 14 hours, the amount of uranium-240 remaining is approximately 49.87 grams.

Radioactive Decay: A Real-World Application

Radioactive decay is a fundamental process in chemistry, with numerous real-world applications. Some examples include:

  • Nuclear power plants: Radioactive decay is used to generate electricity in nuclear power plants.
  • Medical applications: Radioactive decay is used in medical applications, such as cancer treatment and imaging.
  • Geology: Radioactive decay is used to determine the age of rocks and fossils.

Conclusion

In conclusion, the exponential function $A(t) = 100e^{-0.0493t}$ describes the amount of uranium-240 remaining after $t$ hours. The function has two key components: the initial amount and the decay rate. The half-life of uranium-240 is 14 hours, meaning that every 14 hours, the amount of the substance remaining will decrease by half. Radioactive decay is a fundamental process in chemistry, with numerous real-world applications.

References

Further Reading

Introduction

In our previous article, we explored the exponential function that describes the amount of uranium-240 remaining after a certain period of time. In this article, we will answer some frequently asked questions about uranium-240 and radioactive decay.

Q: What is the half-life of uranium-240?

A: The half-life of uranium-240 is 14 hours, meaning that every 14 hours, the amount of the substance remaining will decrease by half.

Q: How does the exponential function describe the amount of uranium-240 remaining?

A: The exponential function $A(t) = 100e^{-0.0493t}$ describes the amount of uranium-240 remaining after $t$ hours. The function has two key components: the initial amount and the decay rate.

Q: What is the initial amount of uranium-240?

A: The initial amount of uranium-240 is 100 grams.

Q: What is the decay rate of uranium-240?

A: The decay rate of uranium-240 is represented by the coefficient $-0.0493$, which is the natural logarithm of the decay constant.

Q: How does radioactive decay occur?

A: Radioactive decay occurs when unstable atoms lose energy through radiation. This process is spontaneous and occurs at a constant rate.

Q: What are some real-world applications of radioactive decay?

A: Radioactive decay has numerous real-world applications, including:

  • Nuclear power plants: Radioactive decay is used to generate electricity in nuclear power plants.
  • Medical applications: Radioactive decay is used in medical applications, such as cancer treatment and imaging.
  • Geology: Radioactive decay is used to determine the age of rocks and fossils.

Q: Is radioactive decay a safe process?

A: Radioactive decay can be a safe process if handled properly. However, it can also be hazardous if not handled correctly.

Q: How can I learn more about radioactive decay and uranium-240?

A: There are many resources available to learn more about radioactive decay and uranium-240, including:

  • Scientific articles: You can find scientific articles on the topic of radioactive decay and uranium-240 in academic journals.
  • Online resources: There are many online resources available, including websites and tutorials.
  • Textbooks: You can find textbooks on the topic of radioactive decay and uranium-240 in your local library or online.

Conclusion

In conclusion, we have answered some frequently asked questions about uranium-240 and radioactive decay. We hope that this article has provided you with a better understanding of the topic.

References

Further Reading