An Isotope Of An Element Has A Half-life Of Approximately 21 Hours. Initially, There Are 18 Grams Of The Isotope Present.(a) Write The Exponential Function, $Q(t$\], That Relates The Amount Of Substance Remaining As A Function Of $t$,

Introduction

Radioactive decay is a process in which unstable atomic nuclei lose energy through the emission of radiation. This process is characterized by the half-life of an isotope, which is the time it takes for half of the initial amount of the substance to decay. In this article, we will explore how to write an exponential function that relates the amount of substance remaining as a function of time, given the half-life of an isotope.

The Concept of Half-Life

The half-life of an isotope is a fundamental concept in nuclear physics. It is defined as the time it takes for half of the initial amount of the substance to decay. For example, if an isotope has a half-life of 21 hours, it means that after 21 hours, half of the initial amount of the substance will have decayed. This process continues until the entire substance has decayed.

Writing the Exponential Function

To write the exponential function that relates the amount of substance remaining as a function of time, we need to use the formula:

$$Q(t) = Q_0 \times 2^{-t/h}$$

where:

• $Q(t)$ is the amount of substance remaining at time $t$
• $Q_0$ is the initial amount of substance
• $h$ is the half-life of the isotope
• $t$ is the time in hours

Applying the Formula

In this problem, we are given that the half-life of the isotope is 21 hours and the initial amount of substance is 18 grams. We can plug these values into the formula to get:

$$Q(t) = 18 \times 2^{-t/21}$$

This is the exponential function that relates the amount of substance remaining as a function of time.

Understanding the Exponential Function

The exponential function $Q(t) = 18 \times 2^{-t/21}$ can be broken down into two parts:

• The initial amount of substance, $Q_0 = 18$ grams
• The decay factor, $2^{-t/21}$, which represents the amount of substance remaining at time $t$

The decay factor is an exponential function that decreases as time increases. This is because the half-life of the isotope is 21 hours, which means that every 21 hours, half of the remaining substance will decay.

Graphing the Exponential Function

To visualize the exponential function, we can graph it using a graphing calculator or a computer program. The graph will show the amount of substance remaining as a function of time.

Interpreting the Graph

The graph of the exponential function will show a rapid decrease in the amount of substance remaining in the first few hours, followed by a slower decrease as time increases. This is because the half-life of the isotope is 21 hours, which means that the substance will decay rapidly at first and then more slowly as time increases.

Conclusion

In this article, we have explored how to write an exponential function that relates the amount of substance remaining as a function of time, given the half-life of an isotope. We have applied the formula to a specific problem and interpreted the results. The exponential function provides a powerful tool for understanding and modeling radioactive decay.

References

• [1] "Radioactive Decay" by Wolfram MathWorld
• [2] "Exponential Functions" by Math Open Reference

Further Reading

• "Radioactive Decay and Half-Life" by Physics Classroom
• "Exponential Functions and Radioactive Decay" by Khan Academy
  

Introduction

In our previous article, we explored how to write an exponential function that relates the amount of substance remaining as a function of time, given the half-life of an isotope. In this article, we will answer some frequently asked questions about radioactive decay and exponential functions.

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

A: The half-life of an isotope is the time it takes for half of the initial amount of the substance to decay. The decay rate, on the other hand, is the rate at which the substance decays. The decay rate is related to the half-life by the formula:

$$\text{Decay Rate} = \frac{\ln(2)}{h}$$

where $h$ is the half-life.

Q: How do I calculate the amount of substance remaining after a certain time?

A: To calculate the amount of substance remaining after a certain time, you can use the exponential function:

$$Q(t) = Q_0 \times 2^{-t/h}$$

where $Q(t)$ is the amount of substance remaining at time $t$, $Q_0$ is the initial amount of substance, $h$ is the half-life, and $t$ is the time.

Q: What is the relationship between the half-life and the decay constant?

A: The decay constant, $\lambda$, is related to the half-life by the formula:

$$\lambda = \frac{\ln(2)}{h}$$

where $h$ is the half-life.

Q: How do I calculate the half-life of an isotope?

A: To calculate the half-life of an isotope, you can use the formula:

$$h = \frac{\ln(2)}{\lambda}$$

where $\lambda$ is the decay constant.

Q: What is the significance of the half-life in radioactive decay?

A: The half-life is a fundamental concept in radioactive decay. It represents the time it takes for half of the initial amount of the substance to decay. The half-life is a measure of the stability of the isotope and is used to predict the amount of substance remaining after a certain time.

Q: Can I use the exponential function to model other types of decay?

A: Yes, the exponential function can be used to model other types of decay, such as chemical reactions and population growth. However, the specific formula and parameters will depend on the type of decay being modeled.

Q: How do I graph the exponential function?

A: To graph the exponential function, you can use a graphing calculator or a computer program. The graph will show the amount of substance remaining as a function of time.

Q: What is the relationship between the exponential function and the natural logarithm?

A: The exponential function is related to the natural logarithm by the formula:

$$Q(t) = Q_0 \times e^{-\lambda t}$$

where $Q(t)$ is the amount of substance remaining at time $t$, $Q_0$ is the initial amount of substance, $\lambda$ is the decay constant, and $t$ is the time.

Conclusion

In this article, we have answered some frequently asked questions about radioactive decay and exponential functions. We hope that this article has provided a better understanding of these concepts and has helped to clarify any confusion.

References

• [1] "Radioactive Decay" by Wolfram MathWorld
• [2] "Exponential Functions" by Math Open Reference
• [3] "Half-Life" by Physics Classroom
• [4] "Exponential Functions and Radioactive Decay" by Khan Academy

Further Reading

• "Radioactive Decay and Half-Life" by Physics Classroom
• "Exponential Functions and Radioactive Decay" by Khan Academy
• "Half-Life and Radioactive Decay" by HyperPhysics