The Radioactive Substance Uranium-240 Has A Half-life Of 14 Hours. The Amount { A(t) $}$ Of A Sample Of Uranium-240 Remaining Is Given By The Following Exponential Function:$[ A(t) = 5600 \left( \frac{1}{2} \right)^{\frac{t}{14}}

The Radioactive Decay of Uranium-240: Understanding the Exponential Function

In the realm of chemistry, radioactive decay is a fundamental concept that helps us understand the behavior of unstable elements. Uranium-240 is one such element that undergoes radioactive decay, resulting in the emission of radiation and the transformation of the nucleus into a more stable form. In this article, we will delve into the world of radioactive decay and explore the exponential function that governs the decay of uranium-240.

The amount of uranium-240 remaining in a sample after a certain period of time is given by the exponential function:

${ A(t) = 5600 \left( \frac{1}{2} \right)^{\frac{t}{14}} }$

where $A(t)$ is the amount of uranium-240 remaining at time $t$, and $t$ is measured in hours. This function is a classic example of an exponential decay function, where the amount of the substance decreases exponentially over time.

The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. In the case of uranium-240, the half-life is 14 hours. This means that every 14 hours, the amount of uranium-240 remaining in the sample will be reduced by half.

To solve the exponential function, we need to isolate the variable $t$. We can do this by taking the logarithm of both sides of the equation:

${ \log(A(t)) = \log(5600) - \frac{t}{14} \log\left(\frac{1}{2}\right) }$

Using the properties of logarithms, we can simplify the equation to:

${ \log(A(t)) = \log(5600) - \frac{t}{14} \log(2) }$

Now, we can solve for $t$ by isolating it on one side of the equation:

${ t = \frac{14}{\log(2)} \left( \log(5600) - \log(A(t)) \right) }$

To visualize the exponential decay of uranium-240, we can graph the function $A(t) = 5600 \left( \frac{1}{2} \right)^{\frac{t}{14}}$. The graph will show a rapid decrease in the amount of uranium-240 remaining over the first few hours, followed by a slower decrease as the substance approaches its half-life.

The exponential decay of uranium-240 has several real-world applications. For example, in nuclear reactors, the decay of uranium-240 is used to regulate the reaction rate. In medicine, the decay of uranium-240 is used to create radioactive isotopes for diagnostic and therapeutic purposes.

In conclusion, the exponential function $A(t) = 5600 \left( \frac{1}{2} \right)^{\frac{t}{14}}$ governs the decay of uranium-240, a radioactive substance with a half-life of 14 hours. By understanding this function, we can gain insights into the behavior of unstable elements and their applications in various fields.

• [1] "Radioactive Decay" by the International Atomic Energy Agency
• [2] "Exponential Decay" by the Wolfram MathWorld
• [3] "Uranium-240" by the Los Alamos National Laboratory

• "Radioactive Decay and Nuclear Reactions" by the American Chemical Society
• "Exponential Functions and Their Applications" by the Khan Academy
• "Uranium-240 and Its Applications" by the Journal of Nuclear Science and Technology
  The Radioactive Decay of Uranium-240: A Q&A Article

In our previous article, we explored the exponential function that governs the decay of uranium-240, a radioactive substance with a half-life of 14 hours. In this article, we will answer some of the most frequently asked questions about the radioactive decay of uranium-240.

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

A: The half-life of uranium-240 is 14 hours. This means that every 14 hours, the amount of uranium-240 remaining in a sample will be reduced by half.

Q: How does the exponential function govern the decay of uranium-240?

A: The exponential function $A(t) = 5600 \left( \frac{1}{2} \right)^{\frac{t}{14}}$ governs the decay of uranium-240. This function shows that the amount of uranium-240 remaining in a sample decreases exponentially over time.

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

A: The half-life is a critical concept in radioactive decay. It represents the time it takes for half of the initial amount of a radioactive substance to decay. In the case of uranium-240, the half-life is 14 hours.

Q: How does the decay of uranium-240 affect its applications?

A: The decay of uranium-240 affects its applications in various fields. For example, in nuclear reactors, the decay of uranium-240 is used to regulate the reaction rate. In medicine, the decay of uranium-240 is used to create radioactive isotopes for diagnostic and therapeutic purposes.

Q: Can uranium-240 be used as a source of energy?

A: Yes, uranium-240 can be used as a source of energy. However, its half-life is relatively short, which makes it less suitable for large-scale energy production. Other isotopes with longer half-lives, such as uranium-235, are more commonly used as fuel in nuclear reactors.

Q: How does the decay of uranium-240 affect the environment?

A: The decay of uranium-240 can have a significant impact on the environment. For example, the radiation emitted by uranium-240 can contaminate soil and water, posing a risk to human health and the environment.

Q: Can uranium-240 be used in medical applications?

A: Yes, uranium-240 can be used in medical applications. For example, it can be used to create radioactive isotopes for diagnostic and therapeutic purposes. However, its short half-life makes it less suitable for long-term medical applications.

Q: How does the decay of uranium-240 compare to other radioactive substances?

A: The decay of uranium-240 is similar to that of other radioactive substances, such as radium-226 and thorium-232. However, its half-life is relatively short, which makes it less suitable for large-scale applications.

In conclusion, the radioactive decay of uranium-240 is a complex process governed by an exponential function. Understanding this process is crucial for various applications, including nuclear reactors, medicine, and environmental science. By answering some of the most frequently asked questions about the decay of uranium-240, we hope to provide a better understanding of this fascinating topic.

• [1] "Radioactive Decay" by the International Atomic Energy Agency
• [2] "Exponential Decay" by the Wolfram MathWorld
• [3] "Uranium-240" by the Los Alamos National Laboratory

• "Radioactive Decay and Nuclear Reactions" by the American Chemical Society
• "Exponential Functions and Their Applications" by the Khan Academy
• "Uranium-240 and Its Applications" by the Journal of Nuclear Science and Technology