Problem 24 - Calculus: Early Transcendental Functions

Question

Involve exponential decay.The half-life of uranium \(^{235} \mathrm{U}\) is approximately \(0.7 \times 10^{9}\) years. If 50 grams are buried at a nuclear waste site, how much will remain after 100 years?

Step-by-Step Solution

Answer

To get the short answer, input the calculated decay constant from step 2 and the amount of time = 100 years in the decay equation \(N(t) = N_0 e^{-\lambda t}\). Solve this equation to find out the remaining Uranium-235.

Step 1: Calculate decay constant

First, calculate the decay constant (\(\lambda\)) using the formula for half-life, \(T_{\frac{1}{2}} = \frac{0.693}{\lambda}\). For Uranium-235, the half-life is given as \(0.7 \times 10^{9}\) years, So, solving the equation for \(\lambda\) we have \(\lambda = \frac{0.693}{T_{\frac{1}{2}}}\)

Step 2: Insert values and solve for decay constant

Replace \(T_{\frac{1}{2}}\) with \(0.7 \times 10^{9}\) years into the equation from step 1 giving us \(\lambda = \frac{0.693}{0.7 \times 10^{9}}\). Calculate to find the value of \(\lambda\)

Step 3: Calculate remaining amount

Now, use the exponential decay formula: \(N(t) = N_0 e^{-\lambda t}\). Substitute the calculated \(\lambda\) from step 2, \(N_0\) = 50 grams (the initial amount) and \(t\) = 100 years into the equation. Solve it to get the remaining amount of Uranium-235 after 100 years.

Key Concepts

Understanding Half-Life

The concept of half-life is fundamental in understanding how radioactive elements like Uranium-235 decay over time. Half-life refers to the time required for a quantity to reduce to half its initial amount naturally. This means, if you start with a specific amount of a radioactive substance, after one half-life, only half of it will remain.

To compute half-life, the equation used is: \[ T_{\frac{1}{2}} = \frac{0.693}{\lambda} \]where:

• \(T_{\frac{1}{2}}\) is the half-life.
• \(\lambda\) is the decay constant.

For Uranium-235, the half-life is known to be approximately \(0.7 \times 10^{9}\) years. This incredibly long half-life shows that Uranium-235 remains radioactive for a very long time.

Uranium-235 and Its Unique Properties

Uranium-235 is a naturally occurring isotope of Uranium and is known for its application in nuclear reactors and atomic bombs. It's one of the few materials that can be used as fuel to sustain a nuclear chain reaction. This unique property makes it vitally important but also requires careful handling and management.

With a half-life of \(0.7 \times 10^{9}\) years, Uranium-235 decays very slowly. Therefore, its radioactive properties can persist for millions of years, which is crucial when considering the disposal and storage of nuclear waste.

In nuclear reactions, Uranium-235 can be split into lighter elements in a process called fission, releasing large amounts of energy. This characteristic is used to produce electricity in nuclear power plants, making the understanding of its decay over time through the concept of half-life and decay constant very important.

The Decay Constant and Its Role

The decay constant, represented by \(\lambda\), plays a pivotal role in understanding the rate at which radioactive isotopes like Uranium-235 undergo decay. It is a probability rate that a single atom will decay in a unit of time.

The relationship between the decay constant and half-life can be shown through the formula: \[ T_{\frac{1}{2}} = \frac{0.693}{\lambda} \]

The decay constant is crucial when calculating how much of a substance will remain after a given period. For example, after calculating \(\lambda\) using the half-life, we can predict how much of Uranium-235 from an original sample (e.g., 50 grams) will remain after a specific time (e.g., 100 years) using the exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \]where:

• \(N(t)\) is the remaining quantity after time \(t\).
• \(N_0\) is the initial quantity.
• \(t\) is the time over which decay is observed.

Understanding the decay constant allows scientists to model how nuclear materials will change over time, ensuring safe handling, storage, and use of radioactive substances.