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. The half-life of U239: Uranium 239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of U239 was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { in minutes } \end{array} & \begin{array}{c} \text { Grams } \\ \text { remaining } \end{array} \\ \hline 0 & 1 \\ \hline 1 & 0.971 \\ \hline 2 & 0.943 \\ \hline 3 & 0.916 \\ \hline 4 & 0.889 \\ \hline 5 & 0.863 \\ \hline \end{array} $$ a. Show that these are exponential data and find an exponential model. (For this problem, round all your answers to three decimal places.) b. What is the percentage decay rate each minute? What does this number mean in practical terms? c. Use functional notation to express the amount remaining after 10 minutes and then calculate that value. d. What is the half-life of U239?

Step by step solution

01

Verify Exponential Decay

To show the data is exponential, observe that for exponential decay, the ratio of the amount at consecutive time points should be constant. Calculate the ratios: \( \frac{0.971}{1} = 0.971 \), \( \frac{0.943}{0.971} \approx 0.971 \), \( \frac{0.916}{0.943} \approx 0.971 \), \( \frac{0.889}{0.916} \approx 0.971 \), and \( \frac{0.863}{0.889} \approx 0.971 \). The repeated ratio suggests an approximate common factor indicating exponential behavior.

02

Find the Exponential Model

Using the initial amount \( A_0 = 1 \) and common ratio \( r \approx 0.971 \), the exponential model is \( A(t) = 1 \times 0.971^t \). Hence, the model is \( A(t) = 0.971^t \).

03

Calculate Percentage Decay Rate

The percentage decay rate each minute is \( (1 - r) \times 100 = (1 - 0.971) \times 100 = 2.9\% \). This means each minute, about 2.9% of Uranium-239 decays.

04

Express Amount Remaining After 10 Minutes

Using the model \( A(t) = 0.971^t \), express the amount remaining after 10 minutes as \( A(10) = 0.971^{10} \).

05

Calculate the Amount After 10 Minutes

Evaluate \( A(10) = 0.971^{10} \approx 0.737 \). Therefore, after 10 minutes, approximately 0.737 grams of U239 is remaining.

06

Determine the Half-Life

The half-life is the time when \( A(t) = \frac{1}{2}A_0 \). Set \( 0.971^t = \frac{1}{2} \) and solve for \( t \). Take logarithms: \( t \log(0.971) = \log(0.5) \), thus \( t = \frac{\log(0.5)}{\log(0.971)} \approx 23.49 \) minutes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life Calculation

When we talk about the half-life of an isotope, we're referring to the time it takes for half of a given amount of the substance to decay. In practical terms, it is how long a sample takes to reduce to 50% of its initial quantity.
For Uranium 239, based on the exercise data, the half-life is calculated using the equation \( 0.971^t = \frac{1}{2} \). This equation states that after time \( t \), the fraction remaining should be half of the original amount. By solving for \( t \), we find \( t \approx 23.49 \) minutes.
This means in roughly 23.5 minutes, only half of the original Uranium 239 exists. Understanding the half-life is crucial for predicting how long a substance remains hazardous or active.

Exponential Modeling

Exponential modeling is a powerful mathematical tool used to describe situations where something grows or decays at a rate proportional to its current value.
In the case of Uranium 239, the decay process can be captured by an exponential function. The model \( A(t) = 0.971^t \) describes how Uranium 239 decreases over time. The initial amount \( A_0 = 1 \) gram and the base \( 0.971 \) represent the factor by which the amount decreases each minute.
By using such models, we can predict future amounts at any given time, making it invaluable in fields such as physics, biology, and finance, where exponential change occurs.

Decay Rate

The decay rate gives us an understanding of how quickly a substance decreases over time. For our Uranium 239 example, the decay rate is 2.9% per minute.
This means with each passing minute, 2.9% of the substance vanishes. To find this, take \( (1 - 0.971) \times 100 \), which calculates the percentage of decay per unit time. The decay rate is crucial since it helps to understand the stability of isotopes and their safety levels, especially in nuclear physics and environmental science.
A higher decay rate signifies a faster decay process, leading to quicker diminution of the substance.

Uranium 239 Isotope

Uranium 239 is a specific and unstable isotope of uranium that decays quickly, making it an excellent example for studying radioactive decay.
This isotope undergoes transformation as it breaks down into other elements due to its instability. Recognizing isotopes like Uranium 239 helps in understanding nuclear reactions, determining age in radiometric dating, and gauging potential risks in nuclear safety.

• Reactivity: High due to rapid decay
• Applications: Often in nuclear physics and studies of radioactive decay chains
• Safety Concerns: Its instability suggests careful handling is necessary

Gaining insights into these properties is essential for scientists working on nuclear energy and environmental conservation.