Problem 14 - College Physics

Question

A radioactive isotope has a half-life of 76.0 min. A sample is prepared that has an initial activity of \(16.0 \times 10^{10}\) Bq. (a) How many radioactive nuclei are initially present in the sample? (b) How many are present after 76.0 min? What is the activity at this time? (c) Repeat part (b) for a time of 152 min after the sample is first prepared.

Step-by-Step Solution

Answer

(a) \(1.05 \times 10^{14}\) nuclei, (b) \(5.25 \times 10^{13}\) nuclei, \(8.0 \times 10^{10}\) Bq, (c) \(2.625 \times 10^{13}\) nuclei, \(4.0 \times 10^{10}\) Bq.

Step 1: Calculate Initial Number of Nuclei

The initial activity \( A_0 \) is given by \( A_0 = \frac{N_0 \ln(2)}{T_{1/2}} \), where \( T_{1/2} \) is the half-life. Rearranging, \( N_0 = \frac{A_0 T_{1/2}}{\ln(2)} \). Plugging in \( A_0 = 16.0 \times 10^{10} \) Bq and \( T_{1/2} = 76.0 \) min, we convert 76.0 min to seconds: \( 76.0 \times 60 = 4560 \) s. Now \( N_0 = \frac{16.0 \times 10^{10} \times 4560}{0.693} \approx 1.05 \times 10^{14} \) nuclei.

Step 2: Calculate Remaining Nuclei After Half-life

After one half-life (76.0 min), the number of remaining nuclei \( N(t) \) is \( N(t) = \frac{N_0}{2} \). Since \( N_0 = 1.05 \times 10^{14} \), \( N(t) = \frac{1.05 \times 10^{14}}{2} = 5.25 \times 10^{13} \) nuclei.

Step 3: Calculate Activity After Half-life

The activity after 76.0 min is half of the initial activity due to the number of remaining nuclei being halved. Therefore, \( A(t) = \frac{A_0}{2} = \frac{16.0 \times 10^{10}}{2} = 8.0 \times 10^{10} \) Bq.

Step 4: Calculate Remaining Nuclei and Activity After 152 min

152 min is two half-lives. Thus, the remaining number of nuclei is \( N(t) = \frac{N_0}{2^2} = \frac{1.05 \times 10^{14}}{4} = 2.625 \times 10^{13} \) nuclei. The activity after two half-lives is \( A(t) = \frac{A_0}{4} = \frac{16.0 \times 10^{10}}{4} = 4.0 \times 10^{10} \) Bq.

Key Concepts

Half-Life Calculation

Half-life is a critical concept in understanding radioactive decay. It refers to the time it takes for half of the radioactive nuclei in a sample to undergo decay. This means that after one half-life, only half of the original radioactive particles remain.
To calculate half-life, we use the formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where:

• \( N(t) \) is the number of undecayed nuclei at time \( t \).
• \( N_0 \) is the initial number of nuclei.
• \( T_{1/2} \) is the half-life of the substance.

In the provided exercise, the half-life of the isotope is 76.0 minutes. This allows us to determine how many nuclei remain after 76 minutes and further after 152 minutes.
By understanding this principle, you can predict the decay pattern of radioactive materials over time.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are atoms that have excess nuclear energy. This energy often leads to the isotopes emitting radiation in the form of alpha, beta, or gamma rays. Radioactive isotopes are crucial in various fields like medicine, archaeology, and energy production.
These isotopes differ from stable isotopes in that they decay over time, losing particles and energy. For example, the exercise considers a specific radioactive isotope whose initial activity is given. This means the rate at which it disintegrates and emits radiation was initially measured to be very high.
Understanding radioactive isotopes is important because they provide insights into energy release processes and are used in technological applications such as radiometric dating and medical imaging.

Nuclear Physics

Nuclear physics explores the components and interactions of atomic nuclei. This field explains the behavior of isotopes like those discussed in the exercise, focusing on nuclear decay processes.
Nuclear decay comes from the unstable nature of certain atomic nuclei. Such nuclei may transform into a more stable configuration by emitting particles, which is a process characterized by half-life studies.
Key contributions of nuclear physics include: * **Nuclear decay theory:** This explains how and why isotopes emit radiation over time. * **Nuclear reactions:** These can release energy or produce new elements, crucial in applications like nuclear power plants and atomic weaponry. Understanding these principles helps to manage applications where controlled nuclear reactions or the study of decay products are vital.

Activity Measurement

Activity measurement refers to quantifying the rate of decay of radioactive isotopes. It is typically measured in becquerels (Bq), where one becquerel is one disintegration per second.
In the given exercise, the initial activity of the sample provides a basis for calculating the number of radioactive nuclei present. Activity helps in determining the remaining radioisotope quantity over time.
The formula for activity \( A \) is:\[ A = \frac{N \ln(2)}{T_{1/2}} \] where:

• \( N \) is the number of radioactive nuclei at time \( t \).
• \( T_{1/2} \) is the half-life of the isotope.

Activity decreases as the nuclides decay, which is evident when observing the outcomes after each half-life period as demonstrated in the exercise.