Problem 29 - University Physics with Modern Physics

Question

The radioactive nuclide \(^{199} \mathrm{Pt}\) has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of \(7.56 \times 10^{11} \mathrm{Bq}\). (a) How many \(^{199} \mathrm{Pt}\) nuclei are initially present in the sample? (b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a time 92.4 minutes after the sample is first prepared.

Step-by-Step Solution

Answer

(a) Initially, there are approximately \(2.02 \times 10^{15}\) nuclei. (b) After 30.8 minutes, \(1.01 \times 10^{15}\) nuclei remain; activity is \(3.78 \times 10^{11}\) Bq. (c) After 92.4 minutes, \(2.53 \times 10^{14}\) nuclei, with activity \(9.45 \times 10^{10}\) Bq.

Step 1: Identify Key Formula

To find the number of nuclei initially present, use the formula for activity: \( A = \lambda N \), where \( A \) is the activity and \( \lambda \) is the decay constant. The activity \( A = 7.56 \times 10^{11} \text{ Bq} \) and the half-life is 30.8 minutes.

Step 2: Calculate Decay Constant

The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by the formula \( \lambda = \frac{0.693}{T_{1/2}} \). Substitute \( T_{1/2} = 30.8 \text{ minutes} = 1848 \text{ seconds} \) to find \( \lambda \). \( \lambda = \frac{0.693}{1848} = 3.75 \times 10^{-4} \text{ s}^{-1} \).

Step 3: Compute Initial Nuclei Count

Using the equation from Step 1, solve for the number of nuclei \( N _0 \) using \( N_0 = \frac{A}{\lambda} \).\( N_0 = \frac{7.56 \times 10^{11}}{3.75 \times 10^{-4}} \approx 2.02 \times 10^{15} \text{ nuclei} \).

Step 4: Determine Nuclei and Activity After One Half-Life

After one half-life (30.8 minutes), the number of nuclei will be halved due to radioactivity decay:\( N = \frac{N_0}{2} = \frac{2.02 \times 10^{15}}{2} = 1.01 \times 10^{15} \text{ nuclei} \).The activity at this time will also be halved:\( A = \frac{A_0}{2} = \frac{7.56 \times 10^{11}}{2} = 3.78 \times 10^{11} \text{ Bq} \).

Step 5: Calculate Nuclei and Activity After Three Half-Lives

For 92.4 minutes, which is three half-lives (30.8 minutes each), the number of decays is 2³:\( N = \frac{N_0}{2^3} = \frac{2.02 \times 10^{15}}{8} = 2.525 \times 10^{14} \text{ nuclei} \).The activity is also reduced by 2³:\( A = \frac{A_0}{2^3} = \frac{7.56 \times 10^{11}}{8} = 9.45 \times 10^{10} \text{ Bq} \).

Key Concepts

Half-life Calculation

The concept of half-life is essential when studying radioactive decay. It represents the time taken for the amount of a radioactive substance to drop to half its initial value. Understanding half-life is crucial for calculating how much of a substance remains after a certain period and is usually denoted by the symbol \( T_{1/2} \). In the case of \( ^{199}\text{Pt} \), with a half-life of 30.8 minutes, this means that in 30.8 minutes, half of the \( ^{199}\text{Pt} \) nuclei in a sample will have decayed into another form.

When faced with exercises about half-life, remember this simple process:

• Determine the starting number of nuclides.
• Identify how many half-life periods have passed.
• For each half-life that passes, divide the remaining amount of substance by two.

This model allows us to predict the number of particles remaining and provides insight into the stability and lifespan of a given radioactive substance.

Decay Constant

The decay constant is a fundamental property of a radioactive material, symbolized by \( \lambda \), and measures the probability of decay of a nucleus per unit time. It connects directly to the half-life through the specific equation: \[ \lambda = \frac{0.693}{T_{1/2}} \] Here, \( 0.693 \) is the natural logarithm of 2, which accounts for the exponential nature of radioactive decay. Knowing the decay constant allows us to understand how quickly or slowly a radioactive material will decay.

For \( ^{199} \text{Pt} \), with a half-life of 30.8 minutes (or 1848 seconds), the decay constant \( \lambda \) can be calculated as follows: \[ \lambda = \frac{0.693}{1848} \approx 3.75 \times 10^{-4} \text{ s}^{-1} \] Once the decay constant is known, it can be used in more complex calculations involving the remaining quantities and decay rates of radioactive materials under different elapsed time.

Radioactive Nuclide Calculation

A central part of any radioactive decay problem is determining how many radioactive nuclei initially exist in the sample. This calculation helps in understanding the starting point for any further calculations related to decay over time.Given the initial activity of a sample, which is commonly expressed in becquerels (Bq), the general formula used is:\[ N_0 = \frac{A}{\lambda} \] where \( N_0 \) is the initial number of nuclei, \( A \) is the activity, and \( \lambda \) is the decay constant. For example, if the activity is \( 7.56 \times 10^{11} \text{ Bq} \) and the decay constant calculated previously was \( 3.75 \times 10^{-4} \text{ s}^{-1} \), the initial number of \( ^{199} \text{Pt} \) nuclei can be expressed as:\[ N_0 = \frac{7.56 \times 10^{11}}{3.75 \times 10^{-4}} \approx 2.02 \times 10^{15} \text{ nuclei} \] Accurately finding \( N_0 \) lays the foundation for further predictions about how many nuclei remain after any given time frame.

Initial Activity Determination

Initial activity refers to the rate at which a radioactive sample undergoes decay at the starting point, often represented by \( A_0 \). This is crucial as it reflects the intensity and strength of the radioactive source when freshly prepared. Activity is measured in becquerels, where one becquerel equals one decay per second.To determine the initial activity of a radioactive sample, you can use the equation relating activity to the decay constant and the number of initial nuclides:\[ A = \lambda N_0 \] This relationship helps ascertain how fast a sample is decaying initially and also serves as a benchmark for understanding future decay rates as time progresses.For \( ^{199} \text{Pt} \) with a high initial activity, \( A_0 = 7.56 \times 10^{11} \text{ Bq} \), this figure will decrease exponentially as the nuclides undergo decay. Knowing the decay constant and initial number of nuclides ensures precise calculations on how the activity changes.