Problem 29 - College Physics

Question

Francium- \(223\left(\begin{array}{c}223 \\ 87\end{array} \mathrm{Fr}\right)\) has a half-life of \(21.8 \mathrm{~min}\). (a) How many nuclei are initially present in a \(25.0-\mathrm{mg}\) sample of \({ }^{223} \mathrm{Fr} ?\) (b) What is its initial activity? (c) How many nuclei will be present \(1 \mathrm{~h}\) and 49 min later? (d) What will be the sample's activity at this later time?

Step-by-Step Solution

Answer

(a) Approximately 6.74 x 10^{19} nuclei. (b) Initial activity: 2.14 x 10^{18} decays/min. (c) After 1 h 49 min: 5.91 x 10^{18} nuclei. (d) Activity at this time: 1.88 x 10^{17} decays/min.

Step 1: Determine the number of Francium-223 atoms in a 25 mg sample

To find the number of nuclei in the sample, we need to convert mass to moles and then to atoms. The molar mass of Francium-223 is approximately 223 g/mol. Calculate the number of moles:\[ \text{Number of moles} = \frac{25 \text{ mg}}{223 \text{ g/mol}} = \frac{0.025 \text{ g}}{223 \text{ g/mol}} \approx 1.12 \times 10^{-4} \text{ moles} \]Next, convert moles to atoms:\[ \text{Number of atoms} = 1.12 \times 10^{-4} \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mole} \approx 6.74 \times 10^{19} \text{ atoms} \]

Step 2: Calculate the initial activity of the sample

Activity, \(A_0\), is given by \(A_0 = \lambda N_0\), where \(\lambda\) is the decay constant and \(N_0\) is the initial number of nuclei. The decay constant can be calculated using the half-life:\[ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{21.8 \text{ min}} = 0.0318 \text{ min}^{-1} \]Therefore, the initial activity is:\[ A_0 = 0.0318 \text{ min}^{-1} \times 6.74 \times 10^{19} \text{ atoms} \approx 2.14 \times 10^{18} \text{ decays/min} \]

Step 3: Determine the remaining nuclei after 1 hour and 49 minutes

First, convert the time to minutes: \(1 \text{ h } 49 \text{ min} = 109 \text{ min}\). Use the decay formula, \(N = N_0 e^{-\lambda t}\), to find the number of remaining nuclei:\[ N = 6.74 \times 10^{19} \times e^{-0.0318 \times 109} \approx 5.91 \times 10^{18} \text{ nuclei} \]

Step 4: Calculate the activity after 1 hour and 49 minutes

The activity \(A\) at time \(t\) is given by \(A = \lambda N\). So, we use the number of nuclei from Step 3:\[ A = 0.0318 \times 5.91 \times 10^{18} \approx 1.88 \times 10^{17} \text{ decays/min} \]

Key Concepts

Half-Life

Half-life is a concept in nuclear physics that describes the time it takes for half of a given amount of a radioactive substance to decay. It is a measure of how quickly or slowly a radioactive element undergoes decay. - The half-life of Francium-223 is 21.8 minutes, which means that every 21.8 minutes, half of the Francium-223 atoms will have decayed into another element. - This property is crucial in calculating the remaining nuclei after a certain time period, as seen in the original exercise. Understanding half-life helps in determining both the activity and remaining quantity of radioactive substances over time. Shorter half-lives indicate a faster rate of decay.

Activity

The activity of a radioactive sample refers to the number of decays per unit time, and it is measured in decays per minute or per second. - Activity can be thought of as a measure of how "active" or "radioactive" a sample is at any given moment. - The initial activity of the Francium-223 sample was calculated using the formula: \[ A_0 = \lambda N_0 \] - Here, \( \lambda \) is the decay constant, and \( N_0 \) is the initial number of nuclei. This equation shows that activity is directly proportional to both the decay constant and the number of undecayed nuclei.Knowing the activity is important for applications that require monitoring radiation levels.

Nuclear Physics

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. It explores phenomena such as radioactivity, which includes concepts like half-life and decay. - It provides the theoretical background required to understand radioactive processes, as explored in the original exercise. - The key focus is on how unstable nuclei transform into stable ones through decay processes. The practical applications range from medical treatments, like radiotherapy, to energy production in nuclear reactors. Understanding nuclear physics is essential for dealing with radioactive materials safely and effectively.

Decay Constant

The decay constant is a probability per unit time that a given nucleus will decay. It is a crucial factor in determining the rate of radioactive decay of a substance.- It is related to half-life via the formula: \[ \lambda = \frac{\ln(2)}{T_{1/2}} \] - This relationship highlights that shorter half-life corresponds to a higher decay constant.- In the example of Francium-223, the decay constant was calculated as \(0.0318\, \text{min}^{-1} \), indicating a relatively quick decay process.The decay constant is an essential parameter for calculating the remaining number of nuclei and the activity of a sample over time.