Problem 48 - Physics for Scientists and Engineers with Modern Physics

Question

(II) \({ }_{55}^{124} \mathrm{Cs}\) has a half-life of \(30.8 \mathrm{~s}\). \((a)\) If we have \(7.8 \mu \mathrm{g}\) initially, how many Cs nuclei are present? (b) How many are present 2.6 min later? \((c)\) What is the activity at this time? \((d)\) After how much time will the activity drop to less than about 1 per second?

Step-by-Step Solution

Answer

(a) \(3.78 \times 10^{16}\); (b) \(3.61 \times 10^{14}\); (c) \(8.13 \times 10^{12}\) decays/s; (d) 360 s.

Step 1: Calculate Initial Number of Cs Nuclei

To find the initial number of Cs nuclei, we use the formula \( N = \frac{m}{M} \times N_A \), where \( m \) is the mass, \( M \) is the molar mass of \( {}_{55}^{124} \mathrm{Cs} \) (which is 124 g/mol), and \( N_A \) is Avogadro's number (\( 6.022 \times 10^{23} \text{mol}^{-1} \)). First, convert the mass from \( \mu \mathrm{g} \) to g: \( 7.8 \mu \mathrm{g} = 7.8 \times 10^{-6} \mathrm{g} \). Then, calculate \( N \):\[N = \frac{7.8 \times 10^{-6}}{124} \times 6.022 \times 10^{23} \approx 3.78 \times 10^{16} \text{ Cs nuclei}\]

Step 2: Calculate Number of Nuclei after 2.6 min

First, convert 2.6 minutes to seconds: \( 2.6 \times 60 = 156 \) seconds. Use the decay formula \( N(t) = N_0 \times e^{-\lambda t} \), where \( \lambda = \frac{\ln(2)}{t_{1/2}} \) is the decay constant. Calculate \( \lambda \):\[\lambda = \frac{\ln(2)}{30.8} \approx 0.0225 \text{s}^{-1}\]Now calculate \( N(t) \):\[N(156) = 3.78 \times 10^{16} \times e^{-0.0225 \times 156} \approx 3.61 \times 10^{14} \text{ Cs nuclei}\]

Step 3: Calculate the Activity at 2.6 min

The activity \( A(t) \) of a radioactive sample is calculated as \( A(t) = \lambda N(t) \). Using \( N(156) \) from the previous step:\[A(156) = 0.0225 \times 3.61 \times 10^{14} \approx 8.13 \times 10^{12} \text{ decays/second}\]

Step 4: Calculate Time for Activity to Drop Below 1 decay/second

To find the time when activity drops below 1 decay/second, solve for \( t \) using \( A(t) = \lambda N_0 e^{-\lambda t} < 1 \). Rearranging gives \( t > \frac{\ln(\lambda N_0)}{\lambda} \). Substituting:\[t > \frac{\ln(0.0225 \times 3.78 \times 10^{16})}{0.0225}\]Calculate the value:\[t > \frac{\ln(8.51 \times 10^{14})}{0.0225} \approx 360 \text{ seconds}\]

Step 5: Final Calculations and Results

Based on the calculations:- (a) The initial number of Cs nuclei is approximately \( 3.78 \times 10^{16} \).- (b) After 2.6 minutes, approximately \( 3.61 \times 10^{14} \) nuclei are left.- (c) The activity at this time is approximately \( 8.13 \times 10^{12} \) decays/second.- (d) The activity drops to less than 1 decay per second after approximately 360 seconds.

Key Concepts

half-life

In the study of radioactive decay, the concept of half-life is crucial. The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei in a sample to decay. This is a probabilistic measure and varies based on the type of substance. For example, in the case of extsuperscript{124}Cs, the half-life is given as 30.8 seconds.

• Half-life helps indicate the rate at which a radioactive substance will diminish.
• This value can be used to determine how long a particular isotope remains active or poses a risk.

The half-life remains constant irrespective of the size of the sample or external conditions. This consistency helps scientists predict how fast a substance will decay. Whenever you're calculating the remaining amount of a radioactive isotope after a certain period, the half-life is the primary parameter to consider. The formula for half-life in calculations often involves determining how many half-lives occur over a given time span.

decay constant

The decay constant, often represented by the Greek letter \(\lambda\), is a crucial factor in understanding radioactive decay. It connects directly to the half-life of a substance and signifies the probability per unit time that a given nucleus will decay. For extsuperscript{124}Cs, the decay constant is calculated using the formula: \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) represents the half-life.

• The decay constant provides an exponential model of decay, highlighting how populations of radioactive nuclei decrease.
• A larger decay constant indicates a faster rate of decay.

In practical terms, the decay constant helps in calculating the number of remaining nuclei at any given time using the exponential decay formula: \( N(t) = N_0 \times e^{-\lambda t} \). Understanding this helps us comprehend the nature of exponential decay which is characteristic of radioactive substances.

activity calculation

Activity in radioactive decay refers to the rate at which a sample of radioactive material undergoes decay, measured in decays per second, often called becquerels (Bq). The activity \( A(t) \) at a given time is computed by the formula \( A(t) = \lambda N(t) \), where \( \lambda \) is the decay constant and \( N(t) \) is the number of undecayed nuclei at time \( t \).

• Activity indicates the intensity of a radioactive source at a particular time.
• This measure helps in assessing safety, determining dosage in medical applications or evaluating the life span of a radioactive material.

For instance, if the number of remaining extsuperscript{124}Cs nuclei is known, one can quickly calculate the current activity to understand how active or potent the material is. Over time, the activity decreases, and this rate of decrease is directly related to both the decay constant and the remaining number of nuclei.