Problem 45 - Fundamentals of Physics

Question

In \(1992,\) Swiss police arrested two men who were attempting to smuggle osmium out of Eastern Europe for a clandestine sale. However, by error, the smugglers had picked up \({ }^{137} \mathrm{Cs}\). Reportedly, each smuggler was carrying a \(1.0 \mathrm{~g}\) sample of \({ }^{137} \mathrm{Cs}\) in a pocket! In (a) bequerels and (b) curies, what was the activity of each sample? The isotope \({ }^{137} \mathrm{Cs}\) has a half-life of \(30.2 \mathrm{y}\). (The activities of radioisotopes commonly used in hospitals range up to a few millicuries.)

Step-by-Step Solution

Answer

The activity of each \(1.0 \mathrm{~g}\) \(^{137} \mathrm{Cs}\) sample is approximately 3.20 trillion becquerels or 86.5 millicuries.

Step 1: Convert Half-life from Years to Seconds

The half-life of \({ }^{137} \mathrm{Cs}\) is given as 30.2 years. First, we need to convert this time period into seconds to use in activity calculations: \\[\text{1 year} = 365 \text{ days} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 31,536,000 \text{ seconds}\] \Now calculate half-life in seconds: \\[t_{1/2} = 30.2 \times 31,536,000 = 9.52 \times 10^8 \text{ seconds}\]

Step 2: Calculate Decay Constant (λ)

Using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\), find the decay constant. \\[\lambda = \frac{0.693}{9.52 \times 10^8} \approx 7.28 \times 10^{-10} \text{ s}^{-1}\]

Step 3: Determine the Number of Atoms in the Sample

To find the number of atoms in \(1.0 \text{ g}\) of \(^{137}\mathrm{Cs}\), calculate as follows: \The molar mass of \(^{137}\mathrm{Cs}\) is approximately 137 g/mol. Hence, \\[\text{Number of moles} = \frac{1.0 \text{ g}}{137 \text{ g/mol}} = 7.3 \times 10^{-3} \text{ moles}\] \Using Avogadro's number (\(6.022 \times 10^{23} \text{ atoms/mol}\)), calculate: \\[N = 7.3 \times 10^{-3} \times 6.022 \times 10^{23} = 4.40 \times 10^{21} \text{ atoms}\]

Step 4: Calculate Activity in Becquerels

Activity \(A\) in becquerels is given by \(A = \lambda N\). \\[A = 7.28 \times 10^{-10} \times 4.40 \times 10^{21} \A \approx 3.20 \times 10^{12} \text{ Becquerels}\]

Step 5: Convert Activity to Curies

To convert from becquerels to curies: \(1 \text{ Curie} = 3.7 \times 10^{10} \text{ Bq}\). \\[A = \frac{3.20 \times 10^{12} \text{ Bq}}{3.7 \times 10^{10} \text{ Bq/Curie}} \A \approx 86.5 \text{ mCi} \\] \Thus, the activity of each sample is approximately 86.5 millicuries.

Key Concepts

Half-Life Calculation

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. A key concept in understanding radioactive decay is the **half-life**, which is the time required for half of the radioactive atoms in a sample to decay. It's a constant value specific to each radioactive isotope.
To calculate the half-life of a substance in different time units, it's essential to understand conversion factors. For instance, the given half-life of \(^{137}\text{Cs}\) is 30.2 years. If you want to convert this into seconds, use:

• 1 year = 31,536,000 seconds
• Therefore, \(t_{1/2} = 30.2 \times 31,536,000 = 9.52 \times 10^8\) seconds

Learning to perform these conversions is crucial for solving many radioactive decay problems. This allows you to apply the half-life to different operational contexts, such as medical applications or environmental studies.

Becquerel and Curie Conversion

In radioactive measurements, the **activity** of a radioactive sample is often expressed in two units: becquerels (Bq) and curies (Ci). These units indicate how many atoms in the sample decay in a given period.

• 1 Becquerel (Bq) is defined as one decay per second.
• 1 Curie (Ci) is equivalent to \(3.7 \times 10^{10}\) becquerels.

Knowing how to convert between these two units is essential when comparing activities.
For instance, if we start with the activity in becquerels, calculated as \(3.20 \times 10^{12}\) Bq for a \(1\, \text{g}\) sample of \(^{137}\text{Cs}\), converting to curies gives:\[A = \frac{3.20 \times 10^{12}\, \text{Bq}}{3.7 \times 10^{10}\, \text{Bq/Ci}} \approx 86.5\, \text{mCi}\]Understanding this conversion helps you to interpret and communicate radioactive levels in a universally accepted way. This is particularly useful in fields like nuclear medicine, where precise measurements are vital for patient safety and treatment efficacy.

Decay Constant

The **decay constant** \(\lambda\) is a probability measure of how quickly a radioactive substance undergoes decay. It's directly related to the half-life and provides a mathematical approach to calculate the activity of a radioactive sample.
The formula is \(\lambda = \frac{\ln(2)}{t_{1/2}}\). This equation arises from the exponential nature of radioactive decay. Here, \(\ln(2) \approx 0.693\) is a constant since radioactive decay follows a first-order kinetic process. This means the process speed is proportional to the number of undecayed nuclei.
For \(^{137}\text{Cs}\), where \(t_{1/2} = 9.52 \times 10^8\) seconds:\[\lambda = \frac{0.693}{9.52 \times 10^8} \approx 7.28 \times 10^{-10} \text{ s}^{-1}\]Having the decay constant allows us to calculate the activity \(A\) of the sample using the formula \(A = \lambda N\), where \(N\) is the number of undecayed atoms. This concept is fundamental in calculations involving decay rates and in predicting how long a substance will remain hazardous.