Problem 54 - College Physics

Question

The radioactive source in most smoke detectors is \(241 \mathrm{Am},\) which has a half-life of 432 years. In a typical detector, only about \(0.100 \mathrm{mg}\) of this material is needed. (a) Write down its alpha decay equation and predict the product nucleus. (b) What is the initial source activity? (c) What would be the source's activity after 40 years in operation? (d) How many \({ }^{241} \mathrm{Am},\) nuclei would have decaved in this 40 -vear period?

Step-by-Step Solution

Answer

(a) \(^{237}\text{Np}\); (b) Calculate using \(A_0 = \lambda N\); (c) Use \(A = A_0 e^{-\lambda t}\); (d) \(\Delta N = N_0 - N\).

Step 1: Alpha Decay Equation and Product Nucleus

The alpha decay of \(^{241}\text{Am}\) involves the emission of an alpha particle \((^4 \text{He})\). The atomic number decreases by 2, and the mass number by 4. Thus, \(^{241}_{95}\text{Am} \rightarrow ^{237}_{93}\text{Np} + ^{4}_{2}\text{He}\). The product nucleus is \(^{237}\text{Np}\).

Step 2: Calculate Initial Source Activity

The initial activity \(A_0\) of a radioactive material is calculated by \(A_0 = \lambda N\), where \(\lambda\) is the decay constant and \(N\) is the number of atoms. First, find \(\lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{432} \text{ years}^{-1}\). Convert 0.100 mg of \(^{241}\text{Am}\) to moles and then to atoms using Avogadro's number.

Step 3: Convert Mass to Atoms

The molar mass of \(^{241}\text{Am}\) is 241 g/mol. \(0.100\text{ mg} = 0.0001 \text{g}\). Number of moles = \(\frac{0.0001}{241}\), and number of atoms \(N = \text{Moles} \times 6.022 \times 10^{23}\).

Step 4: Determine Initial Activity

\(A_0 = \lambda N = \left(\frac{0.693}{432}\right) \times N\). Substitute \(N\) from Step 3 to find \(A_0\).

Step 5: Calculate Activity After 40 Years

Use the decay formula \(A = A_0 e^{-\lambda t}\) to find the activity after 40 years. Substitute \(t = 40\) years and the \(\lambda\) calculated in Step 2.

Step 6: Determine Number of Decayed Nuclei

Using the relationship \(\Delta N = N_0 - N = A_0 - A\), where \(N_0\) represents initial and \(N\) current number of atoms. The number of decayed nuclei in 40 years is the difference in the number of atoms calculated using the activities from Steps 4 and 5.

Key Concepts

Alpha Decay

Alpha decay is a common form of radioactive decay where an unstable nucleus emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, essentially a helium nucleus. This process results in the original element transforming into another element with its atomic number decreased by 2 and its mass number decreased by 4.
For example, in the case of Americium-241 (^{241} ext{Am}), an alpha particle is emitted during decay, forming Neptunium-237 (^{237} ext{Np}). The decay equation is thus:\[^{241}_{95} ext{Am} ightarrow ^{237}_{93} ext{Np} + ^{4}_{2} ext{He}\]This equation indicates that Americium sheds an alpha particle and transmutes into Neptunium.

Half-life Calculation

The half-life of a radioactive substance is the time it takes for half of the substance to decay. It's a fundamental property of radioactive materials. Knowing the half-life of a material helps to predict how long it'll remain active and how its activity decreases over time.
The halflife (T_{1/2}) is related to the decay constant (\lambda) through the formula:\[\lambda = \frac{0.693}{T_{1/2}}\]For ^{241} ext{Am}, the half-life is 432 years. This means over each 432-year period, half of the Americium-241 will have decayed into another substance.

Activity of Radioactive Source

The activity of a radioactive source is a measure of how many nuclei within it decay over a certain period. It's often measured in becquerels (Bq), where 1 Bq is equivalent to 1 decay per second. The activity helps determine how powerful or dangerous a radioactive material can be.
To find the initial activity (A_0), you multiply the decay constant (\lambda) by the number of atoms (N):\[A_0 = \lambda N\]For Americium-241 in smoke detectors, after calculating the number of atoms from the mass, you can use the decay constant from its half-life to find the initial activity. Moreover, activities change over time, and the relationship between initial and current activity at time (t) is given by:\[A = A_0 e^{-\lambda t}\]This allows us to calculate the activity of Americium-241 after 40 years, when its level of radioactivity would decrease following exponential decay.

Decay Constant

The decay constant (\lambda) is a measure of how quickly a radioactive isotope will decay. It represents the probability per unit time that a single atom will decay. The greater the decay constant, the faster the radioactive decay occurs.
The relationship between the decay constant and half-life is given by:\[\lambda = \frac{0.693}{T_{1/2}}\]This formula shows that knowledge of the half-life allows you to calculate the decay constant, providing insight into the rate of decay. In practical terms, knowing the decay constant can help with calculations involving the remaining number of radioactive atoms, as well as the activity of the source over time.