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Chapter 19: Problem 32

Americium-241 is widely used in smoke detectors. The radiation released by this element ionizes particles that are then detected by a charged-particle collector. The half-life of \(^{24} \mathrm{Am}\) is 433 years, and it decays by emitting \(\alpha\) particles. How many \(\alpha\) particles are emitted each second by a 5.00 -g sample of \(^{241} \mathrm{Am}\) ?

Short Answer

Expert verified
The number of α particles emitted per second by a 5.00 g sample of \(^{241}Am\) can be determined as follows: 1. Calculate the number of \(^{241}Am\) atoms in the sample: Number of atoms = \( \frac{5.00 \: g}{241 \: g/mol} \) × (6.022 × 10^23 atoms/mol) 2. Determine the decay constant, λ: λ (decays/s) = \( \frac{0.693}{433 \: years} \) × \( \frac{1 \: year}{3.15 \times 10^7 \: s}\) 3. Calculate the decay rate (decays/s): Decay rate (decays/s) = λ × Number of americium-241 atoms 4. Determine the number of α particles emitted per second: Number of α particles emitted each second = Decay rate (decays/s)

Step by step solution

01

1. Determine the number of americium-241 atoms in the sample

First, we need to calculate the number of moles of americium-241 in the given 5.00 g sample. To do this, we will use the molar mass of americium-241, which is 241 g/mol. Moles of \(^{241}Am\) = \( \frac{mass}{molar \: mass} \) = \( \frac{5.00 \: g}{241 \: g/mol} \) Now, we will use Avogadro's number ( 6.022 x 10^23 atoms/mol) to convert moles into the number of americium-241 atoms: Number of atoms = Moles × Avogadro's number = \( \frac{5.00 \: g}{241 \: g/mol} \) × ( 6.022 x 10^23 atoms/mol)
02

2. Calculate the decay constant

Next, we need to find the decay constant, λ (lambda), using the half-life formula: Half-life (t1/2) = \( \frac{0.693}{λ} \) Solve for λ: λ = \( \frac{0.693}{t_{1/2}} \) = \( \frac{0.693}{433 \: years} \) Now, we need to convert λ to decays per second: λ (decays/s) = \( λ \frac{1 \: year}{3.15 \times 10^7 \: s}\)
03

3. Determine the number of decays per second by americium-241

Now that we have the decay constant, we can use it to calculate the decay rate of the sample. The decay rate is the product of the decay constant and the number of \(^{241}Am\) atoms. Decay rate (decays/s) = λ × Number of americium-241 atoms
04

4. Calculate the number of α particles emitted per second

Since americium-241 decays by emitting α particles, the number of α particles emitted per second will be equal to the decay rate per second found in step 3. Number of α particles emitted each second = Decay rate (decays/s)

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