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Chapter 16: Problem 124

Given four nuclei: \({ }_{10}^{20} \mathrm{~A},{ }_{9}^{17} \mathrm{~B},{ }_{12}^{28} \mathrm{C}\), and \({ }_{86}^{226} \mathrm{D}\), which is likely to emit a \(\beta^{-}\) particle, and which is likely to emit a \(\beta^{+}\) particle? Explain your choices.

Short Answer

Expert verified
Nucleus C is the most likely to emit a β- particle, as it has the highest neutron-to-proton ratio (N/Z > 1), suggesting more neutrons relative to protons. Either nucleus B or D could potentially emit a β+ particle, as both have N/Z ratios below 1, indicating more protons relative to neutrons. However, it's not possible to determine which one is more likely to emit a β+ particle based solely on N/Z ratios.

Step by step solution

01

Calculate the neutron-to-proton ratios of each nucleus

To calculate the neutron-to-proton (N/Z) ratio for each nucleus, divide the number of neutrons by the number of protons: 1. For nucleus A: \( N/Z_A = \frac{20-10}{10} = 1 \) 2. For nucleus B: \( N/Z_B = \frac{17-9}{9} = \frac{8}{9} \) 3. For nucleus C: \( N/Z_C = \frac{28-12}{12} = \frac{4}{3} \) 4. For nucleus D: \( N/Z_D = \frac{226-86}{86} = \frac{35}{43} \)
02

Compare the N/Z ratios

We can compare the N/Z ratios to determine which nuclei are likely to undergo beta decay: 1. Nucleus A has N/Z = 1, which generally indicates stability 2. Nucleus B has N/Z < 1, suggesting it has more protons relative to neutrons, which increases the likelihood of β+ decay 3. Nucleus C has N/Z > 1, suggesting it has more neutrons relative to protons, which increases the likelihood of β- decay 4. Nucleus D has N/Z < 1, suggesting it has more protons relative to neutrons, so it's also a candidate for β+ decay
03

Determine the most likely candidates for beta decay

Based on the N/Z ratios and the rules for beta decay, we identify the most likely candidates for beta-minus and beta-plus decay: - Nucleus C has the highest N/Z ratio, suggesting it's the most likely candidate to undergo β- decay - Nucleus B and D both have N/Z ratios below 1 so could be candidates for β+ decay, although it's not possible to select which one based solely on N/Z ratios as other factors come into play In conclusion, nucleus C is the most likely to emit a β- particle, and either nucleus B or D could potentially emit a β+ particle. However, we cannot determine which of these two is more likely to emit a β+ particle based solely on the neutron-to-proton ratios.

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