Calculate the neutron - proton ratio for each of the following radioactive nuclides.
a. `._(6)C^(14)` b. `._(82)U^(238)` c. `._(82)U^(232)`

Explanation

To calculate the neutron-proton ratio for the given radioactive nuclides, we will follow these steps:

### Step 1: Understand the notation
The notation for a nuclide is represented as \( _{Z}^{A}X \), where:
- \( A \) = mass number (total number of protons and neutrons)
- \( Z \) = atomic number (number of protons)
- \( X \) = chemical symbol of the element

### Step 2: Calculate for each nuclide

#### a. For \( _{6}^{14}C \)
1. Identify the mass number \( A = 14 \) and atomic number \( Z = 6 \).
2. Calculate the number of protons (which is \( Z \)):
- Protons = 6
3. Calculate the number of neutrons using the formula:
- Neutrons = Mass number - Protons = \( 14 - 6 = 8 \)
4. Calculate the neutron-proton ratio:
- Neutron-Proton Ratio = Neutrons / Protons = \( 8 / 6 = 1.33 \)

#### b. For \( _{82}^{238}U \)
1. Identify the mass number \( A = 238 \) and atomic number \( Z = 82 \).
2. Calculate the number of protons:
- Protons = 82
3. Calculate the number of neutrons:
- Neutrons = Mass number - Protons = \( 238 - 82 = 156 \)
4. Calculate the neutron-proton ratio:
- Neutron-Proton Ratio = Neutrons / Protons = \( 156 / 82 \approx 1.902 \)

#### c. For \( _{82}^{232}U \)
1. Identify the mass number \( A = 232 \) and atomic number \( Z = 82 \).
2. Calculate the number of protons:
- Protons = 82
3. Calculate the number of neutrons:
- Neutrons = Mass number - Protons = \( 232 - 82 = 150 \)
4. Calculate the neutron-proton ratio:
- Neutron-Proton Ratio = Neutrons / Protons = \( 150 / 82 \approx 1.829 \)

### Final Results:
- For \( _{6}^{14}C \): Neutron-Proton Ratio = 1.33
- For \( _{82}^{238}U \): Neutron-Proton Ratio ≈ 1.902
- For \( _{82}^{232}U \): Neutron-Proton Ratio ≈ 1.829

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