Which of the following has magic number of protons and neutrons?

Options

A. a. `._(82)Pb^(208)`
B. b. `._(2)He^(3)`
C. c. `._(50)Sn^(120)`
D. d. `._(82)Pb^(206)`

Explanation

To determine which of the given elements has a magic number of protons and neutrons, we need to follow these steps:

### Step 1: Understand Magic Numbers
Magic numbers are specific numbers of protons or neutrons that result in a more stable nucleus. The known magic numbers for protons and neutrons are:
- Protons: 2, 8, 20, 28, 50, 82
- Neutrons: 2, 8, 20, 28, 50, 82, 126

### Step 2: Analyze Each Element
We will analyze the given elements one by one to check their number of protons and neutrons.

1. **Lead (Pb) - 82, 208**
- Protons (Z) = 82 (Atomic number)
- Neutrons (N) = Mass number (A) - Atomic number (Z) = 208 - 82 = 126
- Magic Numbers: Protons = 82 (magic), Neutrons = 126 (magic)

2. **Helium (He) - 2, 3**
- Protons (Z) = 2 (Atomic number)
- Neutrons (N) = Mass number (A) - Atomic number (Z) = 3 - 2 = 1
- Magic Numbers: Protons = 2 (magic), Neutrons = 1 (not magic)

3. **Tin (Sn) - 50, 120**
- Protons (Z) = 50 (Atomic number)
- Neutrons (N) = Mass number (A) - Atomic number (Z) = 120 - 50 = 70
- Magic Numbers: Protons = 50 (magic), Neutrons = 70 (not magic)

4. **Lead (Pb) - 82, 206**
- Protons (Z) = 82 (Atomic number)
- Neutrons (N) = Mass number (A) - Atomic number (Z) = 206 - 82 = 124
- Magic Numbers: Protons = 82 (magic), Neutrons = 124 (not magic)

### Step 3: Conclusion
From the analysis:
- The first lead isotope (82, 208) has both magic numbers for protons (82) and neutrons (126).
- The other elements do not have both protons and neutrons as magic numbers.

Thus, the element that has magic numbers of protons and neutrons is **Lead (82, 208)**.

### Final Answer
The answer is **Option A: Lead (82, 208)**.

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