Maximum number of electrons in `n^(th)` shell is

Options

A. Option 1. `n^(2)`
B. Option 2. 2(l+1)
C. Option 3. `2n^(2)`
D. Option 4. `(l + 1)^(2)`

Answer

To determine the maximum number of electrons that can be accommodated in the \( n^{th} \) shell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Electron Shells**:
- Each electron shell can hold a specific maximum number of electrons. The shells are designated by the principal quantum number \( n \).

2. **Identifying the Pattern**:
- The first shell (\( n = 1 \)) can hold 2 electrons.
- The second shell (\( n = 2 \)) can hold 8 electrons.
- The third shell (\( n = 3 \)) can hold 18 electrons.
- The fourth shell (\( n = 4 \)) can hold 32 electrons.

3. **Finding the General Formula**:
- From the observations, we can see a pattern in the maximum number of electrons that each shell can hold. The maximum number of electrons in the \( n^{th} \) shell can be derived from the formula:
\[ \text{Maximum electrons} = 2n^2 \]
- Here, \( n \) represents the principal quantum number (the shell number).

4. **Conclusion**:
- Therefore, the maximum number of electrons in the \( n^{th} \) shell is given by the formula \( 2n^2 \).

### Final Answer:
The maximum number of electrons in the \( n^{th} \) shell is \( 2n^2 \).

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