In a hypothetical atom, the number of electrons in shell and subshell are x and y, respectively, where each electron has three possible spin states represented as `s = +1//2, -1//2 and 0`. Then the value of x + y is

Options

A. `2n^2 +4l + 2`
B. `3n^2+3l+1`
C. `3n^2 +6l + 3`
D. `6n^2 +3l+3`

Answer

To solve the problem, we need to determine the values of \( x \) (the number of electrons in shells) and \( y \) (the number of electrons in subshells) in a hypothetical atom, where each electron has three possible spin states.

### Step-by-Step Solution:

1. **Understanding the Shells**:
- The number of electrons in a shell is given by the formula \( x = 3n^2 \), where \( n \) is the principal quantum number. This accounts for the three possible spin states of each electron.

2. **Understanding the Subshells**:
- The number of electrons in a subshell is given by the formula \( y = 3(2l + 1) \), where \( l \) is the azimuthal quantum number. The term \( 2l + 1 \) represents the number of orbitals in a subshell, and since each orbital can hold three spin states, we multiply by 3.

3. **Combining the Values**:
- Now, we need to find \( x + y \):
\[ x + y = 3n^2 + 3(2l + 1) \]

4. **Simplifying the Expression**:
- Factor out the common term:
\[ x + y = 3(n^2 + 2l + 1) \]

5. **Conclusion**:
- The expression \( x + y \) is dependent on the values of \( n \) and \( l \). However, since the problem does not specify particular values for \( n \) and \( l \), we can conclude that \( x + y \) can be expressed in the form \( 3(n^2 + 2l + 1) \).

### Final Answer:
The value of \( x + y \) is \( 3(n^2 + 2l + 1) \).

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