Which of the following arrangement is possible ?

Options

A. `{:(n, l, m, s),(5,2,2,+(1)/(2)):}`
B. `{:(n, l, m, s),(2,2,0,-(1)/(2)):}`
C. `{:(n, l, m, s),(3,-2,1,+(1)/(2)):}`
D. `{:(n, l, m, s),(0,0,1,+(1)/(2)):}`

Explanation

To solve the question regarding which arrangement of quantum numbers is possible, we need to analyze the given sets of quantum numbers based on the rules governing them. The quantum numbers are:

1. **Principal Quantum Number (n)**: This can take any positive integer value (n = 1, 2, 3,...).
2. **Azimuthal Quantum Number (l)**: This can take values from 0 to (n-1).
3. **Magnetic Quantum Number (m)**: This can take values from -l to +l.
4. **Spin Quantum Number (s)**: This can take values of +1/2 or -1/2.

Now, let's analyze each arrangement provided in the options.

### Step-by-Step Solution:

1. **Option 1: n = 5, l = 2, m = 2, s = +1/2**
- For n = 5, possible values for l are 0, 1, 2, 3, 4 (since l can be from 0 to n-1).
- l = 2 is valid.
- For l = 2, possible values for m are -2, -1, 0, +1, +2 (since m can be from -l to +l).
- m = 2 is valid.
- s = +1/2 is valid.
- **Conclusion**: This arrangement is possible.

2. **Option 2: n = 2, l = 2, m = 0, s = -1/2**
- For n = 2, possible values for l are 0, 1 (since l can be from 0 to n-1).
- l = 2 is not valid (it exceeds the maximum value of n-1).
- **Conclusion**: This arrangement is not possible.

3. **Option 3: n = 3, l = -2, m = 0, s = -1/2**
- For n = 3, possible values for l are 0, 1, 2.
- l = -2 is not valid (l cannot be negative).
- **Conclusion**: This arrangement is not possible.

4. **Option 4: n = 0, l = 0, m = 0, s = -1/2**
- The principal quantum number n cannot be 0 (it must be a positive integer).
- **Conclusion**: This arrangement is not possible.

### Final Answer:
The only possible arrangement is **Option 1: n = 5, l = 2, m = 2, s = +1/2**.

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