Which sets of quantum number are consitent with the theory ?

Options

A. `n = 2,l = 1,m = 0,s = -1//2`
B. `n = 4,l = 3,m = -2,s = -1//2`
C. `n = 3,l = 2,m = -3,s = +1//2`
D. `n = 4,l = 3,m = -3,s = +1//2`

Explanation

To determine which sets of quantum numbers are consistent with the quantum mechanical model of the atom, we need to analyze each set of quantum numbers based on the rules governing their values. 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 integer values from 0 to (n-1).
3. **Magnetic Quantum Number (m)**: This can take integer values from -l to +l, including zero.
4. **Spin Quantum Number (s)**: This can take values of +1/2 or -1/2.

Now, let's analyze each option provided in the question:

### Step-by-Step Analysis

**Option A: (n = 2, l = 1, m = 0, s = -1/2)**
1. **n = 2**: Valid, as n can be any positive integer.
2. **l = 1**: Valid, as l can be 0 or 1 for n = 2.
3. **m = 0**: Valid, as m can range from -l to +l (here, -1 to +1).
4. **s = -1/2**: Valid, as s can be either +1/2 or -1/2.

**Conclusion**: Option A is correct.

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**Option B: (n = 4, l = 3, m = -2, s = -1/2)**
1. **n = 4**: Valid.
2. **l = 3**: Valid, as l can be 0, 1, 2, or 3 for n = 4.
3. **m = -2**: Valid, as m can range from -3 to +3.
4. **s = -1/2**: Valid.

**Conclusion**: Option B is correct.

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**Option C: (n = 3, l = 2, m = -3, s = -1/2)**
1. **n = 3**: Valid.
2. **l = 2**: Valid, as l can be 0, 1, or 2 for n = 3.
3. **m = -3**: Invalid, as m can only range from -2 to +2 (since l = 2).
4. **s = -1/2**: Valid.

**Conclusion**: Option C is incorrect.

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**Option D: (n = 4, l = 3, m = -3, s = +1/2)**
1. **n = 4**: Valid.
2. **l = 3**: Valid.
3. **m = -3**: Valid, as m can range from -3 to +3.
4. **s = +1/2**: Valid.

**Conclusion**: Option D is correct.

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### Final Summary
The sets of quantum numbers that are consistent with the theory are:
- Option A
- Option B
- Option D

**Incorrect Option**: Option C