Which of the following sets of quantum number is allowable

Options

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

Answer

To determine which set of quantum numbers is allowable, we need to analyze each quantum number based on the rules governing them. Quantum numbers include the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m), and spin quantum number (s).

### Step-by-step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number (n) can take any positive integer value (1, 2, 3,...).
- In the given options, we have n = 2. This is valid for all options.

2. **Determine the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number (l) can take integer values from 0 to (n-1).
- Since n = 2, l can be 0 or 1 (i.e., l = 0, 1).
- Check each option for l:
- If l = 1: This is valid.
- If l = 2: This is invalid (as it exceeds n-1).
- If l = -2: This is invalid (l cannot be negative).
- If l = 0: This is valid.

3. **Evaluate the Magnetic Quantum Number (m)**:
- The magnetic quantum number (m) can take values from -l to +l.
- For l = 1, m can be -1, 0, or +1.
- Check each option for m:
- If m = 0: This is valid.
- If m = 1: This is valid.
- If m = -1: This is valid.
- Any value outside this range is invalid.

4. **Assess the Spin Quantum Number (s)**:
- The spin quantum number (s) can only be +1/2 or -1/2.
- Check each option for s:
- If s = +1/2: This is valid.
- If s = -1/2: This is valid.
- If s = 0: This is invalid.

5. **Conclusion**:
- After analyzing all quantum numbers, we can conclude that the only valid set of quantum numbers is the one that adheres to the rules outlined above. Therefore, the correct option is the one that has:
- n = 2
- l = 0 or 1
- m = -1, 0, or +1
- s = +1/2 or -1/2

### Final Answer:
The correct option is option number 1 (assuming it meets the criteria above).