Which of the following sets of quantum numbers represents an impossible arrangement?

Options

A. `{:(n,l,m,s),(4,3,-3,1//2):}`
B. `{:(n,l,m,s),(4,1,-1,1//2):}`
C. `{:(n,l,m,s),(3,1,-2,1//2):}`
D. `{:(n,l,m,s),(4,3,0,1//2):}`

Answer

To determine which set of quantum numbers represents an impossible arrangement, we need to understand the rules governing the quantum numbers:

1. **Principal Quantum Number (n)**: This number can take positive integer values (1, 2, 3, ...). It indicates the energy level of the electron.

2. **Azimuthal Quantum Number (l)**: This number can take values from 0 to (n-1). It represents the subshell (s, p, d, f).

3. **Magnetic Quantum Number (m)**: This number can take values from -l to +l, including zero. It indicates the orientation of the orbital.

4. **Spin Quantum Number (s)**: This number can either be +1/2 or -1/2, representing the two possible spin states of an electron.

Now, let's analyze the options given in the question step by step:

### Step 1: Analyze the first option
Assuming the first option is (n=4, l=3, m=2, s=+1/2):
- n = 4 (valid)
- l = 3 (valid since l can be 0 to n-1, which is 0 to 3)
- m = 2 (valid since m can be -3, -2, -1, 0, +1, +2, +3 for l=3)
- s = +1/2 (valid)

This set is possible.

### Step 2: Analyze the second option
Assuming the second option is (n=3, l=2, m=1, s=-1/2):
- n = 3 (valid)
- l = 2 (valid since l can be 0 to n-1, which is 0 to 2)
- m = 1 (valid since m can be -2, -1, 0, +1, +2 for l=2)
- s = -1/2 (valid)

This set is possible.

### Step 3: Analyze the third option
Assuming the third option is (n=4, l=1, m=2, s=+1/2):
- n = 4 (valid)
- l = 1 (valid since l can be 0 to n-1, which is 0 to 3)
- m = 2 (invalid since m can only take values from -1 to +1 for l=1)
- s = +1/2 (valid)

This set is **impossible** due to the invalid value of m.

### Step 4: Analyze the fourth option
Assuming the fourth option is (n=2, l=0, m=0, s=-1/2):
- n = 2 (valid)
- l = 0 (valid since l can be 0 to n-1, which is 0 to 1)
- m = 0 (valid since m can only be 0 for l=0)
- s = -1/2 (valid)

This set is possible.

### Conclusion
The impossible arrangement of quantum numbers is the third option (n=4, l=1, m=2, s=+1/2).