For which of the following sets of quantum numbers, an electrons will have the highest energy ?

Options

A. `{:(n,l,m,s),(3,2,1,-1//2):}`
B. `{:(n,l,m,s),(4,3,-1,+1//2):}`
C. `{:(n,l,m,s),(4,1,-1,+1//2):}`
D. `{:(n,l,m,s),(5,0,0,-1//2):}`

Answer

To determine which set of quantum numbers corresponds to the electron with the highest energy, we follow these steps:

### Step 1: Understand the Quantum Numbers
The quantum numbers are:
- **n**: Principal quantum number, indicates the energy level or shell.
- **l**: Azimuthal quantum number, indicates the subshell (0 for s, 1 for p, 2 for d, 3 for f).
- **m**: Magnetic quantum number, indicates the orientation of the orbital.
- **s**: Spin quantum number, indicates the spin of the electron (+1/2 or -1/2).

### Step 2: Calculate n + l for Each Set
To find the energy of the electron, we calculate the sum of n and l (n + l) for each set of quantum numbers. The higher the n + l value, the higher the energy.

1. **Set 1**: (n=3, l=2)
- n + l = 3 + 2 = **5** (3d orbital)

2. **Set 2**: (n=4, l=3)
- n + l = 4 + 3 = **7** (4f orbital)

3. **Set 3**: (n=4, l=1)
- n + l = 4 + 1 = **5** (4p orbital)

4. **Set 4**: (n=5, l=0)
- n + l = 5 + 0 = **5** (5s orbital)

### Step 3: Compare n + l Values
Now we compare the n + l values calculated:
- Set 1: 5
- Set 2: 7
- Set 3: 5
- Set 4: 5

The highest n + l value is **7** from **Set 2** (4, 3, -1, +1/2).

### Conclusion
The set of quantum numbers that corresponds to the electron with the highest energy is **Set 2: (4, 3, -1, +1/2)**.

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