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

Options

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

Answer

To determine which set of four quantum numbers corresponds to an electron with the highest energy, we will use the n + l rule. This rule states that the energy of an electron in an atom can be estimated by the sum of its principal quantum number (n) and its azimuthal quantum number (l). The higher the value of n + l, the higher the energy of the electron. If two sets have the same n + l value, the one with the higher n value has the higher energy.

### Step-by-Step Solution:

1. **Identify the Quantum Numbers**:
We need to look at the four quantum numbers provided for each option. The quantum numbers are:
- Principal quantum number (n)
- Azimuthal quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

2. **Calculate n + l for Each Option**:
For each set of quantum numbers, calculate the sum of n and l.
- **Option 1**: (n=3, l=2) → n + l = 3 + 2 = 5
- **Option 2**: (n=4, l=2) → n + l = 4 + 2 = 6
- **Option 3**: (n=4, l=1) → n + l = 4 + 1 = 5
- **Option 4**: (n=5, l=0) → n + l = 5 + 0 = 5

3. **Compare n + l Values**:
Now we compare the n + l values calculated:
- Option 1: 5
- Option 2: 6
- Option 3: 5
- Option 4: 5

The highest n + l value is 6 from Option 2.

4. **Determine the Highest Energy**:
Since Option 2 has the highest n + l value (6), it corresponds to the electron with the highest energy.

### Conclusion:
The set of quantum numbers that corresponds to the electron with the highest energy is **Option 2**.