Select correct set of quantum numbers of electron have lowest energy

Answer

To determine the correct set of quantum numbers for the electron with the lowest energy, we will apply the n + l rule. Here’s the step-by-step solution:

### Step 1: Understand the Quantum Numbers
Quantum numbers are represented as:
- **n**: Principal quantum number (indicates the energy level)
- **l**: Azimuthal quantum number (indicates the subshell)
- **m_l**: Magnetic quantum number (indicates the orientation of the orbital)
- **m_s**: Spin quantum number (indicates the spin of the electron)

### Step 2: Apply the n + l Rule
According to the n + l rule, the energy of an electron in an atom can be determined by calculating the sum of n and l:
- The subshell with the lowest n + l value has the lowest energy.
- If two or more subshells have the same n + l value, the one with the lower n value has lower energy.

### Step 3: Calculate n + l for Each Set of Quantum Numbers
Let’s denote the sets of quantum numbers as follows:
- **Set A**: (n=3, l=2)
- **Set B**: (n=4, l=2)
- **Set C**: (n=4, l=1)
- **Set D**: (n=5, l=0)

Now, we will calculate n + l for each set:
- For **Set A**: n + l = 3 + 2 = 5
- For **Set B**: n + l = 4 + 2 = 6
- For **Set C**: n + l = 4 + 1 = 5
- For **Set D**: n + l = 5 + 0 = 5

### Step 4: Compare the n + l Values
Now we compare the n + l values:
- Set A: n + l = 5
- Set B: n + l = 6
- Set C: n + l = 5
- Set D: n + l = 5

Sets A, C, and D all have the same n + l value of 5.

### Step 5: Determine the Lowest Energy
Among sets A, C, and D, we need to look at the principal quantum number (n):
- Set A: n = 3
- Set C: n = 4
- Set D: n = 5

Since Set A has the lowest n value (n = 3), it has the lowest energy.

### Conclusion
The correct set of quantum numbers for the electron with the lowest energy is **Set A**: (n=3, l=2).

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