The energy of an electron in an atomic orbital of a multi electron atom depends on

Options

A. the principal quantum number only
B. the principal and Azimuthal quantum number only
C. the principal azimuthal and magnetic quantum number only
D. The principal, Azimuthal, magnetic and spin quantum numbers

Answer

To determine the energy of an electron in an atomic orbital of a multi-electron atom, we need to analyze how various quantum numbers influence this energy. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Quantum Numbers
The quantum numbers that describe the state of an electron in an atom are:
- Principal quantum number (n)
- Azimuthal quantum number (l)
- Magnetic quantum number (m)
- Spin quantum number (s)

### Step 2: Understand the Role of Each Quantum Number
1. **Principal Quantum Number (n)**: This number indicates the main energy level or shell of the electron. Higher values of n correspond to higher energy levels and larger atomic orbitals.
2. **Azimuthal Quantum Number (l)**: This number defines the shape of the orbital (s, p, d, f). Different shapes have different energy levels due to electron-electron repulsions in multi-electron atoms.
3. **Magnetic Quantum Number (m)**: This number describes the orientation of the orbital in space. However, it does not significantly affect the energy in multi-electron atoms.
4. **Spin Quantum Number (s)**: This number describes the spin of the electron. While it is important for determining the overall state of the electron, it does not directly influence the energy levels in the context of this question.

### Step 3: Determine Which Quantum Numbers Affect Energy
In a multi-electron atom, the energy of an electron primarily depends on:
- The principal quantum number (n)
- The azimuthal quantum number (l)

The energy levels are influenced by both n and l due to the combined effects of electron shielding and penetration.

### Step 4: Apply the n + l Rule
The n + l rule is used to determine the relative energies of orbitals:
- The orbital with the lower n + l value has lower energy.
- If two orbitals have the same n + l value, the one with the higher n value has higher energy.

### Conclusion
Based on the analysis, the energy of an electron in an atomic orbital of a multi-electron atom depends on the principal quantum number (n) and the azimuthal quantum number (l) only.

### Final Answer
Thus, the correct option is:
**Option 2: The principal and azimuthal quantum number only.**

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