An electron in an atom has the following set of quantum numbers:
`n=3,l=2,m_(l)=+1,m_(s)=+1//2`
According to the quantum mechanical picture of the atom, which quantum number(s) could be different for electons in this same atom that have exactly the same energy?

Options

A. `n,l,m_(l)andm_(S)`
B. only `l and m_(l)`
C. only `l,m_(l)andm_(S)`
D. only `m_(l)andm_(S)`

Answer

To determine which quantum numbers could be different for electrons in the same atom that have exactly the same energy, we need to analyze the given quantum numbers and their implications. The quantum numbers provided are:

- Principal quantum number (n) = 3
- Azimuthal quantum number (l) = 2
- Magnetic quantum number (m_l) = +1
- Spin quantum number (m_s) = +1/2

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- The principal quantum number (n) indicates the energy level of the electron. For n=3, the electron is in the third energy level.
- The azimuthal quantum number (l) defines the subshell (s, p, d, f). Here, l=2 corresponds to the d subshell.
- The magnetic quantum number (m_l) specifies the orientation of the orbital within the subshell. For l=2, m_l can take values from -2 to +2.
- The spin quantum number (m_s) indicates the spin orientation of the electron, which can be +1/2 or -1/2.

2. **Identifying Quantum Numbers that Can Change**:
- **Principal Quantum Number (n)**: Changing n would place the electron in a different energy level, thus changing its energy. Therefore, n cannot change if the energy is to remain the same.
- **Azimuthal Quantum Number (l)**: For n=3, l can be 0, 1, or 2 (s, p, or d subshells). If l changes, the electron would be in a different subshell, which would change its energy. Therefore, l cannot change.
- **Magnetic Quantum Number (m_l)**: For l=2, m_l can take values of -2, -1, 0, +1, or +2. Changing m_l does not affect the energy of the electron as all orbitals within the same subshell have the same energy. Therefore, m_l can change.
- **Spin Quantum Number (m_s)**: The spin can be +1/2 or -1/2. Changing the spin does not affect the energy of the electron. Therefore, m_s can change.

3. **Conclusion**:
- The quantum numbers that could be different for electrons in the same atom that have exactly the same energy are:
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)

### Final Answer:
The quantum numbers that could be different while maintaining the same energy are **m_l (magnetic quantum number)** and **m_s (spin quantum number)**.

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