Which of the following sets of quantum numbers represents the highest energy of an atom?

Options

A. `n=3,l=1,m=1,s=+1//2`
B. `n=3,l=2,m=1,s=+1//2`
C. `n=4,l=0,m=0,s=+1//2`
D. `n=3,l=0,m=0,s=+1//2`

Answer

To determine which set of quantum numbers represents the highest energy of an atom, we need to analyze the given quantum numbers based on their principal quantum number (n) and azimuthal quantum number (l).

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- Each electron in an atom is described by a set of four quantum numbers:
- Principal quantum number (n): Indicates the energy level and size of the orbital.
- Azimuthal quantum number (l): Indicates the shape of the orbital (s, p, d, f).
- Magnetic quantum number (m_l): Indicates the orientation of the orbital.
- Spin quantum number (m_s): Indicates the spin of the electron.

2. **Identifying the Options**:
- Let's assume we have four sets of quantum numbers:
1. (n=3, l=0) → 3s
2. (n=3, l=2) → 3d
3. (n=4, l=0) → 4s
4. (n=3, l=1) → 3p

3. **Evaluating Energy Levels**:
- The energy of an electron in an atom generally increases with increasing principal quantum number (n). However, for the same principal quantum number, the energy also depends on the azimuthal quantum number (l). The order of energy levels for a given n is typically:
- s < p < d < f
- Therefore, for n=3, the energy levels are:
- 3s < 3p < 3d
- For n=4:
- 4s is lower in energy than 3d.

4. **Comparing the Options**:
- From the options:
- 3s (n=3, l=0) has lower energy.
- 3p (n=3, l=1) has higher energy than 3s but lower than 3d.
- 3d (n=3, l=2) has the highest energy among the n=3 orbitals.
- 4s (n=4, l=0) is lower in energy than 3d.

5. **Conclusion**:
- The highest energy set of quantum numbers among the given options is (n=3, l=2), which corresponds to the 3d orbital. Therefore, the answer is option 2 (3d).

### Final Answer:
The set of quantum numbers that represents the highest energy of an atom is (n=3, l=2) → 3d.