Chapter 6: Problem 69
Why do the \(3 s, 3 p\), and \(3 d\) orbitals have the same energy in a hydrogen atom but different energies in a manyelectron atom?
Short Answer
Expert verified
In hydrogen, all orbitals of the same energy level (\( n \)) are degenerate. In many-electron atoms, electron-electron interactions cause energy differences.
Step by step solution
01
Understanding Energy Levels in Hydrogen
In a hydrogen atom, there is only one electron orbiting the nucleus. The only interaction is between this single electron and the nucleus. For hydrogen, the energy of the electron depends solely on the principal quantum number \( n \), not on the types of orbitals \( (s, p, d) \). Thus, all orbitals with the same \( n \) have the same energy, making them degenerate.
02
Introducing Electron-Electron Interactions
In many-electron atoms, in addition to the nucleus-electron attraction, there are electron-electron repulsions. These repulsions affect electron energy, making it depend on both \( n \) and the type of orbital (quantum numbers \( l \) and \( m_l \)). This causes orbitals with the same \( n \) to split into different energy levels.
03
Orbital Penetration and Shielding Effects
In many-electron atoms, orbitals experience penetration and shielding effects. Electrons in \( s \) orbitals penetrate more towards the nucleus and are less shielded by other electrons, resulting in relatively lower energy than \( p \) and \( d \) orbitals. This difference affects the splitting of energy levels for \( 3s, 3p, \) and \( 3d \) orbitals.
04
Conclusion
In hydrogen, all orbitals of the same principal quantum number have the same energy due to the absence of electron-electron interactions. In many-electron atoms, these interactions and additional factors like penetration and shielding lead to different energies for \( 3s, 3p, \) and \( 3d \) orbitals.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hydrogen Atom
The hydrogen atom is the simplest atom we know, consisting of just one proton in the nucleus and a single electron in its orbit. Because it has only one electron, the complexity found in many-electron systems does not arise. The energy levels in hydrogen depend solely on the principal quantum number, denoted as \( n \). This means that all orbitals with the same value of \( n \) have identical energies, a property known as degeneracy. Consequently, in a hydrogen atom, the \( 3s, 3p, \) and \( 3d \) orbitals all have the same energy, as they share the same principal quantum number, \( n = 3 \). This trait simplifies the hydrogen atom's electron configuration compared to atoms with more electrons.
Electron-Electron Interactions
In many-electron atoms, the story becomes more intricate due to electron-electron interactions. These interactions arise because electrons have the same charge and repel each other. This electron-electron repulsion impacts the energy levels of the atom by making them dependent on the orbital angular momentum quantum number \( l \), in addition to the principal quantum number \( n \). In such systems, even orbitals that share the same \( n \) might not possess the same energy. The specific distribution of electrons in different orbitals and the shielding effect from inner electrons cause these energy variations. Thus, in more complex atoms, you will find that \( 3s, 3p, \) and \( 3d \) orbitals are not degenerate and have distinct energy levels.
Energy Levels
Energy levels in atoms define where electrons can exist around the nucleus. In a hydrogen atom, energy levels are influenced only by the principal quantum number \( n \), because it lacks additional electron-electron interaction. However, in many-electron systems, this simplicity vanishes.
- The principal quantum number \( n \) dictates the shell or primary energy level.
- Angular momentum \( l \) adds a finer distinction, separating the shells into subshells like \( s, p, d, \) and \( f \).
- Each subshell then contains orbitals further defined by magnetic quantum number \( m_l \).
Penetration and Shielding Effects
Penetration and shielding are key factors that impact how electron energies are distributed in many-electron atoms. When an electron penetrates closer to the nucleus, like those in \( s \) orbitals, it experiences a greater electrostatic pull from the nucleus and less shielding from other electrons. This penetration results in lower energy for these electrons compared to others in the same shell but different subshells, such as \( p \) and \( d \) orbitals.
- Penetration: Refers to how deep within the electron cloud an electron can move towards the nucleus.
- Shielding: Occurs when inner electrons reduce the effective nuclear charge felt by outer electrons.