Question

Which orbital in each of the following pairs is lower in energy in a many- electron atom: (a) \(2 s, 2 p\); (b) \(3 p, 3 d\) (c) \(3 s, 4 s ;\) d) \(4 d, 5 f\) ?

Short answer

(a) 2s; (b) 3p; (c) 3s; (d) 4d.

Step-by-step solution

Understanding Orbital Energy Hierarchy

In a many-electron atom, the energy of orbitals is determined by both the principal quantum number ( ) and the azimuthal quantum number ( ext{l} ). Lower corresponds to lower energy, but among orbitals with the same , lower ext{l} corresponds to lower energy.

Comparing 2s and 2p Distributions

For part (a): Both orbitals have the same principal quantum number ( 2 ), but different ext{l} values: 2s ( ext{l} = 0 ) is lower in energy compared to 2p ( ext{l} = 1 ) because ext{l} = 0 represents the lower energy subshell.

Examining 3p and 3d Energies

For part (b): Both orbitals share the principal quantum number 3 . However, 3p ( ext{l} = 1 ) is lower in energy than 3d ( ext{l} = 2 ) in a many-electron atom.

Comparative Analysis of 3s and 4s

For part (c): The 3s orbital has a lower principal quantum number than 4s . Hence, 3s is lower in energy.

Evaluating 4d and 5f Energies

For part (d): 4d and 5f orbitals differ in their principal quantum numbers, where 4d ( 4 ) is lower than 5f ( 5 ). Therefore, 4d is lower in energy.

Key concepts

Many-Electron Atom

Atoms can have many electrons orbiting around the nucleus, forming what we call a many-electron atom. Unlike the simpler hydrogen atom, these atoms contain more than one electron, leading to interactions between them. This makes the understanding of their structure more complex.

• Electrons in these atoms do not move in simple circular orbits.
• The electrons interact with each other and the nucleus.
• Their energies and arrangements can be described using quantum theory.

In a many-electron atom, the energy profiles of the electrons require detailed analysis, which include the effects of electron-electron repulsion. Due to these interactions, energy levels are split and follow specific rules defined by quantum numbers. All these interactions help define the overall behavior and energy of the atom's electrons.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is one of the four quantum numbers used to describe the quantum state of an electron. It fundamentally identifies the shell or energy level an electron resides in an atom.

• The value of \(n\) can be any positive integer: 1, 2, 3, and so on.
• It signifies the distance of the electron from the nucleus and the size of the orbital.
• Higher \(n\) values indicate that the electron is further away from the nucleus and has higher energy.

The principal quantum number is critical for determining the energy level of an electron. It is the primary determinant of an electron's energy, with higher numbers representing higher energy states.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number and symbolized as \(l\), offers insight into the shape of an electron's orbital and its subshell.

• This number can range from 0 up to \(n-1\), where \(n\) is the principal quantum number.
• The value of \(l\) determines the orbital type: \(l = 0\) represents an s orbital, \(l = 1\) is a p orbital, \(l = 2\) is a d orbital, and \(l = 3\) is an f orbital.
• Higher \(l\) values correlate with more complex shapes of orbitals.

Understanding \(l\) is essential when comparing orbital energies in a many-electron system. Even with the same \(n\), different values of \(l\) can cause differences in energy levels due to variations in electron-electron interactions.

Orbital Hierarchy

Orbital hierarchy is an essential concept for understanding the energy arrangements of electrons in many-electron systems. It represents the order and ranking of orbitals based on their energy in an atom.

• In a single electron atom, energy levels depend solely on \(n\), the principal quantum number.
• In a many-electron system, both the principal quantum number \(n\) and the azimuthal quantum number \(l\) influence energy.
• Orbitals with lower \(n\) generally have lower energies but for the same \(n\), the order is typically \(s < p < d < f\) due to differing \(l\) values.

This hierarchy considers both number systems, which is why it is critical to assess the dependencies of both \(n\) and \(l\) to determine the actual energy order of orbitals in more complex atoms.