Question

List the quantum-mechanical orbitals through \(5 s\), in the correct energy order for multi-electron atoms.

Short answer

The order of orbitals up to 5s for multi-electron atoms is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s.

Step-by-step solution

Understanding Quantum Numbers

To list quantum-mechanical orbitals in the correct energy order for multi-electron atoms, it's important to understand the relevant quantum numbers. The principal quantum number (n) indicates the energy level, the azimuthal quantum number (l) defines the subshell, and for each value of l, there are a number of orbitals. For multi-electron atoms, the energy of an orbital increases with n and, for the same n, with l, due to electron-electron interactions and shielding.

Listing Orbitals by increasing energy (n+1 rule)

Using the (n+1) rule, where energy increases with the sum of n and l, list the orbitals from lowest to highest energy while considering the Madelung rule for the filling order of electrons. For each principal quantum number n, list the types of orbitals (s, p, d, f) up to n - 1.

Order of the Orbitals

Start with the lowest energy orbital (1s), and continue listing them in order according to the sum of n and l and by increasing n for the same type of orbital: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s.

Key concepts

Principal Quantum Number

The principal quantum number, symbolized as 'n', is fundamental in determining the energy levels of electrons in an atom. It is an integer that begins at 1 and increases outward from the nucleus, signifying greater energy levels and distances from the nucleus. Think of it as the address indicating the neighborhood an electron resides in; the higher the number, the further away and more energetically the electron orbits. As 'n' increases, the electron's energy and orbital size also increase, allowing for more complex electron motion.

When sorted by the principal quantum number, electron orbitals follow a typical pattern, starting from the lowest energy level (n=1) and moving up. In multi-electron systems, energy levels can start to overlap due to electron-electron interactions, which is why we need to consider additional quantum numbers to fully understand orbital energy order.

Azimuthal Quantum Number

The azimuthal quantum number, designated by 'l', defines the shape of an electron's orbital and it contributes to the energy of that orbital, especially in multi-electron atoms. It ranges from 0 up to one less than the principal quantum number (that is, from 0 to n-1). Each value of 'l' correlates to a different type of orbital: 0 corresponds to an s orbital, 1 to a p orbital, 2 to a d orbital, and 3 to an f orbital.

Different shapes mean varying degrees of distance from the nucleus and differing electron densities in regions around the nucleus. Orbital shapes are important because they impact how electrons interact with each other and with the nucleus, ultimately influencing the chemical behavior of the atom.

Orbital Energy Order

The orbital energy order is the hierarchy that arranges the orbitals from lowest to highest energy based on the principal quantum number (n) and the azimuthal quantum number (l). While one might assume that orbitals fill sequentially based purely on their principal quantum number, this is not the case in multi-electron atoms. Instead, due to electron shielding and electron-electron interactions, orbitals with the same 'n' can have different energy levels.

For instance, in a hydrogen atom, orbital energies only depend on 'n' (since there is only one electron, thus eliminating electron-electron interactions). However, in multi-electron atoms like carbon or oxygen, the '2p' orbitals will have a higher energy than the '2s', even though they have the same principal quantum number, due to the additional energy from the azimuthal quantum number.

Electron-Electron Interactions

Electron-electron interactions are the forces between electrons within an atom. They have a significant impact on an atom's energy level structure, as each electron in an atom can repel every other electron, affecting the distribution and energy levels of all electrons. This repulsion leads to a phenomenon known as shielding, where inner electrons block some of the nuclear charge from affecting outer electrons.

These interactions complicate the simple picture introduced by the Bohr model of the atom and necessitate a more complex understanding facilitated by quantum mechanics. The interactions need to be accounted for when considering the filling of orbitals, as they can lead to unexpected differences in energy between orbitals that might otherwise appear to be comparable.

Madelung Rule

The Madelung rule, also known as the Aufbau principle, guides us in predicting the order in which subshells are filled with electrons in an atom. According to this rule, electrons fill orbitals starting with the lowest available energy states before moving to higher ones. The order can be predicted using the (n+l) rule, which states that the lower the sum of the principal quantum number (n) and the azimuthal quantum number (l), the lower the energy of the orbital.

Applying this rule results in the following order for filling orbitals: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, and so on. However, there are some exceptions to Madelung's rule, especially when dealing with heavier elements, due to the influence of electron-electron interactions and relativistic effects, but it provides a solid foundation for understanding electron configuration in multi-electron atoms.