Which of the following orbital designations are incorrect: \(1 s\), \(1 p, 7 d, 9 s, 3 f, 4 f, 2 d ?\)

Atomic orbitals are a fundamental concept in quantum mechanics describing the probability distribution of an electron around an atom's nucleus. These orbitals are not fixed paths, as one might picture from classical orbits, but rather areas where the probability of finding an electron is highest. Each orbital is defined by a unique set of quantum numbers; it can be visualized as a cloud or a shell where electrons tend to be.

Major types of orbitals are designated as s, p, d, and f, which stand for sharp, principal, diffuse, and fundamental respectively. The shape and orientation of these orbitals vary: s-orbitals are spherical, p-orbitals are dumbbell-shaped, d-orbitals are mostly clover-shaped, and f-orbitals have even more complex shapes. As electrons fill these orbitals, they follow a set pattern described by the Aufbau principle which assists in predicting the electronic structure of an atom.

Quantum numbers are the critical components that define the state of an electron within an atom. These include the principal quantum number (), azimuthal or angular momentum quantum number (), magnetic quantum number (), and the spin quantum number ().

Principal Quantum Number ()

The principal quantum number, denoted as 'n', determines the energy level and size of the orbital; it can only be a positive integer. As 'n' increases, the orbital becomes larger and the electron is further from the nucleus.

Azimuthal Quantum Number ()

This number, denoted as 'l', defines the shape of the orbital and can have values from 0 to (-1). Each value of 'l' corresponds to a different type of orbital (s, p, d, f).

Magnetic Quantum Number ()

Designated as 'm_l', it describes the orientation of the orbital in space and ranges from -l to +l.

Spin Quantum Number ()

Represented by 's', it describes the spin of the electron and has only two possible values, +1/2 or -1/2, corresponding to the two possible spin states of an electron. Together, these numbers define the properties and location of electrons within an atom.

Electron configuration is the arrangement of electrons in the orbitals of an atom or molecule. According to quantum mechanics, electrons fill up atomic orbitals in a specific order, often conceptualized by the Aufbau principle, which posits that electrons occupy the lowest-energy orbitals first. The order in which these energy levels are filled is not strictly sequential, and electrons will fill 4s orbitals before the 3d orbitals, illustrating this exception.

The electron configuration can be written using the notation that includes the energy level, the type of orbital, and the number of electrons in that orbital as superscripts. For example, the electron configuration of carbon is , indicating that there are two electrons in the 1s orbital, two electrons in the 2s orbital, and two electrons in the 2p orbitals.

Knowing an element's electron configuration allows for the prediction of its chemical properties, including the types of chemical bonds it can form and its position in the periodic table. This information is critical for understanding the behavior of atoms in both isolated and combined states.