AP Chemistry 1.5-1.6: Electron Configuration and PES

Quantum-mechanical model of the atom

The quantum-mechanical model describes electrons using three-dimensional wavefunctions and probability distributions rather than definite circular paths. It predicts allowed energies and the regions where electrons are likely to be found.

Atomic orbital

An atomic orbital is a three-dimensional region described by a wavefunction in which an electron has a high probability of being found. It is not a fixed path around the nucleus.

Wavefunction, $\psi$

A wavefunction is a mathematical function that describes the quantum state of an electron. Its squared magnitude, $|\psi|^2$, gives the probability density of finding the electron at a location.

Principal quantum number, $n$

The principal quantum number specifies an electron's main energy level or shell. It can have values $n=1,2,3,\ldots$; larger $n$ generally corresponds to a larger orbital region and, in hydrogen-like species, higher energy.

How are shells and subshells distinguished?

A shell is identified by the principal quantum number $n$ and contains subshells identified by the angular momentum quantum number $\ell$. For a given $n$, the allowed values are $\ell=0$ through $n-1$.

Angular momentum quantum number, $\ell$

The angular momentum quantum number identifies an electron's subshell and orbital shape. For a given principal quantum number, its allowed values are integers from $0$ to $n-1$.

How are subshell labels related to $\ell$?

The correspondence is $\ell=0$ for an $s$ subshell, $\ell=1$ for $p$, $\ell=2$ for $d$, and $\ell=3$ for $f$. For example, $n=5$ and $\ell=3$ describes the $5f$ subshell.

Magnetic quantum number, $m_\ell$

The magnetic quantum number specifies the spatial orientation of an orbital within a subshell. For a given $\ell$, it can have every integer value from $-\ell$ to $+\ell$.

How many orbitals are in a subshell with angular momentum quantum number $\ell$?

A subshell contains $2\ell+1$ orbitals because $m_\ell$ has that many allowed values. Thus, $s$, $p$, $d$, and $f$ subshells contain 1, 3, 5, and 7 orbitals, respectively.

What are the typical shapes of $s$, $p$, and more complex orbitals?

$s$ orbitals are spherical, $p$ orbitals have two-lobed dumbbell shapes, and $d$ and $f$ orbitals have more complex three-dimensional shapes.

Spin quantum number, $m_s$

The spin quantum number describes an electron's intrinsic quantum property and can have only two values: $m_s=+\frac{1}{2}$ or $m_s=-\frac{1}{2}$. It does not represent literal classical rotation in space.

What information is specified by the four quantum numbers?

$n$ identifies the shell, $\ell$ identifies the subshell and general shape, $m_\ell$ identifies the orbital orientation, and $m_s$ identifies the electron's spin state.

Pauli exclusion principle

No two electrons in the same atom may have identical values of all four quantum numbers: $n$, $\ell$, $m_\ell$, and $m_s$.

Why does each orbital hold a maximum of two electrons?

The Pauli exclusion principle requires two electrons in the same orbital to have different spin quantum numbers. Since only $m_s=+\frac{1}{2}$ and $m_s=-\frac{1}{2}$ are possible, an orbital can hold at most two electrons with opposite spins.

What is the maximum number of electrons in a shell with principal quantum number $n$?

A shell contains $n^2$ orbitals, and each orbital holds two electrons, so the maximum number is $2n^2$ electrons.

Determine the allowed quantum numbers for a $4d$ electron.

For $4d$, $n=4$ and $\ell=2$. Therefore, $m_\ell=-2,-1,0,+1,+2$, and each electron can have $m_s=+\frac{1}{2}$ or $-\frac{1}{2}$.

Can an electron have the quantum numbers $n=2$ and $\ell=2$?

No. For a given $n$, $\ell$ must range from 0 to $n-1$. When $n=2$, only $\ell=0$ and $\ell=1$ are allowed, corresponding to $2s$ and $2p$.

What are degenerate orbitals?

Degenerate orbitals have the same energy. In multielectron atoms, orbitals within the same subshell are generally degenerate, while subshells with the same principal quantum number may have different energies because of shielding and electron-electron repulsion.

Aufbau principle

The Aufbau principle states that electrons occupy the available orbitals of lowest energy before occupying higher-energy orbitals.

What is the usual orbital-filling order for electron configurations?

The usual order is $1s$, $2s$, $2p$, $3s$, $3p$, $4s$, $3d$, $4p$, $5s$, $4d$, $5p$, $6s$, $4f$, $5d$, $6p$, $7s$, $5f$, $6d$, $7p$.

Hund's rule

Hund's rule states that electrons occupy degenerate orbitals singly with parallel spins before any pairing occurs. This minimizes electron-electron repulsion within the subshell.

How do you determine the electron configuration of a neutral atom?

Use the atom's atomic number as the total number of electrons, fill subshells in increasing energy order according to the Aufbau principle, place at most two electrons in each orbital according to Pauli exclusion, and apply Hund's rule to degenerate orbitals.

How are electron configurations written for ions?

For anions, add electrons to the neutral atom's configuration. For cations, remove electrons from the occupied subshell with the highest principal quantum number first. For transition-metal cations, electrons are removed from $4s$ before $3d$.

What is the ground-state electron configuration of phosphorus, $Z=15$?

Phosphorus has 15 electrons, so its configuration is $1s^2 2s^2 2p^6 3s^2 3p^3$, or $[Ne]3s^2 3p^3$.

What is the electron configuration of $\mathrm{Mg^{2+}}$?

Neutral magnesium is $[Ne]3s^2$. Removing its two $3s$ electrons gives $\mathrm{Mg^{2+}}: [Ne]$.

What is photoelectron spectroscopy (PES)?

PES measures the kinetic energy of electrons ejected when photons of known energy strike atoms. The measured electron energies reveal the binding energies of electrons in different subshells.

What equation relates photon energy, electron binding energy, and photoelectron kinetic energy?

The relationship is $E_{\text{photon}}=\text{binding energy}+KE$, or $KE=E_{\text{photon}}-\text{binding energy}$.

How does a PES peak's position relate to electron binding energy?

A peak at higher binding energy represents electrons held more strongly by the nucleus. These electrons are generally closer to the nucleus or experience less shielding. If the spectrum's horizontal axis is kinetic energy instead, higher binding energy corresponds to lower kinetic energy.

What does the relative area or intensity of a PES peak indicate?

The relative area or intensity of a peak indicates the relative number of electrons in that subshell. For example, a peak representing six electrons should have approximately twice the area of a peak representing three electrons.

How can a PES spectrum be used to determine an atom's electron configuration?

Group peaks according to subshell binding energies, use their relative areas to determine the number of electrons in each subshell, and order the subshells from highest to lowest binding energy. The resulting electron counts give the electron configuration.

How can PES distinguish electrons in different subshells of the same shell?

In multielectron atoms, subshells with the same $n$ can have different binding energies because of shielding and penetration. A PES spectrum therefore shows separate peaks for subshells such as $2s$ and $2p$.

How can a PES spectrum indicate that a species is an ion?

The total number of electrons represented by all PES peaks gives the species' electron count. Comparing this count with the atomic number identifies whether the species is neutral, a cation, or an anion.