Premed · Premed · General Chemistry 1
Lecture 8: Quantum Mechanics and Orbitals
General Chemistry I
Learning Objectives
By the end of this lecture, students will be able to:
- State the Heisenberg uncertainty principle and explain its significance
- Describe the quantum mechanical model of the atom and the concept of orbitals
- Define the four quantum numbers (n, l, m_l, m_s) and state their allowed values
- Identify the shapes and orientations of s, p, d, and f orbitals
- Apply the Aufbau principle, Pauli exclusion principle, and Hund's rule to write electron configurations
- Write electron configurations using noble gas core notation and orbital diagrams
Lecture Content
I. The Heisenberg Uncertainty Principle
In 1927, Werner Heisenberg established that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle is expressed mathematically as delta_x delta_(mv) >= h / (4pi), where delta_x is the uncertainty in position and delta_(mv) is the uncertainty in momentum. The more precisely one quantity is known, the less precisely the other can be determined. For macroscopic objects, this uncertainty is negligibly small and has no practical consequences. For electrons, however, the uncertainty is significant -- we simply cannot pinpoint an electron's trajectory. As a direct consequence, the concept of well-defined orbits must be abandoned in favor of probability distributions that describe where an electron is likely to be found.
II. The Quantum Mechanical Model of the Atom
In 1926, Erwin Schrodinger formulated a wave equation (Hpsi = Epsi) that treats the electron as a standing wave rather than a classical particle. The solutions to this equation, called wave functions (psi), describe the electron's wave-like behavior mathematically. The square of the wave function, |psi|^2, gives the probability density -- the probability of finding the electron at any given point in space. An orbital is defined as the region of space where there is a high probability (roughly 90%) of finding an electron. This quantum mechanical picture replaces Bohr's fixed orbits with probability clouds, and each orbital is uniquely defined by a set of quantum numbers.
III. The Four Quantum Numbers
A. Principal Quantum Number (n)
The principal quantum number takes positive integer values (n = 1, 2, 3, 4, ...) and determines both the energy level (shell) and the average distance of the electron from the nucleus. Higher values of n correspond to higher energy and larger orbital size. The maximum number of electrons that can occupy shell n is 2n^2.
B. Angular Momentum (Azimuthal) Quantum Number (l)
The angular momentum quantum number ranges from 0 to n - 1 and determines the shape of the orbital, defining what is called the subshell. When l = 0, the orbital is an s orbital (spherical). When l = 1, the orbitals are p orbitals (dumbbell or figure-eight shape). When l = 2, the orbitals are d orbitals (cloverleaf and related shapes), and when l = 3, the orbitals are f orbitals (complex, multi-lobed shapes). The number of subshells within a given shell n is equal to n.
C. Magnetic Quantum Number (m_l)
The magnetic quantum number takes integer values from -l to +l, including zero, and specifies the orientation of the orbital in space. The number of orbitals per subshell is 2l + 1, giving one s orbital, three p orbitals, five d orbitals, and seven f orbitals.
D. Spin Quantum Number (m_s)
The spin quantum number has only two possible values: +1/2 (spin up) and -1/2 (spin down). It describes the intrinsic angular momentum of the electron. Because each orbital can accommodate at most two electrons, those two electrons must have opposite spins.
<image>A comprehensive table and diagram of quantum numbers and orbital shapes. Top: a table listing each quantum number (n, l, m_l, m_s), its name, allowed values, and what property it determines. Bottom: four rows of 3D orbital shape illustrations. Row 1 (l=0): a single spherical 1s orbital and a larger 2s orbital with a nodal sphere, showing increasing size with n. Row 2 (l=1): three mutually perpendicular p orbitals (p_x, p_y, p_z), each shown as a dumbbell along the respective axis, with nodal planes highlighted. Row 3 (l=2): five d orbitals (d_xy, d_xz, d_yz as cloverleafs between axes; d_x2-y2 as cloverleaf along axes; d_z2 as a dumbbell with a torus). Row 4 (l=3): a simplified representation of one f orbital showing its complex multi-lobed shape.</image>
IV. Orbital Shapes in Detail
A. s Orbitals
The s orbitals are spherically symmetric around the nucleus. The 1s orbital is a single sphere, the 2s is a larger sphere containing one radial node, and the 3s is even larger with two radial nodes. In general, the number of radial nodes for any orbital is n - l - 1.
B. p Orbitals
The p orbitals have a characteristic dumbbell shape with a nodal plane passing through the nucleus. The three p orbitals in any subshell are oriented along the x, y, and z axes (p_x, p_y, p_z). They first appear at n = 2 as the 2p subshell.
C. d Orbitals
The d orbitals possess more complex shapes. Four of the five (d_xy, d_xz, d_yz, and d_x2-y2) have cloverleaf patterns, while the fifth (d_z2) has a dumbbell shape with a surrounding donut (torus). They first appear at n = 3 as the 3d subshell.
D. f Orbitals
The f orbitals have seven orientations with complex, multi-lobed shapes. They first appear at n = 4 as the 4f subshell and are particularly important in the chemistry of the lanthanides and actinides.
V. Nodes
A node is a region where the probability of finding an electron is zero (psi = 0). Radial (spherical) nodes are spherical surfaces of zero probability, and their number for any orbital is n - l - 1. Angular (planar or conical) nodes are planes or cones passing through the nucleus, and their number equals l. The total number of nodes for any orbital is therefore n - 1.
VI. Electron Configuration: Filling Rules
A. Aufbau Principle
The Aufbau principle states that electrons fill orbitals starting from the lowest energy and proceeding to higher energy. The filling order is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. A useful mnemonic is the diagonal rule: write the subshells in columns by principal quantum number and draw diagonals from upper right to lower left to read off the filling order.
B. Pauli Exclusion Principle
The Pauli exclusion principle states that no two electrons in the same atom can share the same set of four quantum numbers. The practical consequence is that each orbital holds a maximum of two electrons, which must have opposite spins (they are paired).
C. Hund's Rule
Hund's rule governs the filling of degenerate orbitals (orbitals of equal energy). Electrons first occupy each degenerate orbital singly, all with parallel spins, before any pairing occurs. This maximizes total spin and minimizes electron-electron repulsion. For example, the 2p^2 configuration of carbon places one electron in p_x and one in p_y (both spin up), rather than pairing two electrons in the same p orbital.
VII. Writing Electron Configurations
In full notation, all occupied subshells are listed with superscripts indicating the number of electrons in each. For example: H is 1s^1, He is 1s^2, Li is 1s^2 2s^1, C is 1s^2 2s^2 2p^2, Ne is 1s^2 2s^2 2p^6, Na is 1s^2 2s^2 2p^6 3s^1, and Fe is 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6. Noble gas (core) notation simplifies this by replacing the inner electrons with the symbol of the preceding noble gas in brackets: Na becomes [Ne] 3s^1, Fe becomes [Ar] 4s^2 3d^6, and Br becomes [Ar] 4s^2 3d^10 4p^5. Orbital diagrams use boxes or lines to represent individual orbitals, with arrows (up and down) representing electrons, making Hund's rule visually explicit.
VIII. Electron Configurations of Ions
When forming cations, electrons are removed from the outermost shell first (the shell with the highest principal quantum number). For transition metals, this means removing electrons from the ns subshell before the (n-1)d subshell. Thus Fe^2+ has the configuration [Ar] 3d^6, formed by removing both 4s electrons rather than any 3d electrons. Fe^3+ is [Ar] 3d^5. When forming anions, electrons are added to the lowest available subshell. O^2- becomes 1s^2 2s^2 2p^6, which is [Ne] (isoelectronic with neon), and Cl^- becomes [Ar] (isoelectronic with argon).
IX. Notable Exceptions to Aufbau Filling
Chromium is expected to have the configuration [Ar] 4s^2 3d^4, but its actual configuration is [Ar] 4s^1 3d^5, with a half-filled d subshell. Copper is expected to be [Ar] 4s^2 3d^9, but it is actually [Ar] 4s^1 3d^10, with a fully filled d subshell. Similar exceptions occur in heavier transition metals such as Mo, Ag, and Au. The extra stability of half-filled and fully filled d subshells arises from exchange energy and the minimization of electron-electron repulsion.
<image>An orbital diagram comparison for chromium. Left side (Expected configuration - INCORRECT): shows [Ar] core, then 4s orbital with two up/down arrows (4s^2), and five 3d orbital boxes with four having one up arrow each and one empty (3d^4). Right side (Actual configuration - CORRECT): shows [Ar] core, then 4s orbital with one up arrow (4s^1), and five 3d orbital boxes each with one up arrow (3d^5). A note explains: "Half-filled subshells (d^5) provide extra stability due to maximized exchange energy. Each 3d orbital contains exactly one electron with parallel spin." An energy-level comparison shows the actual configuration is lower in energy.</image>
