Quantum Numbers, Shells, Orbitals, Pauli Exclusion

Course and Lab Announcements

• The next quiz is scheduled for Monday and covers PLQs 9–11.
• There will be no in-person lab for the next two weeks. Experiment 5 is an Excel-based worksheet, due at the next in-person lab on November 17. Lab 6 and its pre-lab materials are also posted.
• The instructor said the class would return to PLQ 10, question 8, which had not yet been discussed.

Light and Electrons: Wave–Particle Duality

• Light shows wave behavior, such as refraction and diffraction. Its wavelength and frequency are related by λν = c, and a photon’s energy is E = hν.
• The photoelectric effect provides evidence that light also behaves as particles called photons. The absence of a threshold-frequency requirement and of a time lag supports the idea that each photon delivers energy individually.
• Electrons also show both kinds of behavior:
  • Their mass and the discrete lines in atomic spectra are evidence of particle-like properties.
  • Their diffraction pattern in the double-slit experiment is evidence of wave-like properties.
• The instructor emphasized distinguishing photons (particles of electromagnetic radiation) from electrons and protons.
• The de Broglie relation gives the wavelength associated with matter: λ = h/(mv). The instructor explained that this wavelength is significant for very small particles, such as electrons, but negligibly small for ordinary large objects.

Uncertainty and the Wave-Function Model

• Heisenberg’s Uncertainty Principle says that an electron’s position and velocity cannot both be known precisely at the same time. Consequently, electrons cannot be described as occupying exact, fixed paths like the orbits in the Bohr model.
• Schrödinger’s wave-function model provides a mathematical way to predict where an electron is likely to be, based on its energy.
• The wave function is written as ψ; the instructor presented the Schrödinger equation as Hψ = Eψ. The probability of finding an electron is associated with ψ².
• Rather than giving an exact location, the model describes regions of probability. A node is a region where the probability of finding the electron is zero.

The First Three Quantum Numbers

The instructor introduced three quantum numbers derived from the Schrödinger model. Together, they describe an electron’s shell, subshell, and orbital orientation.

1. Principal quantum number, n — shell

   • A positive integer: 1, 2, 3, …
   • Indicates the shell and, broadly, the electron’s distance from the nucleus and energy level.
   • Higher n generally means farther from the nucleus and higher potential energy.

2. Angular momentum quantum number, l — subshell and shape

   • For a given n, l can be any integer from 0 through n − 1.
   • The values are commonly represented by letters:
     • l = 0 → s
     • l = 1 → p
     • l = 2 → d
     • l = 3 → f
   • Thus, the first shell has only an s subshell; the second has s and p; the third has s, p, and d; and the fourth has s, p, d, and f.

3. Magnetic quantum number, mₗ — orientation

   • For a given l, mₗ runs from −l through 0 to +l, in integer steps.
   • It specifies the possible orientations of a subshell. The number of orientations is 2l + 1:
     • s: 1 orientation
     • p: 3
     • d: 5
     • f: 7
   • The number of orientations is the same for a given subshell type at different shell numbers. For example, 3p and 4p each have three orientations.
   • A combination such as 1d or 3f is not allowed because the required l value exceeds the permitted range for that n.

Orbitals and Their Shapes

• An orbital refers to a particular shell–subshell–orientation combination.
• The instructor described the common probability shapes:
  • s orbitals are spherical.
  • p orbitals have two lobes, often compared to dumbbells, and have three mutually perpendicular orientations.
  • d orbitals are generally more complex, often described as clover-like; there are five orientations.
  • f orbitals are more complex still, with seven orientations.
• These shapes represent probability distributions, not fixed tracks that electrons follow. The instructor said students should be able to draw the basic s and p shapes; detailed d and f drawings were not required.
• The lecture compared orbitals within the same shell: the instructor explained that, in the course’s model, orbitals with more nodes have higher potential energy. When comparing different shells with the same subshell type, the higher shell number was described as having higher potential energy.

The Fourth Quantum Number and Electron Capacity

• The fourth quantum number is the spin quantum number, mₛ, with two possible values: +½ or −½.
• The instructor linked this to the Stern–Gerlach experiment, in which an electron beam split into two parts in a magnetic field, and described electrons as having two spin states.
• The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of all four quantum numbers.
• As a result, an orbital can hold at most two electrons. They can share the same n, l, and mₗ values, but must have opposite mₛ values.

Electron Configurations and Filling Order

• Electrons fill available subshells from lower to higher potential energy.
• An electron configuration records the shell, subshell letter, and number of electrons as a superscript. For example:
  • Hydrogen: 1s¹
  • Helium: 1s²
• The first shell contains only 1s and can hold two electrons. The second shell includes 2s and 2p: 2s can hold two electrons, while the three 2p orbitals can hold six in total.
• The instructor said the next lessons would continue with configurations, orbital box diagrams, the Aufbau principle, and Hund’s rule.

Speakers and Sources Featured

• Primary speaker: The instructor, identified in the video metadata as Mark Kontulis.
• Student participants in the classroom/chat: Miriam, April, Chet (also captioned as “Chat” in places), Peng/Pang, Aiden, Liam, Jenny, Mark, Nikolai, and William; other unnamed students also contributed.
• Scientific figures, principles, and experiments mentioned: Louis de Broglie, Albert Einstein, Max Planck, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, J. J. Thomson, and the Stern–Gerlach experiment.