Study Notes on Chapter 2: Electronic Structure of Atoms

Chapter 2: Electronic Structure of Atoms

2.1 Quantum Numbers and Atomic Orbitals

  • Definition of Atomic Orbital: Specified by three quantum numbers: n, l, and ml.

  • Principal Quantum Number (n):

    • Refers to the relative size of the orbital and the distance from the nucleus.

    • Possible values: n = 1, 2, 3, 4, … (positive integers only).

    • Determines the atom's energy levels (shells): smaller n corresponds to lower energy levels and higher probability of the electron being closer to the nucleus.

  • Angular Momentum Quantum Number (l):

    • Refers to the orbital shape; also known as the orbital-shape quantum number.

    • Values: 0 to (n-1).

    • l = 0: sublevel is s (sharp).

    • l = 1: sublevel is p (principal).

    • l = 2: sublevel is d (diffuse).

    • l = 3: sublevel is f (fundamental).

    • Sublevels with l > 3 are designated alphabetically (g, h, etc.).

    • Naming convention: sublevel is named by combining n value and subshell letter (e.g. 3s).

  • Magnetic Quantum Number (ml):

    • Refers to orbital orientation; also known as the orbital-orientation quantum number.

    • Values of ml range from -l to +l, including 0.

    • Total number of values for ml indicates the number of orbitals.

Summary of Quantum Numbers for Atomic Orbitals

  • Quantum Numbers and Their Properties:

    • Principal (n): Positive integers (1, 2, 3, …)

    • Angular Momentum (l): 0 to n-1

    • Magnetic (ml): -l to +l

2.2 The Quantum-Mechanical Model and the Periodic Table

  • Definition of an Orbital: An orbital is a mathematical function that describes the probability distribution of an electron in an atom. The square of the wave function (Ψ²) indicates the volume of space around a nucleus where an electron is likely to be found, typically 90% to 95% of its time.

Sample Problems and Explanations

Sample Problem 1: Determining Quantum Numbers for an Energy Level

  • Problem: What values of l and ml are allowed for n = 3?

    • For n = 3, possible l values are 0, 1, 2.

    • For l = 0, ml = 0.

    • For l = 1, ml = -1, 0, +1.

    • For l = 2, ml = -2, -1, 0, +1, +2.

    • Total allowed orbitals with n = 3: 9 orbitals.

Sample Problem 2: Determining Sublevel Names and Orbital Quantum Numbers

  • Problem: Given the quantum numbers, provide the sublevel/suborbital name, possible ml values, and number of orbitals.

    • (a) n = 3, l = 2: 3d; possible ml values: -2, -1, 0, +1, +2; 5 orbitals.

    • (b) n = 2, l = 0: 2s; ml values: 0; 1 orbital.

    • (c) n = 5, l = 1: 5p; ml values: -1, 0, +1; 3 orbitals.

    • (d) n = 4, l = 3: 4f; ml values: -3, -2, -1, 0, +1, +2, +3; 7 orbitals.

Shapes of Atomic Orbitals

  • s Orbital: Spherical in shape, with the nucleus at the center.

  • p Orbitals: Dumbbell-shaped with three orientations (px, py, pz), separated by nodes.

  • d Orbitals:

    • There are five d orbitals.

    • Four have a cloverleaf shape (four lobes).

    • The dz2 orbital has a donut shape with girdles around the center.

Electron Spin and Quantum Number

  • Spin Quantum Number (ms): Describes the intrinsic spin of the electron, allowed values are +1/2 or -1/2.

  • Stern-Gerlach Experiment: Demonstrates electron spin; a beam of hydrogen atoms splits in the presence of a magnetic field, showing that electrons behave as if they have tiny magnetic fields due to their spin.

Summary of Quantum Numbers for Electrons in Atoms

  • Principal (n): Positive integers.

  • Angular Momentum (l): From 0 to n-1 (corresponding to s, p, d, f).

  • Magnetic (ml): From -l to +l (orbital orientation).

  • Spin (ms): +1/2 or -1/2.

  • Exclusion Principle: No two electrons in the same atom can have the same set of four quantum numbers (Pauli's Exclusion Principle).

Rules for Orbital Occupancies in Ground-State Electron Configuration

  1. Aufbau Principle: Lowest-energy orbitals fill first: order of filling is 1s < 2s < 2p < 3s < 3p < 4s < 3d.

  2. Pauli's Exclusion Principle: Only two electrons can occupy an orbital with opposite spins.

  3. Hund's Rule: Within equal-energy orbitals, electrons will occupy each orbital singly with parallel spins before pairing.

Sample Problems on Quantum Numbers from Orbital Diagrams

  • F Atom Quantum Numbers:

    • 3rd electron in 2s: n = 2, l = 0, ml = 0, ms = +1/2.

    • 8th electron in 2p: n = 2, l = 1, ml = -1, ms = -1/2.

Electron Configurations for Elements

  • Element H: 1s¹

  • Element He: 1s²

  • Element Li: 1s² 2s¹

  • Element Ne: 1s² 2s² 2p⁶

  • Element K: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹

  • Element Mo: [Kr] 5s¹ 4d⁵

  • Element Pb: [Xe] 6s² 4f¹⁵ 5d¹⁰ 6p²

Periodic Trends in Key Atomic Properties

Atomic Size

  1. Decreases across a period (left to right) due to increased effective nuclear charge.

  2. Increases down a group due to the increase in principal quantum number (n).

Trends in Ionization Energy (IE)

  1. Defined as the energy required to remove one mole of electrons from one mole of gaseous atoms or ions.

  2. IE trends: IE increases across a period (due to decreasing atomic size) and decreases down a group (increasing atomic size).

Trends in Electron Affinity (EA)

  1. Defined as the energy change when adding an electron to a gaseous atom.

  2. EA trends: Generally goes down a group and increases across a period, but there are specific irregularities.

Summary of Periodic Trends

  • Atomic size increases down a group and decreases across a period.

  • Ionization energy increases across a period and decreases down a group.

  • Electron affinity trends are less consistent compared to atomic size and ionization energy.