Chemistry 119 - Chapter 2: Orbital Shapes, Energies, Electron Spin, and Pauli Principle

Fundamentals of Atomic Models and Quantum Mechanics

• Bohr Model vs. Quantum Mechanical Model:

  • The Bohr model depicts electrons orbiting the nucleus in fixed, quantized circular shells.
  • The Bohr model is limited and only works accurately for the hydrogen atom (single-electron system).
  • Quantum mechanics provides a comprehensive, mathematically complete description of multi-electron atoms.

• Wavefunctions and Probability Distributions:

  • The square of the wavefunction ($\psi^2$) represents the probability density or probability distribution of finding an electron in a specific region of space around the nucleus.
  • This three-dimensional probability distribution defines the spatial volume known as an atomic orbital shape.

• Electronic State Assignment:

  • Electronic states within an atom are uniquely specified and assigned using a set of quantum numbers.

Quantum Numbers and Orbital Structure

• Summary of Quantum Numbers and Interrelationships:

  • Principal Quantum Number ($n$):
  • Values: $n = 1, 2, 3, 4, \dots$
  • Indicates the main energy level or principal shell.
  • Number of subshells in shell $n$ equals $n$.
  • Total number of orbitals in shell $n$ equals $n^2$.
  • Angular Momentum Quantum Number ($l$):
  • Values: $l = 0, 1, 2, \dots, n - 1$
  • Defines the subshell type and three-dimensional shape of the orbital.
  • Subshell designations: $l = 0$ ($s$), $l = 1$ ($p$), $l = 2$ ($d$), $l = 3$ ($f$).
  • Magnetic Quantum Number ($m_l$):
  • Values: $m_l = +l, \dots, 0, \dots, -l$
  • Specifies the spatial orientation of an orbital within a given subshell.
  • Total number of orbitals in a subshell equals $2l + 1$.
  • Spin Quantum Number ($m_s$):
  • Values: $m_s = \pm \frac{1}{2}$
  • Describes the intrinsic spin orientation of an individual electron ($+\frac{1}{2}$ for spin-up $\uparrow$, $-\frac{1}{2}$ for spin-down $\downarrow$).

• Shell and Subshell Breakdown:

  • $n = 1$ Shell:
  • Subshell: $l = 0$ ($1s$)
  • Magnetic quantum number: $m_l = 0$
  • Total orbitals: 1 orbital ($1s$) of 1 type; holds up to 2 electrons.
  • $n = 2$ Shell:
  • Subshell $l = 0$ ($2s$): $m_l = 0$ (one $2s$ orbital, 2 electrons).
  • Subshell $l = 1$ ($2p$): $m_l = +1, 0, -1$ (three $2p$ orbitals, 6 electrons).
  • Total orbitals: 4 orbitals ($n^2 = 2^2 = 4$) of 2 types; holds up to 8 electrons.
  • $n = 3$ Shell:
  • Subshell $l = 0$ ($3s$): $m_l = 0$ (one $3s$ orbital).
  • Subshell $l = 1$ ($3p$): $m_l = +1, 0, -1$ (three $3p$ orbitals).
  • Subshell $l = 2$ ($3d$): $m_l = +2, +1, 0, -1, -2$ (five $3d$ orbitals).
  • Total orbitals: 9 orbitals ($n^2 = 3^2 = 9$) of 3 types; holds up to 18 electrons.
  • $n = 4$ Shell:
  • Subshell $l = 0$ ($4s$): $m_l = 0$ (one $4s$ orbital).
  • Subshell $l = 1$ ($4p$): $m_l = +1, 0, -1$ (three $4p$ orbitals).
  • Subshell $l = 2$ ($4d$): $m_l = +2, +1, 0, -1, -2$ (five $4d$ orbitals).
  • Subshell $l = 3$ ($4f$): $m_l = +3, +2, +1, 0, -1, -2, -3$ (seven $4f$ orbitals).
  • Total orbitals: 16 orbitals ($n^2 = 4^2 = 16$) of 4 types; holds up to 32 electrons.

Characteristics and Shapes of Atomic Orbitals

• $s$ Orbitals ($l = 0, m_l = 0$):

  • Subshell orbital count: $2l + 1 = 2(0) + 1 = 1$ orbital.
  • Extends radially from the nucleus to form a spherical shape.
  • Spherical Nodes:
  • All $s$ orbitals possess $n - 1$ spherical radial nodes.
  • A $1s$ orbital has $1 - 1 = 0$ nodes.
  • A $2s$ orbital has $2 - 1 = 1$ node.
  • A $3s$ orbital has $3 - 1 = 2$ nodes, and so forth.
  • A spherical node is a concentric surface of zero probability for finding an electron.

• $p$ Orbitals ($l = 1$):

  • Subshell orbital count: $2l + 1 = 2(1) + 1 = 3$ degenerate orbitals ($p_x, p_y, p_z$).
  • Dumbbell-shaped, oriented mutually perpendicular along the $x$, $y$, and $z$ axes at $90^\circ$ relative to one another.
  • Each $p$ orbital contains 1 nodal plane ($l = 1$) passing through the nucleus.

• $d$ Orbitals ($l = 2$):

  • Subshell orbital count: $2l + 1 = 2(2) + 1 = 5$ degenerate orbitals.
  • Orbitals possess 2 nodal planes ($l = 2$).
  • The five $d$ orbital orientations are $d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$, and $d_{z^2}$.
  • The $d_{z^2}$ orbital exhibits a unique shape consisting of two lobes along the $z$-axis with a donut-shaped ring (torus) in the $xy$-plane, and is noted for being one of the most reactive orbital configurations.

Electron Spin, Magnetism, and the Pauli Exclusion Principle

• The Stern-Gerlach Experiment:
  • Conducted by passing a beam of neutral silver atoms from a furnace through an inhomogeneous magnetic field toward a detector screen.
  • Classical Physics Prediction: Predicted a continuous vertical spread/smear of atoms based on classical magnetic moment continuum.
  • Experimental Observation: The atomic beam split into two discrete pathways (two distinct spots).
  • Conclusion: Proved quantization of intrinsic electron angular momentum (electron spin). The two deflection paths demonstrate two distinct spin states affected by the magnetic field (one spinning to the right/clockwise, one to the left/counterclockwise).

• Spin Quantum Number ($m_s$):

  • Assigned values: $m_s = \pm \frac{1}{2}$.
  • $m_s = +\frac{1}{2}$ designates spin-up ($\uparrow$).
  • $m_s = -\frac{1}{2}$ designates spin-down ($\downarrow$).

• Magnetic Properties of Matter:

  • Diamagnetic Substances: Contain no unpaired electrons (all electrons paired). They are NOT attracted to a magnetic field and are weakly repelled.
  • Paramagnetic Substances: Contain one or more unpaired electrons. They ARE attracted into a magnetic field.

• Pauli Exclusion Principle:

  • Formulated by Wolfgang Pauli in 1925.
  • Definition: No two electrons in the same atom can possess identical sets of all four quantum numbers ($n, l, m_l, m_s$).
  • Consequence: An individual orbital defined by $(n, l, m_l)$ can accommodate a maximum of 2 electrons, and they must have opposite spins ($+\frac{1}{2}$ and $-\frac{1}{2}$).

Polyelectronic Atoms, Penetration Effect, and Effective Nuclear Charge

• Polyelectronic Atoms:

  • Refers to any atom possessing more than one electron.
  • Electron Correlation Problem: Because exact electron trajectories cannot be known in quantum mechanics, inter-electronic repulsions cannot be calculated with absolute precision.

• Subshell Energy Splitting in Multi-Electron Systems:

  • Unlike hydrogen (where subshells in the same principal energy level are degenerate), multi-electron atoms experience energy splitting among subshells within a principal quantum level $n$:      $E_{ns} < E_{np} < E_{nd} < E_{nf}$

• The Penetration Effect:

  • An electron in a $2s$ orbital penetrates closer to the nucleus than an electron in a $2p$ orbital.
  • Higher nuclear penetration causes the $2s$ electron to experience a stronger nuclear electrostatic attraction.
  • Consequently, the $2s$ orbital is lower in energy than $2p$ orbitals in a multi-electron atom. The same penetration phenomenon applies across higher quantum levels (e.g., $3s < 3p < 3d$).

• Effective Nuclear Charge ($Z^*$):

  • Definition: The net positive nuclear charge experienced by a specific electron in a multi-electron atom, reflecting a balance between attraction to the nucleus and repulsions from other (shielding) electrons.
  • Formula: $Z^* = Z - S$
  • $Z$ = atomic number (total proton charge of nucleus).
  • $S$ = screening/shielding constant (approximated by the number of inner core electrons).
  • Trend across a period: $Z^*$ increases progressively across a period from left to right due to incomplete shielding of nuclear charge by valence electrons.
  • Examples for 2s electron nuclear charge experience:
  • Lithium ($\text{Li}$, $Z = 3$): $Z^* \approx 3 - 2 = 1$
  • Beryllium ($\text{Be}$, $Z = 4$): $Z^* \approx 4 - 2 = 2$
  • Boron ($\text{B}$, $Z = 5$): $Z^* \approx 5 - 2 = 3$

• Single-Electron vs. Multi-Electron Energy Level Hierarchy:

  • Single-Electron Atom (Hydrogen): Energy depends solely on $n$. Orbitals within the same shell are degenerate:          $1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f$
  • Multi-Electron Atom: Screening and penetration cause orbital energies to depend on both $n$ and $l$. The energy order follows the $n + l$ rule:
  • $4s$ vs. $3d$:
    • For $4s$: $n + l = 4 + 0 = 4$
    • For $3d$: $n + l = 3 + 2 = 5$
    • Because $4 + 0 < 3 + 2$, the $4s$ orbital is lower in energy than the $3d$ orbital during ground-state electron filling.

History and Organization of the Periodic Table

• Historical Development:
  • The periodic table was originally constructed empirically to categorize patterns in observed chemical properties of elements.
  • Mendeleev's Periodic Table:
  • Emphasized using elemental periodic patterns to predict the existence and properties of undiscovered elements.
  • Corrected accepted values for several atomic masses.
  • Modern Periodic Table:
  • Organizes elements sequentially by atomic number ($Z$) rather than atomic mass.

The Aufbau Principle and Electron Configurations

• **The Aufbau Principle (