Electronic Structure of an Atom Part 2 Study Notes

Heisenberg’s Uncertainty Principle

• Definition: Heisenberg’s principle or the uncertainty principle states that it is inherently impossible to know both the exact momentum of the electron and its exact location in space simultaneously.

• Mathematical Equation: The principle is represented by the inequality:   $\Delta x \cdot \Delta(mu) \geq \frac{h}{4\pi}$

• Alternative Form (Position and Velocity): Given the condition $\Delta mu = m\Delta u$, the equation for uncertainty in position ($\Delta x$) can be rewritten as:   $\Delta x = \frac{h}{4\pi m\Delta u}$

• Key Variables:

  • $\Delta x$: Uncertainty in position.

  • $\Delta(mu)$: Uncertainty in momentum.

  • $h$: Planck’s constant.

  • $m$: Mass of the particle.

  • $\Delta u$: Uncertainty in velocity.

• Macro vs. Micro World Application: The uncertainty principle is critically important when discussing electrons and other subatomic particles because their masses are extremely small. However, it is not applicable to the macroscopic world because the mass of macroscopic objects is too large for the uncertainty to be detectable or significant.

Quantum Mechanics and Wave Functions

• Background: Schrodinger (Austrian, 1887–1961) proposed an equation that incorporates both the wavelike behavior and the particle-like behavior of electrons.

• The Wave Function (Psi): Solving Schrodinger’s equation for the Hydrogen ($H_2$) atom leads to mathematical functions called wave functions, represented by the Greek lower case symbol $\Psi$.

• Significance of Wave Functions:

  • Although $\Psi$ itself has no direct physical meaning, the square of the function, $\Psi^2$, provides information about an electron’s location when the electron is in an allowed energy state.

  • Probability Density: $\Psi^2$ represents the probability ($P$) of finding an electron at a given point in space. It is also referred to as electron density.

• Quantum (Wave) Mechanics Overview:

  • Developed to describe the motion of small particles confined to very small portions of space.

  • Deals strictly with the probability of finding a particle within a given region of space at a given instant.

  • Schrodinger derived an equation to calculate the amplitude (height) $y$ of an electron wave at various points in space.

Quantum Numbers and Atomic Orbitals

• Orbitals: In the quantum mechanical model, wave functions are called orbitals. Each orbital describes the distribution of electron density in space as determined by the orbital probability density.

• Orbital Characteristics: Every orbital possesses a characteristic energy and shape.

• Conceptual Shift: Unlike the Bohr model, the Quantum Mechanical model does not refer to "orbits" because the motion of electrons cannot be precisely measured or tracked.

• The Three Primary Quantum Numbers: Used to describe the size, shape, and orientation in space of orbitals.

  1. Principal Quantum Number ($n$):

  • Describes the size and energy of the orbital.

  • Allowed values: Integers starting from 1 ($n = 1, 2, 3, \dots$). $n \neq 0$.

  • As $n$ increases, the orbital becomes larger, the electron has higher energy, and is less tightly bound to the nucleus.

  • Energy formula (same as Bohr model): $E_n = \left(-\frac{1}{n^2}\right)(2.18 \times 10^{-18})\,\text{J}$.

  1. Angular Momentum Quantum Number ($l$):

  • Describes the shape of the orbital.

  • Allowed values: Integers from 0 to $n - 1$ for each value of $n$.

  • Subshell Designations:

    • $l = 0$: s orbital (spherical).

    • $l = 1$: p orbital (polar/dumbbell).

    • $l = 2$: d orbital (cloverleaf).

    • $l = 3$: f orbital (complex shapes).

  1. Magnetic Quantum Number ($m_l$):

  • Describes the orientation in space of a particular orbital.

  • Allowed values: Integers from $-l$ to $+l$, including zero.

  • The name arises because orbital orientations were first distinguished in the presence of a magnetic field.

Shells and Subshells

• Electron Shell: A collection of orbitals with the same principal quantum number ($n$). For example, all orbitals with $n=3$ belong to the third shell.

• Subshell: A set of orbitals that share both the same $n$ and $l$ values. Subshells are designated by the number ($n$) and a letter ($s, p, d, f$) corresponding to $l$.

  • Example: An orbital with $n=3$ and $l=2$ is part of the 3d subshell.

• Review of n, l, and m Relationships:

  • n = 1: $l = 0$ (1s); $m_l = 0$; 1 orbital in subshell; 1 total orbital in shell.

  • n = 2:

  • $l = 0$ (2s); $m_l = 0$; 1 orbital in subshell.

  • $l = 1$ (2p); $m_l = 1, 0, -1$; 3 orbitals in subshell.

  • Total orbitals in shell = 4.

  • n = 3:

  • $l = 0$ (3s); $m_l = 0$; 1 orbital.

  • $l = 1$ (3p); $m_l = 1, 0, -1$; 3 orbitals.

  • $l = 2$ (3d); $m_l = 2, 1, 0, -1, -2$; 5 orbitals.

  • Total orbitals in shell = 9.

  • n = 4:

  • $l = 0, 1, 2, 3$ (4s, 4p, 4d, 4f).

  • Number of orbitals per subshell: $s=1, p=3, d=5, f=7$.

  • Total orbitals in shell = 16.

• Mathematical Summary of Quantum Numbers:

  • Number of subshells in a shell = $n$.

  • Number of orbitals in a subshell = $2l + 1$.

  • Total number of orbitals in a shell = $n^2$.

Orbital Shapes and Physical Representations

• s Orbitals ($l=0$):

  • Spherically symmetric around the nucleus.

  • Electron density is the same in all directions for a given distance.

  • Radial Probability Function: Shows the most likely distance to find an electron. As $n$ increases, the most likely distance moves further from the nucleus.

  • Nodes: Points where the probability of finding an electron is zero. The number of nodes is equal to $n - 1$.

  • Maxima: The number of peaks in the radial probability function is equal to $n$.

  • Contour Representation: Usually drawn to represent a 90% probability of finding the electron.

• p Orbitals ($l=1$):

  • Dumbbell-shaped with two lobes.

  • For any given $n$, there are three p orbitals ($p_x, p_y, p_z$) oriented along the Cartesian axes.

  • Probability is greater in the interior of the lobe than on the edges.

• d Orbitals ($l=2$):

  • Five equivalent d orbitals exist for $n \geq 3$.

  • Four have a four-leaf clover shape ($d_{xy}, d_{xz}, d_{yz}, d_{x^2-y^2}$).

  • $d_{xy}, d_{xz}, d_{yz}$ lie in their respective planes with lobes between axes.

  • $d_{x^2-y^2}$ lies in the xy plane with lobes along the x and y axes.

  • $d_{z^2}$ consists of two lobes along the z-axis and a doughnut-shaped ring in the xy plane.

• f Orbitals ($l=3$):

  • Seven equivalent f orbitals exist for $n \geq 4$. Shapes are more complex than d orbitals.

Electron Spin and the Pauli Exclusion Principle

• Electron Spin: An intrinsic property where electrons behave as if spinning on an axis, generating a magnetic field.

• Spin Magnetic Quantum Number ($m_s$):

  • Has two possible values: $+1/2$ (spin-up, $\uparrow$) and $-1/2$ (spin-down, $\downarrow$).

  • These opposite spins produce two opposite magnetic fields, which lead to the splitting of spectral lines.

• Stern-Gerlach Experiment: Demonstrated that electrons are deflected in two different directions when passing through a magnetic field based on their spin quantum number.

• Additional Definitions:

  • Spin pair energy: Energy required to pair an electron with another in the same orbital.

  • Spin-spin coupling: Communication of nuclear spin information between nuclei (visible in NMR spectroscopy).

• The Pauli Exclusion Principle: States that no two electrons in an atom can have the same exact set of four quantum numbers ($n, l, m_l, m_s$).

  • Consequence: An orbital can hold a maximum of two electrons, and they must have opposite spins.

Electron Configurations and Filing Rules

• Electron Configuration: The distribution of electrons among the various orbitals of an atom.

• Ground State: The most stable configuration where electrons occupy the lowest possible energy states.

• Ordering of Orbital Energies:

  • In a single-electron system (Hydrogen), all orbitals with the same $n$ have the same energy (degenerate).

  • In many-electron atoms, for a given $n$, energy increases with $l$: ns < np < nd < nf.

  • General Filling Order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p < 8s \dots

• Hund’s Rule: Electrons occupy degenerate orbitals (orbitals with the same energy) in a way that maximizes the number of electrons with the same spin. Electrons are placed singly in each orbital before pairing begins.

• Orbital Diagrams: Represented by boxes (orbitals) and half-arrows (electrons). Unpaired electrons are always shown with spin up.

• Condensed Electron Configuration: Uses a noble-gas core abbreviation (the nearest noble gas with a lower atomic number in brackets).

  • Core Electrons: Electrons represented by the noble-gas symbol.

  • Outer-shell / Valence Electrons: Electrons involved in chemical bonding. For elements with atomic number $\leq 30$, all outer-shell electrons are valence electrons.

The Periodic Table and Blocks

• s-block: Groups 1 (alkali metals) and 2 (alkaline earth metals) where valence s orbitals are filled.

• p-block: Groups 13 through 18 where valence p orbitals are filled.

• Representative Elements: The combined s-block and p-block elements (main-group elements).

• d-block: Sections containing transition metals where the five d orbitals are being filled (10 columns).

• f-block: Sections containing lanthanides and actinides where the seven f orbitals are being filled (14 columns).

  • Lanthanides: Rare earth elements (atomic numbers 57–70) where 4f orbitals are partially occupied.

  • Actinides: Radioactive elements in the final row where 7s and 5f orbitals are involved; most are not found in nature.

Questions & Discussion

• Q1: Difference between "located at a particular point" and "high probability of being located at a point"?

  • Quantum mechanics forbids specifying an exact location (Heisenberg Principle); it only allows for the statistical probability of finding an electron in a certain volume (electron density).

• Q2: Difference between an orbit and an orbital?

  • An orbit (Bohr model) is a predefined circular path. An orbital (Quantum model) is a mathematical wave function ($\Psi$) representing a 3D region of space where there is a high probability of finding an electron.

• Q3: For the 4s orbital of the hydrogen atom, how many maxima and nodes?

  • Number of peaks (maxima) = $n = 4$.

  • Number of nodes = $n - 1 = 3$.

• Q5: Orbital diagram for Oxygen ($Z=8$)? How many unpaired electrons?

  • Configuration: $1s^2 2s^2 2p^4$.

  • Diagram: $1s:[\uparrow\downarrow] \, 2s:[\uparrow\downarrow] \, 2p:[\uparrow\downarrow][\uparrow][\uparrow]$.

  • Oxygen has 2 unpaired electrons.

• Q6: Electron configuration for Phosphorus ($Z=15$)?

  • $1s^2 2s^2 2p^6 3s^2 3p^3$ or $[Ne] 3s^2 3p^3$.

• Q7: Characteristic valence configuration of Group 17 (Halogens)?

  • $ns^2 np^5$.

• Q8: Bismuth ($Z=83$) Configuration details:

  • (a) Electron configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10} 5p^6 6s^2 4f^{14} 5d^{10} 6p^3$.

  • (b) Condensed configuration: $[Xe] 6s^2 4f^{14} 5d^{10} 6p^3$.

  • (c) Unpaired electrons: 3 unpaired electrons (in the $6p$ subshell).

• Q9: Cobalt ($Co, Z=27$) Configuration details:

  • (a) Electronic configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^7$.

  • (b) Condensed: $[Ar] 4s^2 3d^7$.

  • (c) Unpaired electrons: 3 unpaired electrons (following Hund's rule for $d^7$: $[\uparrow\downarrow][\uparrow\downarrow][\uparrow][\uparrow][\uparrow]$).

• Q10: Condensed configurations for selected atoms:

  • (a) Cs: $[Xe] 6s^1$

  • (b) Ni: $[Ar] 4s^2 3d^8$

  • (c) Se: $[Ar] 4s^2 3d^{10} 4p^4$

  • (d) Cd: $[Kr] 5s^2 4d^{10}$

  • (e) Pb: $[Xe] 6s^2 4f^{14} 5d^{10} 6p^2$