1.4 Quantum Mechanical Model of Atoms Notes

Quantum Mechanical Model of Atoms

Identify the four quantum numbers (n, l, ml, ms), their potential range of values, and their relationship to the electron they represent.
Compare the orbital diagram for a neutral atom (e.g., sulfur) to an ion (e.g., S²⁻).
Differentiate between paramagnetic and diamagnetic compounds.
Determine the number of valence electrons in a given atom.

Bohr's Model Inadequacy: The Bohr model, while a significant advancement, failed to explain the structure and behavior of atoms with more than one electron due to its failure to account for electron repulsion.
Modern Quantum Mechanics: A more rigorous study of the electronic structure of atoms.
Key Difference:
- Bohr: Electrons follow defined circular pathways (orbits) at fixed distances.
- Quantum Mechanics: Electrons move rapidly and are localized within regions of space called orbitals.

Probability: The quantum mechanical model suggests describing the probability of finding an electron within a region of space.
Uncertainty Principle: It is impossible to simultaneously determine with perfect accuracy both the momentum and the position of an electron.
- Assessing position requires stopping the electron (removing momentum).
- Assessing momentum requires movement (changing position).
Mathematical Representation: The uncertainty principle can be expressed as: (\Delta x \cdot \Delta p \geq \frac{h}{4\pi}), where \Delta x is the uncertainty in position and \Delta p is the uncertainty in momentum.

Four Quantum Numbers: Any electron in an atom is completely described by four quantum numbers:
- n (principal quantum number)
- l (azimuthal quantum number)
- ml (magnetic quantum number)
- ms (spin quantum number)
Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.
Energy State: The position and energy of an electron described by its quantum numbers.
Hierarchy: The value of n limits the values of l, which limits the values of ml.
Information Provided: Quantum numbers provide information about the size, shape, and orientation of orbitals.
MCAT Expertise Analogy: Quantum numbers become more specific (state → city → street → house number).

Definition: The principal quantum number, denoted by n, can take on any positive integer value.
Energy Level/Radius: Larger n implies a higher energy level and radius of the electron shell.
Electron Capacity: The maximum number of electrons within a shell is given by: 2n^2
Energy Difference: The energy difference between shells decreases with increasing distance from the nucleus.
- E{n=3} - E{n=4} < E{n=1} - E{n=2}
Bridge to Physics: Analogous to gravitational potential energy, where higher objects have greater potential energy.

Definition: The azimuthal (angular momentum) quantum number, denoted by l, determines the shape and number of subshells within a principal energy level (shell).
Importance: Significant implications for chemical bonding and bond angles.
Range of Values: For a given n, l ranges from 0 to n-1.
Number of Subshells: The n value also indicates the number of possible subshells.
Key Concept: For principal quantum number N, there will be N possible values for L, ranging from zero to n-1.
Spectroscopic Notation: Shorthand representation of principal and azimuthal quantum numbers.
- l = 0: s subshell
- l = 1: p subshell
- l = 2: d subshell
- l = 3: f subshell
Example: An electron in n = 4 and l = 2 is in the 4d subshell.
Maximum Electrons in Subshell: Given by: 4l + 2
Subshell Energy: The energies of subshells increase with increasing l value, but overlap can occur between different principal energy levels (e.g., 4s < 3d).

Definition: The magnetic quantum number, denoted by ml, specifies the particular orbital within a subshell.
Electron Capacity per Orbital: Each orbital can hold a maximum of two electrons.
Possible Values: ml ranges from -l to +l, including 0.
Examples:
- s subshell (l = 0): ml = 0 (one orbital)
- p subshell (l = 1): ml = -1, 0, +1 (three orbitals)
- d subshell: five orbitals
- f subshell: seven orbitals
Orbital Shapes:
- s orbitals: spherical
- p orbitals: dumbbell-shaped, aligned along x, y, and z axes (px, py, pz)
Key Concept:
- For any l, there will be 2l + 1 possible values for ml.
- For any n, this produces n^2 orbitals.
- For any value of n, there will be a maximum of 2n^2 electrons (two per orbital).
Probability Density: The likelihood of finding an electron in a particular region of space.
Periodic Table Blocks:
- p-block: contains six groups of elements.
- s-block: contains two elements.
- d-block: contains 10 elements.
- f-block: contains 14 elements.

Definition: The spin quantum number, denoted by ms, describes the intrinsic angular momentum of an electron, which is quantized.
Spin Orientations: Electrons have two spin orientations, +1/2 and -1/2.
Paired Electrons: Electrons in the same orbital must have opposite spins (paired).
Parallel Spins: Electrons in different orbitals with the same ms value have parallel spins.

Definition: The pattern by which subshells are filled, and the number of electrons within each principal energy level and subshell.
Spectroscopic Notation:
- Number: Principal energy level (n)
- Letter: Subshell (l)
- Superscript: Number of electrons in that subshell
Example: 2p^4 indicates four electrons in the 2p subshell of the second principal energy level.
Aufbau Principle: Electrons fill subshells from lower to higher energy.
n + l Rule: The lower the sum of n + l, the lower the energy of the subshell. If two subshells have the same n + l, the lower n fills first.
Periodic Table Method: An alternative way to approach electron configurations is through simply reading the periodic table.

Hund's Rule: Orbitals are filled such that there are a maximum number of half-filled orbitals with parallel spins within a given subshell.
Electron Repulsion: Electrons prefer to occupy their own orbital before doubling up, due to electron repulsion.
Stability: Half-filled and fully filled orbitals have lower energies (higher stability).
Exceptions: Chromium (Cr) and copper (Cu) deviate from expected configurations to achieve half-filled or fully filled d subshells.

Paramagnetism: Materials with unpaired electrons are weakly attracted to a magnetic field because unpaired electrons align their spins with the field.
Diamagnetism: Materials with only paired electrons are slightly repelled by a magnetic field.
*Real world example maglev trains use diamagnetic materials to levitate the train.

Definition: Electrons in the outermost energy shell, most easily removed, and available for bonding.
Groups I/II: Only the highest s subshell electrons are valence electrons.
Groups III-VIII: The highest s and p subshell electrons are valence electrons.
Transition Elements: Valence electrons are those in the highest s and p subshells.
Lanthanides/Actinides: Valence electrons are those in the highest s and f subshells.
Period 3 Elements: Can accept electrons into their d subshell, exceeding the octet rule.