Quantum Mechanics, Electron Configurations, and Periodic Trends

Historical Context and Theoretical Origins:
- The properties of electrons discussed do not originate from the work of Schrödinger or Heisenberg. They involve a different system that incorporates Einstein's work.
- Quantum mechanics is a probabilistic theory. While Einstein famously rejected this, stating, "God doesn't play dice," he was eventually proven wrong by experimental evidence. Quantum mechanics is currently considered the most successful physical theory established.
The Property of Spin:
- Spin is a fundamental property that is technically described as a magnetic moment or angular momentum. There is no literal spinning occurring, as electrons are considered point particles.
- The mathematics for spin are well-understood, but its physical nature remains unintuitive.
- Matter Particles (Fermions): Electrons and other matter particles have a spin of $\frac{1}{2}$ or $-\frac{1}{2}$ .
- Rotational Symmetry: In a mathematical sense (analogous to a "movie script"), an electron must "go around twice" (rotate $720^{\circ}$ ) to return to its original state. This is contrasted with photons or bosons (force-mediating particles), which return to the same place with one full turn (spin of $1$ , $0$ , or $-1$ ).

The Principle Explained: No two electrons can possess the exact same quantum mechanical properties (the same quantum state) if they occupy the same space.
Physical Repulsion: This principle is the reason physical objects do not pass through each other. When you "touch" something, the electrons on the periphery of your atoms repel the electrons of the other object. Technically, at a quantum level, you never actually "touch" another person; your electrons simply repel theirs.
Orbital Occupation: Each orbital can occupy a maximum of only two electrons. To exist in the same orbital, these electrons must have opposite spins, categorized as spin up ( $\uparrow$ ) and spin down ( $\downarrow$ ).

Definition: The Aufbau Principle (from the German for "building up") dictates that electrons fill the lowest energy levels first before moving to higher levels.
Orbital Sequence and Capacity:
- $n=1$ : Contains the $1s$ orbital (capacity: $2$ electrons).
- $n=2$ : Contains $2s$ (capacity: $2$ ) and three $2p$ orbitals (total capacity: $6$ electrons).
- $n=3$ : Contains $3s$ , $3p$ , and $3d$ orbitals.
- $n=4$ : Contains $4s$ , $4p$ , $4d$ , and $4f$ orbitals.
- The sequence generally stops being predictive in nature after the $4f$ subshell.
Subshell capacities and Periodic Table Blocks:
- $s$ -block: 1 orbital, $2$ electrons, corresponds to $2$ columns on the periodic table.
- $p$ -block: 3 orbitals, $6$ electrons, corresponds to $6$ columns on the periodic table.
- $d$ -block: 5 orbitals, $10$ electrons, corresponds to $10$ columns on the periodic table.
- $f$ -block: 7 orbitals, $14$ electrons, corresponds to $14$ columns on the periodic table.

Mapping Schema (The Triangular System/Madelung Rule):
- To determine the order of energetic filling, use the diagonal arrow chart:
- $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s$
- Note the energy crossover: The $4s$ orbital is slightly lower in energy than the $3d$ orbital because its "penetration" to the nucleus is marginally more effective.
Specific Examples:
- Helium ( $He$ ): $\text{Proton count } = 2$ . Configuration: $1s^2$ .
- Lithium ( $Li$ ): $\text{Proton count } = 3$ . Configuration: $1s^2\,2s^1$ .
- Fluorine ( $F$ ): $\text{Proton count } = 9$ . Configuration: $1s^2\,2s^2\,2p^5$ .
- Chlorine ( $Cl$ ): $\text{Proton count } = 17$ . Configuration: $1s^2\,2s^2\,2p^6\,3s^2\,3p^5$ .
- Titanium ( $Ti$ ): $\text{Proton count } = 22$ . Configuration: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^2$ .
- Gallium ( $Ga$ ): $\text{Proton count } = 31$ . Configuration: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^{10}\,4p^1$ .
Valence Similarity: Elements such as Fluorine and Chlorine behave similarly because they share essentially the same outer electron configuration (e.g., $s^2\,p^5$ ).

Coulombic Attraction Formula: $E = k \frac{Q_1 \times Q_2}{d}$ , where $Q$ represents charge and $d$ represents distance. Higher charges (more protons) increase the strength of interaction.
Horizontal Trend (Rows/Periods):
- As you move to the right across a row, the principal quantum number ( $n$ ) remains the same, but the number of protons increases.
- This higher nuclear charge pulls electrons closer, making the atoms smaller.
Vertical Trend (Columns/Groups):
- As you move down a column, the principal quantum number ( $n$ ) increases, creating a larger electron shell/space.
- Atoms become significantly larger as you move down.
Comparative Influence: A change in column has a much more significant effect on atomic size than a change in row.
Extremes:
- Largest Atom: Francium ( $Fr$ ) (bottom-left of the table).
- Smallest Atom: Helium ( $He$ ) (top-right of the table).
Ease of Ionization: Smaller atoms hold onto their electrons more tightly, making them harder to remove.

Chemical Substitutions: Elements in the same group can often mimic each other in biological processes due to their similar electronic configurations.
The Chernobyl Case Study:
- Following the Chernobyl disaster, radioactive Strontium ( $Sr$ ) was released.
- Because Strontium is in the same column as Calcium ( $Ca$ ), it possesses a similar electronic structure.
- Cattle ingested Strontium as if it were Calcium, leading to its presence in dairy and milk products, which posed a significant health hazard.

Audience Question: "What are you describing exactly? Like layers of… ?"
- Response: The professor explains they are like shells or "nesting dolls" (citing the term "Russian dolls"). Electrons fill these shells starting small and building outward.
Audience Question: "How do I know what I'm adding up? The superscript?"
- Response: Yes, the superscript represents the number of electrons in the orbital. In a neutral atom, these should add up to equal the number of protons (atomic number).
Audience Question: "How do we know how an electron goes into an orbital?"
- Response: This is determined by the electron configuration rules (Aufbau, Pauli, and spin). The periodic table layout (e.g., Magnesium and Calcium being in the same column) confirms these rules experimentally.
Audience Question: "Why is there no $6d$ ?"
- Response: The professor notes this is where complications arise, specifically regarding the Lanthanides and Actinides (the $f$ -blocks). Elements like Uranium ( $U$ ) marks the end of the naturally occurring elements we typically study in this context. These more complex structures will be addressed tomorrow.