Quantum Mechanical Model of Atoms

Bohr's model limitations:
- Inadequate for atoms with more than one electron.
- Did not account for electron repulsion.
Modern quantum mechanics:
- Provides a more rigorous study of electronic structure.
- Electrons are localized within orbitals, not fixed orbits.
- Describes the probability of finding an electron in a region of space.
- Heisenberg uncertainty principle: It is impossible to simultaneously determine with perfect accuracy the momentum and the position of an electron.

Quantum Numbers

Four quantum numbers completely describe any electron in an atom: $n$ , $l$ , $m<em>l$ , and $m</em>s$ .
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.
Value dependencies: $n$ limits $l$ , which limits $m_l$ .
Qualitative information: Quantum numbers give information about the size, shape, and orientation of the orbitals.

Principal Quantum Number

Denoted by $n$ .
Can take on any positive integer value.
Larger $n$ means higher energy level and radius of the electron shell.
Maximum number of electrons within a shell: $2n^2$
Energy difference between shells decreases as distance from the nucleus increases.
Energy difference is a function of $\frac{1}{n<em>i^2} - \frac{1}{n</em>f^2}$ .
- Example: Energy difference between $n = 3$ and $n = 4$ : $\frac{1}{9} - \frac{1}{16}$ , which is less than the energy difference between $n = 1$ and $n = 2$ : $1 - \frac{1}{4}$ .

Azimuthal Quantum Number

Denoted by $l$ .
Refers to the shape and number of subshells within a given principal energy level (shell).
Important for chemical bonding and bond angles.
Value of $n$ limits the value of $l$ :
- For a given $n$ , possible values for $l$ range from 0 to $n - 1$ .
- Example: If $n = 1$ , $l = 0$ . If $n = 2$ , $l = 0$ or 1.
- The $n$ value also indicates the number of possible subshells.

Spectroscopic Notation

Shorthand representation of principal and azimuthal quantum numbers.
- $l = 0$ is called s.
- $l = 1$ is called p.
- $l = 2$ is called d.
- $l = 3$ is called f.
Example: An electron in shell $n = 4$ and subshell $l = 2$ is in the 4d subshell.
Maximum number of electrons within a subshell: $4l + 2$
Energies of subshells increase with increasing $l$ value.
- Subshells from different principal energy levels may overlap (e.g., 4s has lower energy than 3d).

Magnetic Quantum Number

Denoted by $m_l$ .
Specifies the particular orbital within a subshell.
Each orbital can hold a maximum of two electrons.
Possible values of $m_l$ are integers between $-l$ and $+l$ , including zero.
- $l = 0$ (s subshell) limits $m_l$ to 0 (one orbital).
- $l = 1$ (p subshell) limits $m_l$ to -1, 0, +1 (three orbitals).
- d subshell has five orbitals ( $-2$ to $+2$ ).
- f subshell has seven orbitals ( $-3$ to $+3$ ).
Shape of orbitals depends on the subshell:
- s orbitals are spherical.
- p orbitals are dumbbell-shaped and align along the x, y, and z axes (px, py, pz).
Shapes of d and f orbitals are more complex.
2p Block and Periodic Table:
- 2p contains three orbitals.
- Each orbital contains max of 2 electrons, then 6 electrons can be added during the course of filling the 2p orbitals.
- P block contains six groups of elements.
  - S block contains 2 elements.
  - D block contains 10 elements.
  - F block contains 14 elements.

Spin Quantum Number

Denoted by $m_s$ .
Electron has two spin orientations: $+\frac{1}{2}$ and $-\frac{1}{2}$ .
Electrons in the same orbital must have opposite spins (paired).
Electrons in different orbitals with the same $m_s$ values have parallel spins.

Electron Configuration

Pattern by which subshells are filled.
Spectroscopic notation: $\text{number} \rightarrow \text{principal energy level}$ , $\text{letter} \rightarrow \text{subshell}$ , $\text{superscript} \rightarrow \text{number of electrons in that subshell}$ .
- Example: 2p4 indicates four electrons in the p subshell of the second principal energy level.
Aufbau principle (building-up principle): Electrons fill from lower to higher energy subshells.
n + l rule: The lower the sum of $n + l$ , the lower the energy of the subshell. If two subshells have the same $n + l$ value, the subshell with the lower $n$ value has lower energy.
*Example:
* 5d: $n=5$ , $l=2$ , $n+l = 7$
* 6s: $n=6$ , $l = 0$ , $n+l = 6$ . Thus the 6s subshell has lower energy and will fill.
Periodic table reading: The lowest s subshell is 1s, the lowest p subshell is 2p, the lowest d subshell is 3d, and the lowest f subshell is 4f.
Abbreviated configurations: Using noble gases in brackets.
*Example:
* What is the electron configuration of osmium? $Z = 76$
* The noble gas that comes just before osmium is xenon. $Z = 54$ .
* Therefore the configuration begins with Xe.
* Continuing across the periodic table, we pass through the 6s subshell, cesium and barium, the 4f subshell (the lindenide series, and into the 5d subshell).
* Osmium's configuration is: $[Xe]6s^24f^{14}5d^6$
Ions:
- Negatively charged ions (anions) gain electrons that fill according to the same rules.
- Positively charged ions (cations) lose electrons starting with the subshells with the highest $n$ value. If multiple subshells have the same $n$ , electrons are removed from the subshell with the highest $l$ value.
  *Example:
  * What is the electron configuration of $Fe^{3+}$ ?
  * The electron configuration of iron is $[Ar]4s^23d^6$ .
  * Electrons are removed from the 4s subshell before the 3d subshell because it has a higher principal quantum number.
  * Therefore, $Fe^{3+}$ has a configuration of $[Ar]3d^5$ not $[Ar]4s^23d^3$ .

Hund's Rule

In subshells with multiple orbitals (e.g., 2p), orbitals are filled to maximize the number of half-filled orbitals with parallel spins.
Electrons prefer to occupy their own orbital before doubling up due to electron repulsion.
Half-filled and fully filled orbitals have lower energies (higher stability).
Exceptions to electron configuration:
- Chromium (Cr) and other elements in its group: $[Ar]4s^13d^5$ instead of $[Ar]4s^23d^4$ .
- Copper (Cu) and other elements in its group: $[Ar]4s^13d^{10}$ instead of $[Ar]4s^23d^9$ .
Similar shifts can be seen for f subshells but not for p subshells.

Magnetic Properties

Paramagnetic materials: Atoms with unpaired electrons are weakly attracted to a magnetic field.
Diamagnetic materials: Atoms with only paired electrons are slightly repelled by a magnetic field.

Valence Electrons

Electrons in the outermost energy shell, most easily removed, and available for bonding.
Elements in groups 1 and 2: Only the highest s subshell electrons are valence electrons.
Elements in groups 13 through 18: The highest s and p subshell electrons are valence electrons.
Transition elements: The highest s and d subshell electrons are valence electrons.
Lanthanide and actinide series: The highest s and f subshell electrons are valence electrons.
Elements in period 3 and below can accept electrons into their 3d subshell, allowing them to violate the octet rule.
*Example:
* Elemental vanadium: 5 valence electrons (2 in 4s, 3 in 3d).
* Elemental selenium: 6 valence electrons (2 in 4s, 4 in 4p).
* Sulfate ion: 12 valence electrons. The other four electrons have entered the sulfur atom's three d subshell, which is normally empty in elemental sulfur.