Physical and (Bio)Inorganic Chemistry: Atomic Structure, Quantum Theory, and Transition Metals

Atomic Structure and Quantum Theory

The Classical Picture of the Atom

The earliest models of the atom described electrons orbiting the nucleus in a manner analogous to planets orbiting the sun. This model, however, could not explain several experimental observations.

• Electrons were thought to move in fixed orbits around a central nucleus.

• This model is known as the planetary model of the atom.

Failures of Classical Physics: The Photoelectric Effect

Classical wave theory could not explain the photoelectric effect, where light incident on a metal surface causes the emission of electrons.

• Photoelectric Effect: When ultraviolet light (2000–4000 Å) strikes a metal, electrons are emitted with a velocity $v$.

• Key Observation: The energy of emitted electrons depends on the frequency of the light, not its intensity.

• Increasing intensity produces more electrons, but each has the same energy.

Example: Increasing the brightness of monochromatic light increases the number of photoelectrons but not their energy.

Einstein's Quantum Explanation

Einstein proposed that light consists of discrete packets of energy called photons, each with energy $E = nh\nu$, where $\nu$ is the frequency and $n$ is the number of photons.

• Energy Quantization: $E = nh\nu$

• Photon: A quantum of electromagnetic radiation.

The Rutherford Atom

Rutherford's gold foil experiment revealed the existence of a small, dense, positively charged nucleus.

• Most alpha particles passed through the foil undetected.

• A few were deflected at small angles; very few were deflected backward.

• Conclusion: Most of the atom's mass and all positive charge are concentrated in the nucleus.

Bohr Model and Quantized Energy Levels

Bohr introduced the concept of quantized energy levels for electrons in atoms, explaining atomic stability and spectral lines.

• Electrons occupy specific energy levels.

• Energy is quantized; electrons cannot exist between levels.

• Bohr's Formula (for hydrogen): $E_n = -\frac{R_H}{n^2}$, where $R_H$ is the Rydberg constant and $n$ is the principal quantum number.

Wave-Particle Duality and the de Broglie Relationship

Particles such as electrons exhibit both wave-like and particle-like properties.

• de Broglie Equation: $\lambda = \frac{h}{mv}$, where $\lambda$ is wavelength, $h$ is Planck's constant, $m$ is mass, and $v$ is velocity.

• Planck's constant: $h = 6.626 \times 10^{-34}$ Js

Heisenberg's Uncertainty Principle

The position and momentum of an electron cannot both be precisely determined.

• Uncertainty Principle: $\Delta p \Delta x \geq \frac{h}{4\pi}$

• Greater precision in position leads to greater uncertainty in momentum.

The Schrödinger Equation and Atomic Orbitals

The Schrödinger equation describes the behavior of electrons as wavefunctions in atoms.

• Time-independent Schrödinger Equation: $-\frac{h^2}{8\pi^2 m} \nabla^2 \psi + V\psi = E\psi$

• Solutions yield atomic orbitals, which describe the probability of finding an electron at a particular location.

• s-orbitals: Spherical in shape.

• p-orbitals: Dumbbell-shaped.

• d-orbitals: Four-leaf clover shapes.

Atomic Spectra and the Rydberg Formula

Electrons transition between energy levels, emitting or absorbing photons of specific wavelengths, producing atomic spectra.

• Atomic Spectra: Discrete lines corresponding to electron transitions.

• Rydberg Formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

• $R$ is the Rydberg constant ($1.09737 \times 10^7$ m$^{-1}$).

• $n_1$ and $n_2$ are integers representing energy levels.

Covalent Bonding and Molecular Structure

The Covalent Bond

A covalent bond is formed by the sharing of electrons between atoms, resulting in a stable molecule.

• Example: $\mathrm{H}_2$ molecule: $\mathrm{H} : \mathrm{H}$

• Born-Oppenheimer Approximation: Nuclei are treated as stationary while electrons move in their field.

• This allows calculation of molecular potential energy curves as a function of bond length.

Bonding Theories

Bonding involves stabilizing and destabilizing interactions between electrons and nuclei.

• Stabilizing: Electron-nucleus attractions.

• Destabilizing: Electron-electron and nucleus-nucleus repulsions.

Properties of Gases and Gas Laws

Physical Properties of Gases

Gases are characterized by low density, high compressibility, and the ability to expand to fill their containers.

• Most elements that are gases are found toward the upper right of the periodic table.

• Gases can be compressed easily, indicating large intermolecular spaces.

• Gas vs. Vapour: A gas is a substance that is normally gaseous at room temperature; a vapour is the gaseous form of a substance that is liquid or solid at room temperature.

Parameters Describing Gases

• Amount (n): in moles

• Temperature (T): in Kelvin

• Volume (V): in Litres

• Pressure (P): in Atmospheres

Pressure Units

Unit	Value
Pascal (Pa)	1 Pa = 1 N/m2
Bar	1 bar = 105 Pa = 100 kPa
Atmosphere (atm)	1 atm = 1.01325 × 105 Pa = 101.325 kPa = 760 Torr = 14.7 lb/in2

Gas Laws

• Boyle's Law: At constant temperature, the volume of a gas is inversely proportional to its pressure.

  • $P \propto \frac{1}{V}$

  • $P_1V_1 = P_2V_2$

• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature.

  • $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.

  • $V \propto n$

• Ideal Gas Law: Combines the above laws into one equation.

  • $PV = nRT$

  • $R = 0.0821$ L·atm·K−1·mol−1

Transition Metals and Coordination Chemistry

Transition Metals in the Periodic Table

Transition metals are found in the d-block of the periodic table and are characterized by partially filled d-orbitals.

• First row: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn

• General electron configuration: $4s^2 3d^n$

• Exceptions: Chromium and copper have unique configurations due to stability of half-filled and fully-filled d-orbitals.

Electron Configurations and Ion Formation

• For neutral atoms, 4s fills before 3d.

• For +2 ions, electrons are removed from 4s before 3d.

• Removal of s-electrons increases effective nuclear charge felt by d-electrons, stabilizing the ion.

Physical Properties of Transition Metals

• Dense, malleable, ductile, shiny

• Excellent conductors of heat and electricity

• High melting and boiling points

• Variable oxidation states, useful in catalysis and biological systems

Biological Importance of Transition Metals

• Iron: Essential for oxygen transport in hemoglobin and myoglobin

• Copper: Important in electron transport and oxygen-carrying proteins (e.g., hemocyanin)

• Zinc, Cadmium, Mercury: Group 12 elements, lose only s-electrons, not considered true transition metals

Transition Metal Complexes and Coordination Chemistry

Transition metals form complexes with ligands, which are electron pair donors (Lewis bases).

• Coordination Number: Number of ligands directly attached to the metal ion

• Ligands: Can be monodentate (one binding site) or polydentate (multiple binding sites)

• Common Geometries: Octahedral (6 ligands), tetrahedral (4 ligands), square planar (4 ligands)

Table: Common Coordination Numbers and Geometries

Coordination Number	Geometry
6	Octahedral
4	Tetrahedral or Square Planar

Crystal Field Theory

The arrangement of ligands affects the energy of d-orbitals, leading to splitting and color changes in complexes.

• Strong field ligands (e.g., CN−, CO) cause large splitting, favoring square planar geometry.

• Weak field ligands (e.g., Cl−, Br−, I−) favor tetrahedral geometry.

• Color arises from d-d electron transitions, which depend on ligand type and geometry.

Examples of Transition Metal Complexes

• [Fe(CN)6]4−: Six cyanide ligands, Fe center

• [Cr(C2O4)2(H2O)2]−: Oxalate and water ligands, Cr3+ center

• [Ni(CO)4]: Carbon monoxide ligands, Ni center

Additional info: These notes cover foundational concepts in atomic structure, quantum theory, gas laws, and transition metal chemistry, which are essential for understanding physical and inorganic chemistry at the college level.