Physical and (Bio)Inorganic Chemistry: Quantum Theory, Atomic Structure, Bonding, and Properties of Gases

Quantum Theory and Atomic Structure

The Classical Picture of the Atom

Early models of the atom described electrons as orbiting the nucleus in a manner analogous to planets orbiting the sun. This classical view, however, failed to explain several experimental observations.

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

• This model could not account for the stability of atoms or the discrete nature of atomic spectra.

Failures of Classical Physics: The Photoelectric Effect

The photoelectric effect demonstrated the inadequacy of classical wave theory in explaining atomic phenomena.

• When ultraviolet light (2000–4000 Å) strikes a metallic surface, electrons are emitted with a specific velocity (v).

• The energy of ejected electrons depends on the frequency of incident light, not its intensity.

• Higher intensity produces more photoelectrons, but each has the same energy.

• This contradicted classical wave theory, which predicted that higher intensity should increase electron energy.

Einstein and Quantization of Light

Einstein explained the photoelectric effect by proposing that light consists of discrete packets of energy called photons.

• Energy of radiation: $E = nh u$

• n is the number of photons, ν is frequency, h is Planck's constant.

• Radiation is quantized in units of $h u$.

The Rutherford Atom

Rutherford's gold foil experiment revealed the nuclear structure of the atom.

• Most alpha particles passed through the foil; a few were deflected at large angles.

• Conclusion: Most mass and all positive charge are concentrated in a small nucleus, surrounded by electrons.

• This led to the nuclear atom model.

Bohr Model and Quantized Energy Levels

Niels Bohr introduced the concept of quantized energy levels for electrons in atoms.

• Electrons are restricted to specific energy values.

• Energy levels for hydrogen atom: $E_n = -\frac{e^4 M_e}{8 \varepsilon_0^2 h^2 n^2}$ $E_n = -\frac{k}{n^2}$

• This model explained atomic spectra and stability of atoms.

Wave-Particle Duality

Both light and electrons exhibit wave and particle properties.

• Energy of a photon: $E = h\nu$

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

• de Broglie wavelength: $\lambda = \frac{h}{mv}$

Heisenberg's Uncertainty Principle

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

• $\Delta p \Delta x = \frac{h}{4\pi}$

• This principle is fundamental to quantum mechanics.

Wavefunctions and Probability

Electrons are described by wavefunctions (ψ), which give the probability of finding an electron at a particular location.

• The probability is given by $\psi^2$.

The Schrödinger Equation

The behavior of electrons in atoms is described by the Schrödinger equation.

• Time-independent form for hydrogen atom: $-\frac{h^2}{8\pi^2 m} \nabla^2 \psi - \frac{e^2}{4\pi \varepsilon_0 r} \psi = E \psi$

• Solutions yield atomic orbitals, which describe electron probability distributions.

Atomic Orbitals

• s-orbitals: Spherical in shape.

• p-orbitals: Dumbbell-shaped.

• d-orbitals: Four-lobed (cloverleaf) shapes.

Atomic Spectra

Electrons transition between discrete energy levels, emitting or absorbing photons of specific wavelengths.

• Atomic spectra consist of emission, absorption, and continuous spectra.

• Rydberg formula for spectral lines: $\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ Where $R_H$ is the Rydberg constant ($1.09737 \times 10^7$ m-1), $n_2$ is the upper energy level, $n_1$ is the lower energy level.

• Spectral series (Lyman, Balmer, Paschen, Brackett) are important in spectroscopy.

Covalent Bonding and Molecular Structure

The Covalent Bond

A covalent bond is formed by the sharing of electrons between atoms.

• Example: Formation of H2 molecule from two hydrogen atoms.

• Stability arises from electron sharing and energy minimization.

The Born-Oppenheimer Approximation

This approximation treats nuclei as stationary while solving for electron wavefunctions.

• Valid because nuclei are much heavier and move more slowly than electrons.

• Allows calculation of molecular potential energy curves by varying internuclear separation.

• Equilibrium bond length is identified from the minimum of the potential energy curve.

Bonding Theories

Bond formation involves stabilizing and destabilizing interactions.

• Stabilizing: Electron-nucleus attractions.

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

• Overlap of atomic orbitals leads to bond formation (e.g., H2).

Properties of Gases

Physical Properties and Composition of Air

Gases are characterized by their ability to be compressed and their molecular spacing.

• Composition of dry air at sea level:

Constituent	Molar mass (g/mol)	Volume (%)	Mass (%)
Nitrogen, N2	28.02	78.09	75.52
Oxygen, O2	32.00	20.95	23.14
Argon, Ar	39.95	0.93	1.29
Carbon dioxide, CO2	44.01	0.03	0.05

Gases vs. Vapours

• Gas: Substance that is gaseous at room temperature and pressure (e.g., oxygen).

• Vapour: Gaseous form of a substance that is liquid or solid at room temperature (e.g., water vapour).

Parameters Describing Gases

• Amount (n, moles)

• Temperature (T, Kelvin)

• Volume (V, Litres)

• Pressure (P, Atmospheres)

Pressure and Measurement

• Pressure arises from molecular collisions with container walls.

• Measured using a barometer (mercury column height proportional to pressure).

• SI unit: Pascal (Pa); 1 atm = 101325 Pa = 760 Torr = 1.01325 bar

Gas Laws

Boyle's Law

Volume of a confined gas is inversely proportional to pressure at constant temperature.

• $P \propto \frac{1}{V}$

• $P_1 V_1 = P_2 V_2$

• Example: Compressing a gas from 4 L at 4 atm to 1 L requires 16 atm.

Charles's Law

Volume of a gas is directly proportional to absolute temperature at constant pressure.

• $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Example: Balloon volume changes with temperature.

Avogadro's Law

Volume of a gas is directly proportional to the number of moles at constant temperature and pressure.

• $V \propto n$

• $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

Combined Gas Law and Ideal Gas Law

• Combining the three laws: $V \propto \frac{nT}{P}$

• Ideal Gas Law: $PV = nRT$

• Gas constant $R = 0.082$ L·atm·K-1·mol-1

• At STP (0°C, 1 atm), 1 mol of gas occupies 22.414 L (standard molar volume).

Transition Metals and Coordination Chemistry

Transition Metals in the Periodic Table

Transition metals are found between the s-block and p-block elements and exhibit unique properties due to their d-orbitals.

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

• Electron configurations: 4s orbitals fill before 3d, with exceptions (Cr, Cu).

• Transition metal ions often lose s-electrons first, resulting in stable +2 oxidation states.

Physical Properties of Transition Metals

• Malleable, ductile, lustrous

• High melting and boiling points

• Excellent conductors of heat and electricity

• Variable oxidation states (important for catalysis)

Biological Importance of Transition Metals

• Iron: Essential for haemoglobin and myoglobin; involved in oxygen transport and storage.

• Copper: Important in biological systems (e.g., haemocyanin in molluscs and arthropods).

• Zinc, Cadmium, Mercury: Group 12 elements; do not lose d-electrons, not classified as transition metals.

Transition Metal Complexes

Transition metals form complexes with ligands via coordinate (dative covalent) bonds.

• Lewis acids: Metal ions with empty orbitals

• Ligands: Electron pair donors (Lewis bases)

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

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

Geometry of Complexes

• Common coordination numbers: 4 (tetrahedral or square planar), 6 (octahedral)

• Geometry depends on ligand type and orbital interactions

d-Orbital Splitting

• Ligand interactions cause splitting of d-orbital energies

• Octahedral: $d_{x^2-y^2}$ and $d_{z^2}$ at higher energy; $d_{xy}$, $d_{xz}$, $d_{yz}$ at lower energy

• Tetrahedral: Opposite splitting pattern

• Square planar: $d_{x^2-y^2}$ at much higher energy

Colour and Magnetism

• Colour arises from d-d electron transitions, which depend on ligand type

• Complexes may exhibit different magnetic properties

Examples of Complexes

• [Fe(CN)6]4-: Six CN- ligands, Fe2+ center

• [Cr(C2O4)3(H2O)3]: Oxalate and water ligands, Cr3+ center

• [Ni(CO)4]: Carbon monoxide ligands, Ni0 center

Summary

Quantum theory revolutionized our understanding of atomic structure, bonding, and the properties of matter. Transition metals and their complexes play vital roles in chemistry and biology, exhibiting unique electronic, magnetic, and catalytic properties.

Additional info: Some context and explanations have been expanded for clarity and completeness, including definitions, equations, and examples relevant to undergraduate chemistry.