Introduction to Quantum Physics, Thermodynamics, and Molecular Interactions

Classical Physics Foundations

Classification of Classical Physics Classical physics is fundamentally divided into the study of matter and the study of radiation.

• Matter (Particles):     * Deals with discrete entities that follow specific trajectories.     * Classical Mechanics: Pioneered by scientists such as Newton, Lagrange, and Hamilton.     * Statistical Mechanics: Pioneered by Boltzmann and others.     * Classical Thermodynamics.

• Radiation (Waves / Fields):     * Deals with continuous phenomena characterized as waves and fields.     * Classical Electrodynamics: Developed by Maxwell, Faraday, Hertz, and others.

Physical Properties: Waves vs. Particles

The Transition to the Quantum World

In the 20th century, the study of atomic systems necessitated a fundamental revision of classical physical ideas. Physical objects began to exhibit dual characteristics.

1. Light Waves as Particles:     * Observed via the photoelectric effect.     * Light impinging on metals causes the instantaneous emission of electrons in a manner similar to billiard ball impacts.     * Practical Application: This phenomenon is the basis for automatic door openers in grocery stores.

2. Electrons (Particles) as Waves:     * Electrons exhibit electron interference, proving they can pass through each other.     * Practical Application: This forms the basis of electron microscopes.

Architecture of the Quantum View This view, which often contradicts "common sense" regarding unique physical definitions, was constructed by:

• Neils Bohr

• Werner Heisenberg

• Max Planck

• Albert Einstein

• Louis de Broglie

• Erwin Schrödinger

• Wolfgang Pauli

• Paul Dirac

Principles of Wave-Particle Duality

The Nature of Observation

• A true understanding of nature requires accepting that physical objects are neither exclusively particles nor exclusively waves.

• No experiment can measure both aspects simultaneously; therefore, a mixture of both is never observed.

• The behavior observed (particle-like or wave-like) depends entirely on the method of observation.

• Physical objects are described by mathematical functions which represent measures of probability.

Matter Waves (Louis de Broglie, 1923) De Broglie proposed that matter particles should exhibit wave properties. While usually difficult to observe due to extremely small wavelengths, this was confirmed by Davisson and G.P. Thomson using high-energy electrons.

• Electrons scattered from crystals show the same patterns as X-rays of similar wavelengths.

• Example: Electron microscope imaging (e.g., a picture of a fly) utilizes these wave properties.

Summary of Duality

• Light:     * Wave nature: EM wave, used in optical microscopes, exhibits interference.     * Particle nature: Photons, converts light to electric current, exhibits the photoelectric effect.

• Particles (e.g., Electrons):     * Wave nature: Matter waves, used in electron microscopes, results in discrete (quantum) states in confined systems like atoms.     * Particle nature: Electric current, observed via photon-electron collisions.

Quantitative Particle and Wave Character

The Photoelectric Effect (Particle Character of Radiation) When a photon ($h u$) acts as a particle-like projectile hitting a metal:

• Conservation of Energy:     $\frac{1}{2}mv^2 = h u - \phi$

• Variables:     * $\frac{1}{2}mv^2 = E_k$: Kinetic energy of the released photoelectron.     * $h u$: Energy supplied by the photon.     * $\phi$ (or work function): Minimum energy required to remove an electron from the metal to infinity.

• Threshold Rule: The threshold for electron emission does not depend on the intensity of incident radiation.

• Applications: Photoelectron spectroscopy (UPS, XPS) is based on this effect.

Work Function Examples:

• Rubidium: $2.09\,eV$ ($1.69 \times 10^4\,cm^{-1}$, $593\,nm$)

• Potassium: $2.25\,eV$ ($1.81 \times 10^4\,cm^{-1}$, $539\,nm$)

• Sodium: $2.30\,eV$ ($1.86 \times 10^4\,cm^{-1}$, $551\,nm$)

Electron Diffraction (Wave Character of Matter) Diffraction is a characteristic property of waves. Bragg showed constructive interference for X-rays when: $\lambda = 2d \sin(\theta)$ Davisson and Germer showed the same interference phenomenon using electrons.

• De Broglie Relation:     $\lambda = \frac{h}{p}$     (where $p = mv$; linking particle momentum to wavelength).

• Wavelength of an Accelerated Electron:     If an electron is accelerated from rest through a potential difference $V$:     $\lambda = \frac{h}{(2m eV)^{1/2}}$

• Experiment: An appropriate potential difference creates an electron beam that diffracts with the lattice of a Nickel crystal.

Mapping Physical Theories

Quantum Biology and Electron Tunneling

Quantum Tunneling

• Definition: The ability of a subatomic particle to travel through potential energy barriers.

• Mechanism: Due to wave-particle duality, particles like electrons and protons can pass through energy barriers without violating physical laws because of their wave characteristics.

• Biological Application: Enzymes and proteins use quantum tunneling to transfer electrons in the Electron Transfer Chain.

Case Study: Ferritin

• Ferritin stores electrons for several hours, reducing $Fe^{3+}$ to water-soluble $Fe^{2+}$.

• The mechanism for transit through the $2\,nm$ thick protein shell is electron tunneling.

• Distances: Single electron tunneling events occur up to $8\,nm$; sequential electron tunneling can occur up to $12\,nm$ through the ferritin structure.

Non-Covalent Interactions in Biological Systems

Biological structures are maintained by various non-covalent interactions:

1. Ionic Bond (Salt Bridge): Occurs between charged groups (e.g., $Asp$ and $Lys/H_3N^+$).

2. Hydrogen Bond (H-bond): Occurs between polar groups (e.g., between two Serine residues).

3. van der Waals Interactions: Weak forces between neutral particles (e.g., between Leucine and Valine residues).

4. Hydrophobic Interaction: Driven by the exclusion of water by nonpolar surfaces.

The Hydrogen Bond (Detailed)

• Bond Angle in Water: $104.5^\circ$.

• Lengths:     * Covalent $O-H$ bond: $0.0965\,nm$.     * Hydrogen bond ($H \cdots O$): $0.177\,nm$.

• Bond Dissociation Energy:     * H-bond: $23\,kJ/mol$.     * $O-H$ covalent bond: $470\,kJ/mol$.

• Donors/Acceptors: Stronger bonds form between highly electronegative atoms (e.g., $O-H \cdots O$ is stronger than $N-H \cdots N$).

Water as a Unique Molecule

• High resistance to boiling/freezing protects organisms in aquatic environments.

• Electronegativity: Oxygen is the second most electronegative element (after Fluorine), creating polar bonds.

• Dipole: The $104.5^\circ$ bond angle creates a very strong dipole.

• Surface Tension: Water has the highest surface tension of all liquids, excluding mercury.

Solvent Effects and the Hydrophobic Effect

Hydration

• Water molecules orient themselves around ions.

• Sodium ($Na^+$): Water oxygens (partial negative) face the cation.

• Chloride ($Cl^-$): Water hydrogens (partial positive) face the anion.

Entropy and Hydrophobicity

• Ordering water around amphipathic compounds is entropically unfavorable.

• Mechanism: Water expels hydrophobic chains to cluster them together to minimize entropy loss.

• Van der Waals interactions between water and hydrocarbon chains are negligible compared to water-water hydrogen bonds.

• Lipid Bilayers: Formation is driven by water maximizing its own hydrogen-bonding entropy, not by attraction between lipid tails.

Thermodynamic breakdown of a single hydrocarbon chain in water:

• $\Delta H$ (Enthalpy): Approximately $0$ or slightly unfavorable; water-water H-bonds are disrupted with no compensating strong interactions.

• $\Delta S$ (Entropy): Strongly negative; water molecules near the nonpolar surface become ordered, reducing freedom.

• \Delta G = \Delta H - T\Delta S > 0 (Unfavorable).

Packing Parameter and Geometry Determines whether lipids form micelles or bilayers: $P = \frac{v}{a_0 l}$

• Where $v$ = tail volume, $a_0$ = headgroup area, $l$ = tail length.

• Single tail (detergents): P < 1/3 \rightarrow Micelles.

• Double tail (phospholipids): 1/2 < P < 1 \rightarrow Bilayers.

Biological Applications of Physical Principles

Membrane Proteins and Hydrophobic Matching

• Transmembrane helices are rich in hydrophobic residues.

• Insertion: Replaces lipid-water interfaces with protein-lipid interfaces, releasing ordered water for an entropy gain.

• Hydrophobic Mismatch: If protein hydrophobic length does not match bilayer thickness, the membrane deforms and protein function can be modulated.

• Note: Cholesterol alters membrane thickness, regulating channel activity.

Enzyme-Substrate Complexes

• The release of ordered water from the active site and substrate surface favors the formation of the complex ($ES$).

• Interaction is stabilized by H-bonding, ionic bonds, and hydrophobic interactions.

Van der Waals Forces and Potential

Classification of vdW Forces Sum of electrostatic forces between neutral particles over nanometer distances:

1. Keesom Force: Interaction between permanent dipoles.

2. Debye Force: Interaction between a permanent dipole and an induced dipole.

3. London (Dispersion) Force: Interaction between non-polar particles.

Lennard-Jones Interatomic Potential $U_{ij} = 4\epsilon \left[ \left( \frac{\sigma}{R} \right)^{12} - \left( \frac{\sigma}{R} \right)^{6} \right]$

• Repulsive term ($R^{12}$): Strong repulsive forces when R < \sigma.

• Attractive term ($R^6$): Weak attractive forces.

• Separation at minimum energy: $R = 1.12\sigma$.

Questions & Discussion

• Assertion/Reason:     * Assertion (A): Non-covalent interactions are individually weak but collectively strong.     * Reason (R): These interactions are additive and act over short distances.

• Question 1: In biological systems, salt bridges are weakened mainly because of:     * Answer: B. High dielectric constant of water.

• Question 2: Which non-covalent interaction is least affected by changes in temperature?     * Answer: D. Covalent interactions (Note: Interactions like hydrophobic effects are highly temperature-dependent).

• Hydrogen Bonding Check:     * Methane can form a hydrogen bond? False.     * Methanol can form a hydrogen bond? True.

Thermodynamics in Biological Systems

Definitions

• Thermodynamics: The study of energy transformations in matter. Focuses on storage, transformation, and dissipation.

• Objectives: To understand heat/work relationships, energy change influences, temperature effects on equilibrium (e.g., bioreactors), and biochemical processes.

Energy Transformations

• Photosynthesis (Plants): Light energy $\rightarrow$ Chemical energy.

• Neurotransmission (Nerve): Chemical energy $\rightarrow$ Electrical energy.

• Vision (Eye): Light energy $\rightarrow$ Electrical energy.

• Muscle Movement: Chemical energy $\rightarrow$ Mechanical energy.

System Types

1. Open: Exchanges both matter and energy with the environment.

2. Closed: Exchanges only energy with the environment.

3. Isolated: Exchanges neither matter nor energy.

Gibbs Free Energy ($\Delta G$) Developed by Willard Gibbs in the 1870s. $\Delta G = \Delta H - T\Delta S$

• $\Delta H$: Enthalpy (drive toward stability).

• $\Delta S$: Entropy (drive toward disorder).

• Predictive Elements:     * $-\Delta G$: Exergonic, spontaneous (favorable).     * $+\Delta G$: Endergonic, non-spontaneous (unfavorable).     * $\Delta G = 0$: Equilibrium.

Spontaneity Table

Second Law of Thermodynamics and Enthalpy

Second Law (Law of Entropy)

• Developed in the 1850s by Rudolf Clausius.

• Statement: A system and its surroundings always proceed toward a state of maximum disorder (entropy).

• Living Systems: Highly ordered, but they increase the entropy of the universe because conversions are not $100\%$ efficient; some heat is always released.

Enthalpy (\Delta H) Defined as the change in heat content or heat of formation. $\Delta H = \Delta U + P\Delta V$ $\Delta U = Q - W$

• $\Delta U$: Internal energy change.

• $Q$: Heat added to the system.

• $W$: Work done by the system.

Biochemical Reaction Comparison

Atomic and Electronic Spectroscopy

Energy and Wavelength Relationship $E = \frac{hc}{\lambda}$ (Where energy increases as wavelength decreases).

• Spectrum Ranges:     * Gamma rays: < 0.01\,nm     * X-rays: $0.01 - 10\,nm$     * Ultraviolet: $10 - 400\,nm$     * Visible: $400\,nm$ (Violet) to $700\,nm$ (Red).     * Infrared: $700\,nm - 0.01\,cm$     * Radio waves: $1\,m - 100\,m+\,$

Atomic Excitation and De-excitation

• Excitation: An atom acquires energy to promote an electron to a higher energy state than the ground state.

• De-excitation: Atom returns to ground state, often by emitting a photon.

Singlet and Triplet States

• Excited triplet states are less energetic than excited singlet states.

• Probability: Singlet-to-triplet (or reverse) transitions are significantly less probable because they involve a change in electron spin.

Molecular Orbital (MO) Transitions

Ground State and Gaps

• HOMO: Highest Occupied Molecular Orbital (typically bonding orbitals).

• LUMO: Lowest Unoccupied Molecular Orbital (typically antibonding orbitals).

• Transition: Light absorption occurs if photon energy equals the HOMO-LUMO energy gap ($\Delta E$).

Types of Transitions

1. $\sigma \rightarrow \sigma^*$: High energy, short wavelength. Seldom observed as bonds often rupture first.

2. $\pi \rightarrow \pi^<em>$: Narrower gaps.      Ethene: Absorbs at $165\,nm$.     * Conjugated Systems: Narrower energy gaps than isolated bonds.     * 1,3-butadiene: Highly conjugated system with multiple bonding/antibonding orbitals ($\pi_1, \pi_2, \pi_3^<em>, \pi_4^</em>$).

3. $n \rightarrow \pi^<em>$: Transition of a lone pair (non-bonding) electron to an antibonding orbital.      4-methyl-3-penten-2-one: Strong UV absorbance at $236\,nm$ ($\pi \rightarrow \pi^<em>$) and a second absorbance at $314\,nm$ ($n \rightarrow \pi^</em>$).

Photoluminescence and Quantum Yield

• Jablonski Diagram: Displays transitions including non-radiative and radiative processes.     * Non-radiative timescales: $10^{-15}$ to $10^{-10}\,s$.     * Radiative timescales (Fluorescence/Phosphorescence): $10^{-5}$ to $10^1\,s$.

• Quantum Yield (\phi):     $\phi = \frac{N_{photons\,Emitted}}{N_{photons\,Absorbed}}$     $\phi = \frac{k_f}{k_f + \sum k_{nr}}$     $\phi = \frac{k_f}{k_f + k_i + k_{ec} + k_{ic} + k_{pd} + k_d}$     (Where $k_f$ = fluorescence, $k_i$ = intersystem crossing, $k_{ec}$ = external conversion, $k_{ic}$ = internal conversion, $k_{pd}$ = predissociation, $k_d$ = dissociation).