Graduate
  1. Physical Chemistry  1.1. Thermodynamics  1.1.1. Laws of Thermodynamics  1.1.2. Enthalpy and heat capacity  1.1.3. Entropy and free energy  1.1.4. Phase Equilibrium  1.1.5. Statistical thermodynamics  1.1.6. Thermodynamic cycle  1.1.7. Fugacity and activity  1.2. Quantum Chemistry  1.2.1. Principles of quantum mechanics  1.2.2. Schrödinger Equation  1.2.3. Particle in a box  1.2.4. Operators and eigenvalues  1.2.5. Atomic orbitals  1.2.6. Molecular orbital theory  1.2.7. Perturbation theory  1.2.8. Valence bond theory  1.2.9. Hybridization and chemical bonding  1.3. Chemical kinetics  1.3.1. Rate laws and reaction mechanisms  1.3.2. Collision theory  1.3.3. Transition state theory  1.3.4. Enzyme Kinetics  1.3.5. Chain Reactions and Polymerization  1.3.6. Photochemical reactions  1.4. Statistical mechanics  1.4.1. Partitioning functions  1.4.2. Molecular distribution functions  1.4.3. Boltzmann Distribution  1.4.4. Bose–Einstein and Fermi–Dirac statistics  1.5. Spectroscopy  1.5.1. Rotational Spectroscopy  1.5.2. Vibrational Spectroscopy  1.5.3. Electronic Spectroscopy  1.5.4. Nuclear magnetic resonance spectroscopy  1.5.5. Mass Spectrometry  1.5.6. Raman Spectroscopy  1.5.7. Electron paramagnetic resonance spectroscopy  1.6. Surface and colloidal chemistry  1.6.1. Absorption isotherm  1.6.2. Surface tension and wetting  1.6.3. Colloidal Stability  1.6.4. Catalysis  1.6.5. Emulsions and micelles  1.7. Electrochemistry  1.7.1. Nernst equation  1.7.2. Electrochemical cells  1.7.3. Conductivity and mobility  1.7.4. War  1.7.5. Fuel cells  1.7.6. Electrolysis
  2. Organic chemistry  2.1. Reaction mechanism  2.1.1. Nucleophilic substitution reactions  2.1.2. Electrophilic addition reactions  2.1.3. Elimination reactions  2.1.4. Rearrangement reactions  2.1.5. Radical reactions  2.1.6. Pericyclic reactions  2.2. Spectroscopy and structural determination  2.2.1. UV-Vis Spectroscopy  2.2.2. IR Spectroscopy  2.2.3. NMR Spectroscopy  2.2.4. Mass Spectrometry  2.2.5. X-ray crystallography  2.3. Stereoscopic  2.3.1. Chirality and optical activity  2.3.2. Structural analysis  2.3.3. Geometrical isomerism  2.3.4. Dynamic Stereochemistry  2.4. Organometallic Chemistry  2.4.1. Organolithium and organomagnesium reagents  2.4.2. Palladium-catalyzed cross-coupling reactions  2.4.3. Transition Metal Complexes  2.4.4. Metal–carbon bond  2.5. Polymer chemistry  2.5.1. Polymerization mechanism  2.5.2. Polymer Characteristics  2.5.3. Biodegradable Polymer  2.5.4. Conducting polymer  2.5.5. Supramolecular polymers  2.6. Medicinal chemistry  2.6.1. Drug design and development  2.6.2. Pharmacokinetics and pharmacodynamics  2.6.3. Structure-activity relationships  2.6.4. Molecular docking and drug screening
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Crystal field theory  3.1.2. Ligand field theory  3.1.3. Spectrochemical Series  3.1.4. Chelation and stability  3.2. Organometallic Chemistry  3.2.1. Metal Carbonyls  3.2.2. Catalysis by organometallic complexes  3.2.3. Metallocenes  3.3. Bio-inorganic chemistry  3.3.1. Metalloproteins and enzymes  3.3.2. The role of metals in biological systems  3.3.3. Metal ion transport and storage  3.4. Solid state chemistry  3.4.1. Crystal Structures  3.4.2. Band theory of solids  3.4.3. Superconductors  3.4.4. Defects in the crystal  3.5. Lanthanides and Actinides  3.5.1. Electronic configuration in lanthanides and actinides  3.5.2. Coordination chemistry of the lanthanides  3.5.3. Magnetic Properties of Lanthanides
  4. Analytical chemistry  4.1. Chromatography  4.1.1. Gas Chromatography  4.1.2. High Performance Liquid Chromatography  4.1.3. Thin Layer Chromatography  4.2. Spectroscopic Techniques  4.2.1. Atomic absorption spectroscopy  4.2.2. X-ray diffraction  4.2.3. Inductively coupled plasma spectroscopy  4.3. Electroanalytical methods  4.3.1. Potentiometry  4.3.2. Voltammetry  4.3.3. Colometry  4.4. Mass Spectrometry  4.4.1. Ionization Technique  4.4.2. Fragmentation Pattern  4.5. Chemometrics  4.5.1. Multidisciplinary analysis  4.5.2. Machine Learning in Chemistry
  5. Theoretical and Computational Chemistry  5.1. Molecular dynamics simulation  5.1.1. Force fields and energy minimization  5.1.2. Monte Carlo simulation in molecular dynamics simulations  5.2. Quantum chemical methods  5.2.1. Hartree–Fock theory  5.2.2. Density functional theory  5.2.3. Semi-empirical methods  5.3. Computational drug design  5.3.1. Molecular Docking  5.3.2. QSAR Modeling  5.3.3. Virtual Screening
  6. Biochemistry  6.1. Enzyme Kinetics  6.1.1. Michaelis-Menten kinetics  6.1.2. Inhibition mechanism  6.2. Metabolism and Bioenergetics  6.2.1. Glycolysis  6.2.2. Citric acid cycle  6.2.3. Electron transport chain  6.3. molecular Biology  6.3.1. DNA replication and repair  6.3.2. Protein synthesis  6.3.3. Gene Regulation  6.4. Structural Biochemistry  6.4.1. Protein Folding  6.4.2. Membrane Biophysics
  7. Environmental Chemistry  7.1. Atmospheric Chemistry  7.1.1. Greenhouse gases  7.1.2. Ozone depletion  7.1.3. Air pollutants and smoke  7.2. Water Chemistry  7.2.1. pH and water hardness  7.2.2. Wastewater treatment  7.2.3. Aquatic Chemistry  7.3. Soil Chemistry  7.3.1. Heavy metal contamination  7.3.2. Soil pH and Buffering  7.3.3. Nutrient cycling  7.4. Toxicology and Chemical Safety  7.4.1. Toxicokinetics  7.4.2. Risk Assessment in Toxicology and Chemical Safety in Environmental Chemistry  7.4.3. Effects of pollutants on the environment

Graduate → Physical Chemistry ↓


Statistical mechanics


Statistical mechanics is a deep and wide field in physical chemistry that provides a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. It serves as a deep link that connects the laws of thermodynamics with the atomic-scale phenomena described by quantum mechanics and classical physics.

At its core, statistical mechanics aims to predict the properties and behaviour of bulk matter based on the properties of its constituent particles. It does this by using statistical methods to handle the complex and often chaotic movements and interactions of innumerable particles.

Basic concepts of statistical mechanics

Statistical mechanics involves analyzing systems with a large number of components through statistical methods. It mainly revolves around system microstates, macrostates, and their associated probabilities.

Microstates and macrostates

A microstate is a specific detailed configuration of a system, meaning how each molecule within the system is located and moving at any given time. In contrast, a macrostate is a description of the overall properties of a system, without considering its individual particles.

Consider the example of a gas in a container, which is a classical system in statistical mechanics. The macrostate of the gas can be characterized by overall properties such as pressure, volume, and temperature. However, it is composed of many microstates, each of which specifies the exact position and velocity of each molecule of the gas.

Visual representation of microstates and macrostates

Macrostate Micro

Groups in statistical mechanics

In statistical mechanics, an ensemble is a large collection of hypothetical copies of a system, considered together. Each copy represents a different possible microscopic state of the system. The ensemble approach allows a clear understanding of equilibrium and non-equilibrium processes.

There are three main types of groups:

  • Microcanonical ensemble: It represents an isolated system with fixed energy, volume and number of particles.
  • Canonical ensemble: this considers a system in thermal equilibrium with a heat bath at a constant temperature, allowing energy exchange but not particle exchange.
  • Grand canonical ensemble: Here, both energy and particles can be exchanged with a reservoir, allowing changes in the number of particles and energy levels.

Entropy and the second law of thermodynamics

A fundamental concept in statistical mechanics is entropy. Entropy is a measure of the number of distinct ways in which a thermodynamic system can be arranged, often understood as a measure of disorder or randomness in a system. In statistical mechanics, entropy is widely described in terms of probability.

Mathematically, the entropy S can be defined as the number of microstates W available in the system:

        S = k_b log(w)

where k_B is the Boltzmann constant.

Applications of statistical mechanics

Statistical mechanics is used to explain and predict a wide variety of phenomena, including phase transitions, chemical equilibria, and thermal potentials.

Phase transition

Phase transitions, such as the change of water from a liquid to a gas, occur when a substance changes its state. Statistical mechanics explains such changes in terms of the behavior and interactions of particles at the microscopic level.

Chemical equilibrium

Statistical mechanics sheds light on chemical equilibrium, a state where the macroscopic properties of a chemical reaction remain unchanged over time. It does this by evaluating the probabilities of different reaction pathways and using the concept of probability distributions in different microscopic states.

Thermal capacity

The heat capacity of a system can be understood using statistical mechanics, which provides information about how energy is distributed among the particles of a system and how this affects heat absorption or emission.

Boltzmann distribution

The Boltzmann distribution quantitatively describes the distribution of particles between different energy states in thermal equilibrium. It provides a probability distribution function, which defines the probability of a particle occupying a certain energy state at a given temperature.

The probability P_i of a system being in a particular energy state E_i is given by the following expression:

        P_i = (e^(-E_i/k_B T)) / Z

where Z is the partition function, summing over all possible cases:

        ∆Z=∆e^(-E_i/k_B T)

Example: Partition function calculation

For a simple two-level system with energies 0 and ε, the partition function is calculated as:

        Z = e^(0/k_B T) + e^(-ε/k_B T) = 1 + e^(-ε/k_B T)

It gives the possible distribution of particles between two energy levels at a specific temperature.

Conclusion

Statistical mechanics is a cornerstone of physical chemistry, providing a framework for explaining and predicting the physical properties and behaviors of matter from a molecular perspective. With fundamental concepts including microstates, macrostates, ensembles, and entropy, statistical mechanics allows scientists to understand the complexity of matter and energy at both the atomic and macroscopic levels. Understanding statistical mechanics opens avenues for a deeper understanding of complex phenomena in the realm of thermodynamics, quantum mechanics, and chemical kinetics.


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Statistical mechanics

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