⚗️Theoretical Chemistry Unit 9 – Statistical Thermodynamics & Ensembles

Fundamentals of Statistical Thermodynamics

• Bridges microscopic properties of matter (positions, velocities of atoms) to macroscopic thermodynamic properties (temperature, pressure, entropy)
• Assumes large systems (many particles) to derive average behaviors using probability distributions
• Employs ensemble theory, studying large collections of microstates corresponding to a macrostate
• Relates entropy to the number of accessible microstates via Boltzmann's entropy formula: $S = k_B \ln \Omega$
  • $S$: entropy
  • $k_B$: Boltzmann constant
  • $\Omega$: number of microstates
• Enables derivation of thermodynamic quantities from microscopic properties without tracking individual particles
• Provides a framework for understanding phase transitions, chemical reactions, and transport phenomena
• Finds applications in physics, chemistry, biology, and materials science

Probability and Microstates

• Microstate represents a specific configuration of a system (positions and momenta of all particles)
• Macrostate corresponds to observable thermodynamic properties (temperature, volume, pressure)
• Multiple microstates can correspond to the same macrostate
• Probability of a microstate depends on the system's constraints and energy distribution
• Equal a priori probability postulate assumes all accessible microstates are equally likely at equilibrium
• Probability distribution functions (Boltzmann, Fermi-Dirac, Bose-Einstein) describe the likelihood of a particle occupying a specific energy state
• Ensemble average calculates macroscopic properties by averaging over all microstates in an ensemble
  • Example: average energy $\langle E \rangle = \sum_i P_i E_i$, where $P_i$ is the probability of microstate $i$ with energy $E_i$

Ensemble Theory Basics

• Ensemble is a large collection of virtual copies of a system, each in a different microstate
• Represents all possible states a system can occupy under given constraints
• Three primary ensembles in statistical mechanics:
  • Microcanonical ensemble (fixed N, V, E)
  • Canonical ensemble (fixed N, V, T)
  • Grand canonical ensemble (fixed μ, V, T)
• Ensembles enable studying average properties and fluctuations without tracking individual systems
• Ergodic hypothesis assumes time average equals ensemble average for macroscopic properties
• Ensemble theory connects microscopic properties to macroscopic observables through statistical averages
• Partition function $Z$ is a central quantity in ensemble theory, used to calculate thermodynamic properties
  • Example: Helmholtz free energy $F = -k_BT \ln Z$ in the canonical ensemble

Canonical Ensemble

• Models a system in thermal equilibrium with a heat bath at constant temperature $T$
• System exchanges energy with the heat bath, but particle number $N$ and volume $V$ are fixed
• Probability of a microstate $i$ with energy $E_i$ is given by the Boltzmann distribution: $P_i = \frac{e^{-\beta E_i}}{Z}$
  • $\beta = \frac{1}{k_BT}$: inverse temperature
  • $Z = \sum_i e^{-\beta E_i}$: canonical partition function
• Partition function $Z$ is a sum over all microstates, used to normalize probabilities and calculate thermodynamic quantities
• Average energy $\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}$
• Helmholtz free energy $F = -k_BT \ln Z$ connects the partition function to thermodynamics
• Canonical ensemble is widely used in chemistry and biology to model systems at constant temperature

Microcanonical Ensemble

• Describes an isolated system with fixed particle number $N$, volume $V$, and total energy $E$
• All accessible microstates with the same energy $E$ are equally probable
• Microcanonical partition function $\Omega(N, V, E)$ counts the number of microstates with energy $E$
• Entropy is directly related to the microcanonical partition function: $S = k_B \ln \Omega$
• Temperature is defined as $\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{N,V}$
• Pressure is given by $P = T\left(\frac{\partial S}{\partial V}\right)_{N,E}$
• Microcanonical ensemble is useful for studying isolated systems and deriving fundamental relations
• Provides a foundation for understanding the connection between entropy and the number of accessible microstates

Grand Canonical Ensemble

• Models an open system that exchanges both energy and particles with a reservoir
• System is characterized by fixed chemical potential $\mu$, volume $V$, and temperature $T$
• Grand canonical partition function $\Xi$ is a sum over all microstates with varying particle numbers: $\Xi = \sum_{N=0}^\infty \sum_i e^{-\beta(E_i - \mu N_i)}$
• Average particle number $\langle N \rangle = \frac{1}{\beta}\frac{\partial \ln \Xi}{\partial \mu}$
• Grand potential $\Phi = -k_BT \ln \Xi$ is the characteristic thermodynamic potential of the grand canonical ensemble
• Pressure $P = -\left(\frac{\partial \Phi}{\partial V}\right)_{T,\mu}$
• Grand canonical ensemble is useful for studying systems with variable particle numbers, such as adsorption and chemical reactions

Partition Functions and Their Applications

• Partition functions are central to statistical mechanics and connect microscopic properties to macroscopic observables
• Different ensembles have specific partition functions:
  • Canonical: $Z = \sum_i e^{-\beta E_i}$
  • Grand canonical: $\Xi = \sum_{N=0}^\infty \sum_i e^{-\beta(E_i - \mu N_i)}$
• Thermodynamic quantities can be derived from partition functions using partial derivatives
  • Example: Average energy $\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}$ in the canonical ensemble
• Partition functions enable the calculation of heat capacities, compressibilities, and other response functions
• Molecular partition functions can be constructed from translational, rotational, vibrational, and electronic contributions
• Partition functions find applications in adsorption isotherms (Langmuir, BET), chemical equilibria, and phase transitions
• Computational methods (Monte Carlo, molecular dynamics) can be used to evaluate partition functions for complex systems

Connecting Statistical Mechanics to Thermodynamics

• Statistical mechanics provides a microscopic foundation for thermodynamics
• Thermodynamic potentials (internal energy, enthalpy, Helmholtz and Gibbs free energies) can be derived from partition functions
  • Example: Helmholtz free energy $F = -k_BT \ln Z$ in the canonical ensemble
• Entropy is related to the number of accessible microstates via Boltzmann's entropy formula: $S = k_B \ln \Omega$
• Temperature, pressure, and chemical potential emerge as derivatives of the entropy or partition function
• Fluctuation-dissipation theorem relates response functions (heat capacity, compressibility) to equilibrium fluctuations
• Statistical mechanics enables the calculation of phase diagrams, critical exponents, and transport coefficients
• Provides a framework for understanding non-equilibrium phenomena and irreversible processes
• Connects microscopic interactions to macroscopic properties, bridging the gap between molecular simulations and thermodynamic experiments