L4

Course Information Course Title: Statistical Mechanics Instructor: Dr. Gabriele C. Sosso Institution: University of Warwick, Department of Chemistry Contact: G Block, Office 4, G.Sosso@warwick.ac.uk

Lecture Schedule Overview

• Lecture 1 (Nov 9): Connection between micro (quantum mechanics) and macro (thermodynamics)

  • Concept of ensemble and ensemble average

  • Workshop on ensemble averages

• Lecture 2 (Nov 10):

  • Concept of phase space

  • Concept of partition function

  • Microcanonical ensemble (NVE)

• Lecture 3 (Nov 16):

  • Canonical partition function

  • Thermodynamic quantities from canonical partition function

  • Canonical Ensemble (NVT) Workshop

• Lecture 4 (Nov 17):

  • Molecular partition functions

  • Fermi-Dirac, Bose-Einstein, and Boltzmann statistics

• Subsequent Topics:

  • The equilibrium constant, translational, vibrational, and electronic partition functions

  • Approaches to treat homonuclear molecules and Lennard-Jones liquids

Learning Objectives (End of the Course) Understand:

• Formalism of the canonical ensemble

• Molecular partition functions

• Indistinguishable vs. distinguishable particles

• Usage of Boltzmann vs. Bose-Einstein and Fermi-Dirac Statistics

Hamiltonian Models

• Total Hamiltonian

• Total Energy Representation: H_{Total} \approx H_1 + H_2 + ... + H_N

• Example 1: Hamiltonian for a diluted gas:

  • Molecules are non-interacting: H_{Total} \approx N

  • Total energy expressed as contributions from individual molecules

• Example 2: Diatomic molecule

  • Contributions to total Hamiltonian:

    • H_{Translational}, H_{Rotational}, H_{Vibrational}, H_{Electronic}

Energy Contributions

• Diatomic Molecule Contributions:

  • H_{Translational} : Movement along x, y, z

  • H_{Rotational} : Rotation along two axes

  • H_{Vibrational} : Vibration along the bond line

  • H_{Electronic} : Electronic transitions

• Energy Level Approximation:

  • E_{Trans} << E_{Rot} << E_{Vib} << E_{Ele}

Molecular Partition Functions

• Fundamental Concept

  • For distinguishable particles: Q(N,V,T) = q_1(V,T) \cdot q_2(V,T) \cdot ... \cdot q_N(V,T)

  • If particles are identical: Q(N,V,T) = [q(V,T)]^N

  • Special case: For indistinguishable particles:

    • Q(N,V,T) = \frac{[q(V,T)]^N}{N!}

  • Assume independence among particles

Statistical Distributions

• Types of Statistics

  • Fermions:

    • Half-integer spin e.g. electrons

    • No two identical fermions can occupy the same state (Pauli Exclusion Principle)

  • Bosons:

    • Integer spin e.g. photons

    • Identical bosons can occupy the same quantum state

  • Boltzmann Statistics (N >> E):

    • Can apply to non-interacting particles generally

    • Q(N,V,T) defined for indistinguishable particles as above

Practical Applications

• Considerations and Conditions

  • Criteria for Boltzmann statistics:

    • High temperatures (usually > 10 K)

    • Low densities (acceptability in typical gases)

  • Use specific quantum statistical mechanics for fermions and bosons only in respective state

Additional Resources

• Recommended Reading: McQuarrie - Statistical Mechanics, Chapters on:

  • Boltzmann statistics (4.1)

  • Fermi-Dirac and Bose-Einstein statistics (4.2)

Exercises

• Problem Solving

  • Task: Calculate Helmholtz free energy A of a system of independent, distinguishable particles with accessible states consisting of ground and excited states under thermal equilibrium.

  • Importance of practical application in understanding theoretical principles.