Thermodynamics & Statistical Mechanics

From the zeroth law to the partition function — the four laws, and the microscopic story behind them.

About this course

A full course in classical thermodynamics and its statistical foundation, in six modules. Starts from equilibrium, temperature and the ideal gas, builds the first law through heat capacities and adiabatic processes, then develops the second law, entropy and irreversibility. The later modules cover the thermodynamic potentials and Maxwell relations, real engine cycles, phase transitions and real gases, and close by rebuilding entropy from microstates via the Boltzmann distribution and the partition function.

What you will be able to do

  • Apply the zeroth and first laws to work, heat and thermodynamic processes
  • Compute heat capacities, enthalpy changes and adiabatic relations for ideal gases
  • State the second law and bound engine efficiency with the Carnot cycle
  • Reason about entropy, irreversibility and the direction of spontaneous change
  • Choose the right thermodynamic potential and exploit the Maxwell relations
  • Analyse phase transitions with Clausius-Clapeyron and real gases near criticality
  • Derive thermodynamic quantities from the partition function

What this course covers

  • The fundamental relation of thermodynamics
  • The mismatch between natural and controlled variables
  • The Legendre transform and the four thermodynamic potentials
  • Minimisation criteria: which potential governs which conditions
  • Spontaneity as a contest: ΔG = ΔH − TΔS
  • Full Gibbs free energy calculation for isothermal compression
  • Carnot ceiling vs. real engine shortfall
  • Otto cycle structure and efficiency derivation
  • Compression ratio limit and engine knock in gasoline engines
  • Diesel cycle: compression ignition and higher efficiency
  • Rankine cycle and why real power plants reach ~40%
  • Reversed cycles: refrigerators, heat pumps, and COP
  • State Functions vs. Path Functions
  • Internal Energy as a State Function
  • First Law Sign Convention and Asymmetry
  • Cycles, Enclosed Area, and Net Work
  • Rectangular Cycle Worked Problem
  • Heat Engines, Efficiency, and Cycle Direction
  • The Second Law of Thermodynamics
  • The Carnot Cycle
  • Reversibility as the Key to Ideal Engines
  • Deriving the Carnot Efficiency Formula
  • Carnot Ceiling as an Unbreakable Limit
  • Preview of Entropy as a State Variable
  • Microstates and Macrostates
  • Fundamental Postulate of Statistical Mechanics
  • Boltzmann's Entropy Formula S = k log W
  • Entropy of Free Expansion by Microstate Counting
  • Statistical Origin of the Second Law
  • Arrow of Time from Initial Conditions
  • Thermodynamic systems, states, and equilibrium
  • Temperature and the zeroth law
  • Heat, work, and the first law of thermodynamics
  • Entropy and the second law of thermodynamics
  • Carnot efficiency limit
  • Statistical interpretation of entropy (Boltzmann's equation)
  • Mixed Partial Derivatives: The Calculus Foundation
  • Deriving the Four Maxwell Relations
  • The Star Maxwell Relation: Linking Immeasurable to Measurable
  • The TdS Equations
  • Proving Ideal Gas Internal Energy Depends Only on Temperature
  • Real Gas Contrast and Lecture Synthesis
  • Why a gas has two heat capacities
  • Enthalpy as the natural energy at constant pressure
  • Mayer's relation: Cp − Cv = R for an ideal gas
  • The heat-capacity ratio γ = Cp/Cv
  • Equipartition and molecular degrees of freedom
  • Capstone synthesis: heat, internal energy, work and enthalpy
  • Thermal Equilibrium and the Definition of Temperature
  • Specific Heat and the Mass-Weighted Average Final Temperature
  • The Zeroth Law of Thermodynamics
  • Thermometric Scales: Celsius and the Substance-Dependence Problem
  • The Ideal Gas Temperature Scale and Absolute Zero
  • The Kelvin Scale and Its Practical Use
  • Phase Transitions and Latent Heat
  • The Phase Diagram
  • Derivation of the Clausius-Clapeyron Equation
  • Integrated Clausius-Clapeyron Equation for Boiling
  • Applications: Pressure Cookers, Everest, and Denver
  • Water's Anomalous Fusion Curve and Ice Slipperiness
  • Carnot result reveals heat-over-temperature balance
  • Entropy defined as a state function
  • Entropy change formula for an ideal gas
  • Entropy statement of the second law
  • Entropy as the arrow of time
  • Boltzmann factor derivation
  • Temperature as competition in the Boltzmann factor
  • The partition function Z
  • Extracting thermodynamics from Z
  • Two-level paramagnet: full worked example
  • Energy vs. entropy competition in state populations
  • The Ideal Gas Law
  • Pressure Derived from Molecular Collisions
  • Temperature as Molecular Kinetic Energy
  • Root-Mean-Square Speed Calculation
  • Maxwell-Boltzmann Speed Distribution
  • Equipartition of Energy
  • Entropy Production and Irreversibility
  • The Arrow of Time
  • Free Expansion and Entropy Change
  • Entropy of Mixing and the Gibbs Paradox
  • Entropy Change in Heat Flow Between Two Blocks
  • Maxwell's Demon and Second-Law Limits
  • Why ideal gas law fails for real gases
  • The van der Waals equation
  • Van der Waals isotherms and mechanical instability
  • The critical point: definition and conditions
  • Derivation of critical constants from van der Waals equation
  • Law of corresponding states and universal ratio
  • Gibbs Criterion Derived from the Second Law
  • ΔG as an Enthalpy–Entropy Tug-of-War
  • Four Sign Regimes of Spontaneity
  • Crossover Temperature Formula T = ΔH/ΔS
  • Equilibrium and Phase Transition Temperatures
  • Applying the Criterion: Detecting Thermodynamic Impossibility
  • The Adiabatic Condition: No Heat and the First Law
  • Derivation of PV^γ = Constant
  • Diesel Engine as Adiabatic Compression Worked Problem
  • Adiabat vs. Isotherm Steepness on the PV Plane
  • Speed of Sound: Newton's Error and Laplace's Adiabatic Fix
  • Dry Adiabatic Lapse Rate and Mountain Cold
  • The partition function crank: recap and roadmap
  • Ideal gas laws derived from Z
  • Dulong–Petit law and the failure of classical equipartition
  • Einstein model: quantized oscillators and heat capacity
  • Paramagnetism and Curie's law from Z
  • The partition function as a universal master key
  • The Gibbs Paradox and indistinguishability
  • Fermi-Dirac and Bose-Einstein distributions
  • Classical Boltzmann as the dilute limit of quantum statistics
  • The Fermi energy and the degenerate electron gas
  • Physical consequences of quantum statistics
  • Course synthesis and the road ahead
  • Entropy of Mixing
  • Chemical Potential in a Mixture
  • Raoult's Law and Henry's Law
  • Colligative Properties: Particle Counting
  • Osmotic Pressure and the Van't Hoff Equation
  • Second Law Limits on Desalination
  • The Third Law of Thermodynamics
  • Zero Entropy as Perfect Certainty, Not Zero Energy
  • Absolute Entropy via Integration from Zero
  • Debye T³ Law and Copper's Absolute Entropy
  • Self-Consistency of the Third Law and Vanishing Heat Capacity
  • Unattainability of Absolute Zero
  • Work done by a gas: PdV and PV diagram area
  • Path dependence of work
  • Isothermal work: the marquee integral
  • Heat, work, and the First Law of Thermodynamics
  • The four canonical thermodynamic processes
  • Synthesis: state functions vs path functions in problem-solving
  • Opening the system: the chemical potential defined
  • Extending thermodynamic potentials to particle exchange
  • Chemical potential as the equalizing potential at equilibrium
  • Chemical potential of an ideal gas: the logarithmic pressure formula
  • Diffusion driven by chemical potential: the full worked problem
  • Phase coexistence as equality of chemical potentials
  • Flow Work and Enthalpy in Open Systems
  • Steady-Flow Energy Equation
  • Throttling and Isenthalpic Processes
  • Joule-Thomson Effect in Real Gases
  • Engine Cycles, Net Work, and Efficiency
  • The First Law's Limit and the Second Law Cliffhanger

Syllabus

  1. Foundations & Temperature

    1. L1 · The Players & the First Two Laws
    2. L2 · Equilibrium, Temperature & the Zeroth Law
    3. L3 · The Ideal Gas & Kinetic Theory
    4. L4 · Work, Heat & Thermodynamic Processes

  2. The First Law

    1. L5 · Internal Energy & the First Law
    2. L6 · Heat Capacities & Enthalpy
    3. L7 · Adiabatic Processes
    4. L8 · First-Law Applications

  3. The Second Law & Entropy

    1. L9 · The Second Law & the Carnot Cycle
    2. L10 · Entropy
    3. L11 · Irreversibility & the Arrow of Time
    4. L12 · The Third Law & Absolute Entropy

  4. Free Energy & Potentials

    1. L13 · Thermodynamic Potentials
    2. L14 · Maxwell Relations & the TdS Equations
    3. L15 · Free Energy & Spontaneity
    4. L16 · Open Systems & the Chemical Potential

  5. Engines, Refrigerators & Phase Transitions

    1. L17 · Real Cycles — Otto, Diesel & Rankine
    2. L18 · Phase Transitions & Clausius–Clapeyron
    3. L19 · Real Gases & Critical Points
    4. L20 · Mixtures, Solutions & Colligative Properties

  6. The Statistical Bridge

    1. L21 · Microstates & the Boltzmann Entropy
    2. L22 · The Boltzmann Distribution & the Partition Function
    3. L23 · Applications of the Partition Function
    4. L24 · Quantum Statistics & What's Next