Introduction to Thermodynamics

Equilibrium

Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. (A primitive notion is not defined in terms of previously defined concepts, but appears to be immediately understandable and thus taken for granted.) Other concepts (processes, state functions) of thermodynamics are necessarily defined upon thermodynamic equilibrium.

Thermodynamic equilibrium of a system interacting with its surroundings is the unique stable stationary state of the system that is eventually reached over a long time.

The above postulate asserts thermodynamic equilibrium as a unary relation (i.e. property) of a system.

A thermodynamic system, by default, refers to a system at thermodynamic equilibrium.

Thermodynamic state

Notes on Thermodynamic state

A system in thermodynamic equilibrium is further partitioned into equivalence classes by several equivalence (binary) relations.

A contact equilibrium between two systems is defined as the absence of a particular type of (primitive) transfer process when the systems are connected by a path permeable only to this process. As another primitive notion of the theory of thermodynamics, heat is asserted as a distinct transfer process.

A system in thermodynamic equilibrium is then characterized by multiple contact equilibria.

Two systems are in thermodynamic equilibrium if they are simultaneously in all the following contact equilibria:

  1. Thermal equilibrium: No heat flow occurs between two systems when they are connected by a path permeable only to heat.
    • Radiative equilibrium: No radiative heat flux between two systems when they are connected by a path permeable only to heat flux.
  2. Mechanical equilibrium: No mechanical work occurs between two systems when they are connected by a path permeable only to mechanical work.
  3. Chemical Equilibrium (including Phase equilibrium): No chemical reaction (including diffusion) occurs between two systems when they are connected by a path permeable only to its reactants and products.

Contact equilibrium can be interpreted as binary relation. By physical symmetry, every contact equilibrium is a symmetric and reflexive relation. Except thermal equilibrium, the other two equilibria have already been asserted to be Euclidean relations, thus equivalence relations. Together with the Zeroth Law which asserts transitivity, multiple contact equilibria establishes thermodynamic equilibrium as a (finer) equivalence relation.

Thermodynamic state of a system at thermodynamic equilibrium is a equivalence class of its thermodynamic equilibrium.

State variables

Notes on State variables

Transport Phenomena:

  • Momentum transport - viscosity
  • Energy transport - heat conduction
  • Mass transport - molecular diffusion

After assigning direction to a transfer process, the (quotient set of) equivalence classes defined by the contact equilibrium corresponding to this process can be strictly totally ordered. Thus we may assign a specific intensive property to systems at equilibrium for each transfer process.

Temperature is the intensive property of a system that drives heat flow.

In absence of other factors triggering a transfer process, the intensive property corresponding to the process must be spatially uniform in a system at equilibrium. Such is the case of heat transfer and temperature.

The thermodynamic states of a system are thus parameterized into a finite set of intensive variables.

A state variable of a system at thermodynamic equilibrium is a property of the system that only depends on the current thermodynamic equilibrium.

The intensive properties for transfer processes are state variables. Essentially any function of the thermodynamic states of a system is its state variable.

An equation of state is an equation of several state variables.

An equation of state may be expressed as an explicit function mapping a set of state variables to some other state variable.

Topics

  1. Statistical Distributions
    • Maxwell distribution for velocity and speed
    • Boltzmann distribution for number density
    • Maxwell–Boltzmann distribution for energy
    • Equipartition theorem & heat capacity
  2. Phase Transition

Dynamics

Laws of thermodynamics

  1. Zeroth law of Thermodynamics
    • Thermal equilibrium between two systems is a transitive relation.
    • All heat is of the same kind. [Maxwell, J.C. (1871), p. 57.]
  2. First law of Thermodynamics (for closed and open systems)
    • Heat is the transfer of internal energy.
  3. Second law of Thermodynamics
    • Heat can never autonomously transfer along the temperature gradient.
  4. Third law of Thermodynamics
    • As the temperature of any condensed system approaches absolute zero, its entropy change in any process also approaches zero.

Only the second and the third laws are actually empirical laws.

Major implications of the laws of thermodynamics:

  1. The Zeroth Law of Thermodynamics establishes temperature, unlike chemical potentials, as a one-dimensional quantity.
  2. The First Law of Thermodynamics establishes heat as the transfer of a unique form of energy, which thus can transform with other forms of energy but conserves as whole in isolated systems.
  3. The Second Law of Thermodynamics
    • justifies entropy as a state function of a system.
    • implies entropy of a system never decrease in adiabatic processes. [Entropy maximization]
    • implies Carnot's theorem (maximum efficiency of heat engines).
    • justifies an absolute temperature scale.
  4. The Third Law of Thermodynamics justifies (Planck) absolute entropy.

Conjugate variables and thermodynamic potentials

Thermodyanmic equations

Table: State Functions as Conjugate Pairs by Transfer Processes

Intensive Extensive
Mechanical p V
Thermal T S
Material {μ_i} {N_i}

The intensive parameters of a system are not all independent. In thermodynamics, a phase is a form of matter that is homogeneous in chemical composition and state of matter; a component is a chemical species in a specific phase, each with a unique chemical potential. When chemical species and phases are differentiated, chemical potentials of components in each phase are related through the Gibbs-Duhem relation. For a non-reactive system involving c components and p phases at thermodynamic equilibrium (c≥p), the degree of freedom of its thermodynamic equilibria is 2+c-p: this is Gibbs' phase rule. The phase space of the thermodynamic equilibria of a single-species single-phase system is two dimensional.

Common thermodynamic (energy) potentials: Internal energy U, Enthalpy H, Helmholtz free energy F, Gibbs free energy G.

First derivatives of thermodynamic potentials are state variables conjugate to the varying state variable.

Second derivatives of thermodynamic potentials are material properties:

  • Coefficient of thermal expansion, coefficient of thermal pressure;
  • Isothermal compressibility, Adiabatic compressibility;
  • specific heat (at constant pressure, at constant volume).

Dynamic relations

Notes on Dynamic relations

First order equations of thermodynamic potentials: Fundamental thermodynamic relation - conservation of energy and Second Law combined. These equations apply for quasi-static reversible processes.

Second order equations of thermodynamic potentials: Maxwell relations - mixed order second derivatives are equal.

For a particular system undergoing quasi-static reversible processes, all properties of its thermodynamic states can be determined with the fundamental thermodynamic relation and the system's constitutive relations. The fundamental thermodynamic relation is based on physical laws, while constitutive relations are phenomenological. The fundamental thermodynamic relation is one equation, which may be expressed with different natural variables and corresponding thermodynamic potentials. For systems with c components, it has c+1 constitutive relations, mapping between the independent conjugate pairs. You need one additional extensive state variable (e.g. total particle number or volumn) to complete the equations.

Statistical Thermodynamics

Statistical thermodynamics, aka equilibrium statistical mechanics, studies the properties of thermodynamic states of a system of particles. It derives state variables and equations of state from ensemble's probability distribution over micro-states. If not static, the ensemble evolution is given by the Liouville equation for classical particles, which can be derived from particle equations of motion, which in this case are Hamilton's equations. Statistical thermodynamics also studies further the into microscopic level, such as fluctuations. In other words, it takes thermodynamics from phenomenology to mechanistic theory.

Statistical ensemble theory

Notes on Statistical ensemble theory

A statistical ensemble is a massive collection of independent hypothetical systems; each system is a massive collection of equivalent particles and its configuration evolves under a same set of dynamic equations.

Three scales of statistical ensemble model: (micro) particle -> (macro) system -> ensemble.

An ensemble is at statistical equilibrium if for each micro-state in it, the ensemble also contains all the future and past micro-states with equal probabilities.

The fundamental postulates of statistical mechanics: equal a priori probability postulate for isolated systems.

  1. Ergodic hypothesis. [most systems are not ergodic.]
  2. Principle of indifference.
  3. Maximum entropy: the correct ensemble has the largest Gibbs entropy (probability).

Mean values of element states in a system are equal to mean values of corresponding system states in an ensemble.

Note Outline:

  1. Ensemble theory
    1. Fundamental concepts
    2. Microstate
  2. Three equilibrium ensembles
    1. Microcanonical ensemble
    2. Canonical ensemble
    3. Grand canonical ensemble

Boltzmann's entropy formula: A macroscopic state of a system is a distribution on the microstates. Entropy is a measure of this distribution.

Topics

Topics in statistical mechanics

  1. Systems of approximate noninteracting particles
    1. distribution over energy level derived from ensemble theory
    2. most probable distribution over energy level
  2. Quasi-thermodynamic theory of Fluctuations (probability distribution)

🏷 Category=Physics