Thermodynamics

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 of thermodynamics (transport processes, state variables) 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

When two systems are connected, transport processes could happen at the interface. All transport processes are primitive notions in theoretical studies of the corresponding phenomena. Specifically, heat is asserted as a distinct transport process in the theory of thermodynamics.

Transport Phenomena:

• Momentum transport: viscosity;
• Energy transport: heat conduction;
• Mass transport: molecular diffusion;

A contact equilibrium between two systems is defined as the absence of a particular type of transport process when the systems are connected by a path permeable only to that process. Contact equilibrium shall be understood as a macroscopic equilibrium, with (almost) perfectly balanced microscopic exchanges resulting in no macroscopic flow of any kind anywhere.

Two systems are in thermodynamic equilibrium if they are simultaneously in all of 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†.

† Paths permeable to substance is always permeable to convective heat flow, thus chemical equilibrium needs special treatment.

The above definition establishes thermodynamic equilibrium as an equivalence relation of two systems. Since contact equilibrium can be interpreted as binary relation, by physical symmetry, every contact equilibrium is a symmetric and reflexive relation. The two contact equilibria other than thermal equilibrium have already been asserted to be Euclidean relations (similar to transitivity). Together with the Zeroth Law which asserts transitivity for thermal equilibrium, the three contact equilibria are all equivalence relations, establishing thermodynamic equilibrium as a (finer) equivalence relation.

Thermodynamic equilibrium of a system is thus partitioned into equivalence classes by these contact equilibria. The thermodynamic state of a thermodynamic system refers to the equivalence class of its current thermodynamic equilibrium.

State variables

Notes on State variables

If a transport process has been assigned a direction, the (quotient set of) equivalence classes defined by the corresponding contact equilibrium can be strictly totally ordered. To capture this order structure, we can assign a unique number to each equivalence class such that the number monotonically decreases along the direction of the transport process. The quantity such assigned can be interpreted as the driving force of the transport process. This type of quantity are intensive properties of a system, because subdividing the system does not affect their value: any two subsystems of a thermodynamic system are in thermodynamic equilibrium with each other. Temperature is the intensive property of a system that drives heat transfer.

Besides order, metric is often found useful in studying equilibrium states. In case of a one-sided chain with metric, the structure of the quotient set of equivalence classes are typically modeled as non-negative real numbers for convenience. In fact, pressure, temperature, and chemical potentials are all nonnegative.

With one macroscopic intensive property for each transport process, we can identify the thermodynamic states of a system with an \(n\)-tuple: pressure, temperature, and chemical potentials.

A definite number of real variables define the states that are the points of the manifold of equilibria. [Carathéodory, C. (1909)]

Every macroscopic intensive property of a thermodynamic system is balanced. In absence of other factors triggering a transport process, the property corresponding to the process must be spatially uniform in a system at equilibrium. Intensive properties other than temperature may be driven to spatial inhomogeneity by a static long-range force field imposed on it by its surroundings. Such equilibrium inhomogeneity does not occur to temperature because, borrowing primitive notions from mechanics, long-range "force" that drives heat flow is not necessary in thermodynamics.

A state variable of a thermodynamic system is a property of the system that only depends on the current thermodynamic equilibrium. A state function, with subtle difference, refers to a state variable regarded as a function of other state variables. Obviously, the intensive properties for transport processes are state variables. State variables can also be extensive properties: volume, entropy, amount of a chemical species. Essentially any function of the thermodynamic states of a system is its state variable.

A conjugate pair is a pair of intensive and extensive properties whose product refers to work/action/energy, and are analogous to generalized force and generalized coordinate in classical mechanics. One state variable of a pair is called the conjugate variable of the other.

Table: State Variables as Conjugate Pairs by Transport Process

Process	Intensive	Extensive
Mechanical	\(p\)	\(V\)
Thermal	\(T\)	\(S\)
Material	\({μ_i}\)	\({N_i}\)

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

Zeroth law of Thermodynamics:

Thermal equilibrium between two systems is a transitive relation. [Carathéodory, C.]

All heat is of the same kind. [Maxwell, J.C. (1871), p. 57.]

First law of Thermodynamics (for closed and open systems)

Heat is the transfer of (internal) energy.

Second law of Thermodynamics

Heat can never autonomously transfer along the temperature gradient.

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. In fact, Constantin Carathéodory (1909) formulated thermodynamics, up to the second law, on a purely mathematical axiomatic foundation.

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 variable of a system (integral of the ratio of incremental heat transfer divided by temperature in a reversible process).
   • implies the entropy of a system never decrease in adiabatic processes.
   • implies isolated systems always spontaneously increase their entropy.
   • implies Carnot's theorem (maximum efficiency of heat engines).
   • justifies an absolute temperature scale.
4. The Third Law of Thermodynamics justifies (Planck) absolute entropy.

Thermodynamic potentials

Thermodyanmic equations

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 \( \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p \).

Gibbs' phase rule: For a non-reactive thermodynamic system involving \(c\) components and \(p\) phases (\(c \ge p\)), the degree of freedom of its thermodynamic equilibria is \(2+c-p\). The phase space of the thermodynamic equilibria of a single-species single-phase system is two dimensional. Reasoning of Gibbs' phase rule: System at thermodynamic equilibrium has uniform temperature and pressure. If a system consist of multiple components, since it must be homogeneous in each phase at equilibrium, only the amount fractions in each phase count as additional degrees of freedom.

The complete set of thermodynamic (energy) potentials in closed systems: (A closed system has no material exchange with the environment; in other words, conservation of mass holds within the system.) (All the common thermodynamic potentials are energy potentials, but there are also entropy potentials, know as free entropies.)

• Internal energy \(U\): is the capacity to do mechanical work (useful work) plus the capacity to release heat.
• Enthalpy \(H\): the capacity to do non-mechanical work plus the capacity to release heat.
• Helmholtz free energy \(F\): the capacity to do mechanical work.
• Gibbs free energy \(G\): the capacity to do non-mechanical work.

Capacity	work \(-pV\)	non-mechanical work
heat \(TS\)	\(U\)	\(H\)
no heat	\(F\)	\(G\)

Principle of maximum work:

• In isothermal processes, the maximum work a system can do to the environment equals the decrease of its Helmholtz free energy.
• In isothermal isobaric processes, the maximum non-mechanical work a system can do equals the decrease of its Gibbs free energy.

For all thermodynamic processes between the same initial and final states, the delivery of work is a maximum for a reversible process.

Criteria for thermodynamic equilibrium of a closed system:

• If isolated (isovolumetric, constant energy), a thermodynamic system has the maximum entropy.
• In isentropic isovolumetric conditions, a thermodynamic system has the minimum internal energy.
• In isothermal isovolumetric conditions, a thermodynamic system has the minimum Helmholtz free energy.
• In isothermal isobaric conditions, a thermodynamic system has the minimum Gibbs free energy.

All thermodynamic potentials are (non-standard) Legendre transforms of each other, substituting a variable to its conjugate. The Legendre transforms are nonstandard in that it is the difference, rather than the sum, of the two functions that equals the product of the conjugate pair. This strategy is used to shift each of the independent variables between extensive and intensive properties. If there are \(D\) dimensions to the thermodynamic space, then there are \(2^D\) unique thermodynamic potentials. Depending on the variables controlled in a specific situation, one thermodynamic potential will be the easiest to work with and thus the most useful. Most reactions happen at constant \(p\) and \(T\), so the Gibbs free energy is the most useful potential in studies of chemical reactions.

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).

Grand potential in open systems: \( \Psi = F - \sum_i \mu_i N_i \)

Dynamic relations

Notes on Dynamic relations

The fundamental thermodynamic relation is first order equation of a thermodynamic potential, which combines the First and the Second Law (for reversible processes). The fundamental thermodynamic relation is one equation, which may be expressed with different natural variables and corresponding thermodynamic potentials. (Natural variables for a thermodynamic potential refers to the state variables appeared as differentials in the corresponding fundamental thermodynamic relation.)

\[ \begin{align} \mathrm{d} U &= &T \mathrm{d} S - p \mathrm{d} V + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} H &= &T \mathrm{d} S + V \mathrm{d} p + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} F &= &-S \mathrm{d} T - p \mathrm{d} V + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} G &= &-S \mathrm{d} T + V \mathrm{d} p + \sum_i \mu_i \mathrm{d} N_i \\ \mathrm{d} \Psi &= &-S \mathrm{d} T - p \mathrm{d} V - \sum_i N_i \mathrm{d} \mu_i \end{align}\]

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

\[ Z_{xy} = Z_{yx} \]

Euler integrals of the fundamental thermodynamic relations determine thermodynamic potentials up to a reference constant. For example, \[ U = T S - p V + \sum_i \mu_i N_i \]

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.

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. If relating the conjugate pairs of the transport processes, an equation of state is a constitutive relation of a given system. If a particular system involves \(D\) distinct transport processes, its phase space has \(D\) dimensions, and it needs \(D\) constitutive relations to fully specify its thermodynamic property. 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 volume) 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)

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