Introduction to Thermodynamics

Unlike classical dynamics which aims at the motion of mass (think of equations of motion), the main motivation for thermodynamics is to account for heat in energy transform and know the maximum efficiency for mechanical work production or heat extraction (think of principle of maximum work).

Thermodynamics is a study of transport phenomena where a system concept is used, just like fluid mechanics. A system is a hypothetical region, along with the material within it. The environment of a system is everything in the physical space that is not part of the system. A closed system cannot have material exchange with the environment, while an open system can.

## 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.

• 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: No chemical reaction occurs between two systems when they are connected by a path permeable only to its reactants and products†.
• Phase equilibrium: no diffusion.

† 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.

An equivalence relation partitions a set into a collection of equivalence classes, referred to as the quotient set/space of the set by the equivalence relation. The thermodynamic equilibrium of a system is thus partitioned into a quotient set by these contact equilibria, called the (phase) space of thermodynamic states. 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.

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 is typically modeled as nonnegative real numbers for convenience.

This type of quantities 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.

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.

With one macroscopic intensive property for each transport process, we can identify the thermodynamic states of a system with a list of nonnegative numbers: pressure, temperature, and chemical potentials. But not all of them are independent; they are related by the Gibbs–Duhem equation in each phase.

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

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.

### 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, more notes

## 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

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 requires temperature to be homogeneous at equilibrium, unlike one chemical potential for each component.
2. The First Law of Thermodynamics establishes heat as another form of energy transfer, so system energy can be changed by multiple energy transport processes but conserves as whole in isolated systems.
3. The Second Law of Thermodynamics justifies entropy as a state variable of a system, defined up to an arbitrary reference state.
4. The Third Law of Thermodynamics justifies absolute zero as an absolute reference point of entropy, which defines absolute entropy when reference entropy is assigned zero.

### First Law and conservation

The First Law effectively unifies distinct physicochemical processes under the umbrella of energy, a conserved quantity in every transport phenomenon. The First Law states that, when the three types of transport processes of thermodynamics are considered, the change of internal energy of a system equals the sum of mechanical work done to the system by the environment, heat flowing into the system, and energy added by new particles: ($\delta$ denotes changes of process functions, $\mathrm{d}$ denotes changes of state functions.)

$$\mathrm{d} U = \delta Q - \delta W + \sum_i \mu_i\,\mathrm{d}N_i$$

Any type of work can be included in a thermodynamic system:

• Heat (indispensable, dictates the direction of spontaneous processes and criteria of equilibrium);
• Mechanical work: volumetric work, surface work, elastic work, gravitational work;
• Chemical reactions: gas reactions, biochemical reactions;
• Electromagnetic work: electrical work, electromagnetic polarization;

### Second Law and its implications

Clausius inequality as the Second Law: Entropy change of a system in any infinitesimal quasi-static process is no less than the heat flown into the system divided by temperature.

$$\mathrm{d}S \ge \frac{\delta Q}{T}$$

Macroscopic entropy as an extensive state variable of a thermodynamic system is defined, up to a reference value, as the integral of the ratio of incremental heat inflow to temperature in any reversible process from a reference thermodynamic state to the current thermodynamic state:

$$S = S_0 + \int_L \frac{\delta_\text{rev} Q}{T}$$

Compared with other extensive state variables, entropy has a unique importance in thermodynamics, analogous to thermodynamic potentials.

The Second Law has many implications:

• 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.

A quasi-static process is a process throughout which the system is always in thermodynamic equilibrium. A reversible process is a quasi-static process that causes no change in the total entropy of the system and its surroundings. Note that reversible processes are not necessarily isentropic for the system. Transport phenomena are all irreversible.

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

$$\delta W \le T \mathrm{d} S - \mathrm{d} U$$

• In isothermal processes, the maximum work a system can do to the environment equals the decrease of its Helmholtz free energy: $\delta W \le - (\mathrm{d} F)_T$
• In isothermal isobaric processes, the maximum non-mechanical work a system can do equals the decrease of its Gibbs free energy: $\delta W - p \mathrm{d} V \le - (\mathrm{d} G)_{p, T}$

Criteria for thermodynamic equilibrium of a closed system: (principle of maximum entropy; principle of minimum energy)

• 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.

### Dynamic relations

Notes on Dynamic relations

All properties of the thermodynamic states of a system 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, unless derivable from statistical mechanics.

The fundamental thermodynamic relation is the exact differential equation of a thermodynamic potential, combining the First Law and the Second Law under reversible processes:

$$\mathrm{d} U = T \mathrm{d} S - p \mathrm{d} V + \sum_i \mu_i \mathrm{d} N_i$$

The natural variables of a thermodynamic potential refers to the state variables appearing as differentials in its fundamental thermodynamic relation. The natural variables of a thermodynamic potential are important because if a thermodynamic potential can be determined as a function of its natural variables, all the (intensive and extensive) thermodynamic properties of the system can be obtained by taking partial derivatives of the thermodynamic potential with respect to the natural variables.

First order partial derivatives of a thermodynamic potential with respect to its natural variables are the conjugate variables of the natural variables, as the fundamental thermodynamic relation shows. If these partial derivatives are expressed as functions of natural variables, they are called equations of state. All equations of state of a thermodynamic potential forms the constitutive relation of a given system.

$$p_i = \left( \frac{\partial U}{\partial q_i} \right)_{q_j}$$

Second partial 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). Material properties are related by Maxwell relations: mixed second order derivatives are equal (Schwarz's theorem).

$$\left( \frac{\partial^2 U}{\partial q_i \partial q_j} \right)_{q_k} = \left( \frac{\partial^2 U}{\partial q_j \partial q_i} \right)_{q_k}$$

Euler integral of the fundamental thermodynamic relation determines a thermodynamic potential up to a reference constant. Since all the natural variables of internal energy are extensive properties, additivity implies that internal energy is a homogeneous function of degree 1. Following Euler's homogeneous function theorem, at constant values of the intensive properties, the integrated internal energy is: $$U = T S - p V + \sum_i \mu_i N_i$$

The exact differential equation of the above equation gives the Gibbs–Duhem equation:

$$S \mathrm{d}T - V \mathrm{d}p + \sum_{i=1}^I N_i \mathrm{d}\mu_i = 0$$

The dimension of the space of thermodynamic system is determined by Gibbs' phase rule: For a non-reactive thermodynamic system involving $N$ species and $p$ phases, the degree of freedom of its thermodynamic equilibria is $f = 2 + N - p$. If chemical reactions are involved, with $R$ independent reactions, the number of components (组分) is $C = N - R$, giving phase rule $f = 2 + C - p$. For a thermodynamic system with volumetric work and chemical processes only, the number of its natural variables $D$ equals its degree of freedom plus the number of its phases: $D = f + p$. For example, a single-species single-phase thermodynamic system has three natural variables and two degrees of freedom, whose independent state variables may be chosen as pressure and volume. Reasoning of Gibbs' phase rule: System at thermodynamic equilibrium has uniform temperature and pressure. If a system consists 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.

### Legendre transformation and thermodynamic potentials

Starting from internal energy, other thermodynamic potentials can be derived by (non-standard) Legendre transformations, substituting a set of extensive state variables with a set of intensive state variables. The substituting state variables form conjugate pairs, pairs of intensive and extensive properties analogous to generalized forces and generalized coordinates in classical mechanics whose products have the unit of energy. Either state variable of a conjugate pair is called the conjugate variable of the other.

Table: State variables as conjugate pairs of work by transport process [@Alberty1994]

Process Intensive Extensive
Thermal $T$ $S$
Volumetric $-p$ $V$
Surface $\gamma$ $A_s$
Linear $f$ $L$
Gravitational $\psi$ $m$
Electrical $\phi_i$ $Q_i$
Electric polarization $\mathbf{E}$ $\mathbf{v}$
Magnetic polarization $\mathbf{B}$ $\mathbf{m}$
Chemical ${μ_i}$ ${N_i}$

The Legendre transformations are nonstandard in that it is the difference, rather than the sum, of the two functions that equals the product of the conjugate pair. Without providing their names, some common thermodynamic potentials are defined by the following nonstandard Legendre transformations:

\begin{align} U(S, V, \{N_i\}) - F(T, V, \{N_i\}) &= T S \\ U(S, V, \{N_i\}) - H(S, p, \{N_i\}) &= -p V \\ U(S, V, \{N_i\}) - G(T, p, \{N_i\}) &= T S - p V \\ U(S, V, \{N_i\}) - \Phi(T, V, \{\mu_i\}) &= T S + \sum_i \mu_i N_i \end{align}

All thermodynamic potentials contain exactly the same information. And the fundamental thermodynamic relation of the new thermodynamic potentials can be derived from their definitions:

\begin{align} \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} \Phi &= &-S \mathrm{d} T - p \mathrm{d} V - \sum_i N_i \mathrm{d} \mu_i \end{align}

Legendre transformations are used to shift each of the natural variables between extensive and intensive properties, so that one thermodynamic potential will be the easiest to work with given the controlled variables in a specific situation. For example, Gibbs free energy is the most convenient thermodynamic potential in studies of chemical reactions, because most chemical reactions happen at constant pressure and temperature, and are regarded as closed systems.

For a thermodynamic system with $D$ natural variables, there are $2^D$ unique thermodynamic potentials. Closed systems with only volumetric work have 3 natural variables, so there are 8 distinct thermodynamic potentials. Some of these thermodynamic potentials are named and interpreted as follows:

• Internal energy $U$: the total energy of the system (Hamiltonian); the capacity to do work plus the capacity to release heat;
• Enthalpy $H$: the capacity to do non-volumetric work plus the capacity to release heat;
• Helmholtz free energy $F$ (or $A$ in chemistry): the capacity to do work;
• Gibbs free energy $G$: the capacity to do non-volumetric work;
• Grand potential (Landau Potential) $\Phi$: a common thermodynamic potential in open systems;

Thermodynamic potential can be generally denoted as $U[p_i]$, with a comma-separated list of intensive state variables introduced as natural variables by a Legendre transformation of internal energy. [@Callen1985]

All the common thermodynamic potentials are energy potentials, but there are also entropy potentials, know as free entropies.

## 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. Micro-canonical ensemble
2. Canonical ensemble: $F = - k_B T \ln Z$
3. Grand canonical ensemble: $\Phi = - k_B T \ln \mathcal{Z}$
4. Isothermal–isobaric ensemble: $G = - k_B T \ln \Delta$
5. Isoenthalpic–isobaric 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 non-interacting 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)

## Non-equilibrium thermodynamics

Local (Planckian) temperature of a system can be defined such that emission is given by Kirchhoff's law. A system is in local thermodynamic equilibrium if the local kinetic (Maxwellian) temperature is equal to the Planckian temperature of the radiation field. In a system at local thermodynamic equilibrium, the thermodynamic driving forces are varying in space and time so slowly that one can assume thermodynamic equilibrium in some neighborhood about any point in the system. For example, temperature in a cup of water with a melting ice cube.

Onsager reciprocal relations for transport processes of near-equilibrium systems: Linearly relating the gradient of entropic forces and the rate of entropy production due to irreversible processes, the Onsager matrix of kinetic coefficients is positive semi-definite. [@Onsager1931]

Conjecture of minimum entropy production: At near-equilibrium stationary states, a system under purely linear diffusion with negligible inertia has pointwise minimum rate of entropy production. [@Prigogine1945]

Dissipative structures of systems far from thermodynamic equilibrium: importation and dissipation of energy (and matter) could result in dynamical regimes that can be regarded as thermodynamic steady states in a system. [@Prigogine1967]

## Miscellaneous from [@Alberty1994]

The description of the state of the system must include at least one extensive variable.

The criterion for equilibrium is stated in terms of the natural variables after all the constraints have been applied.

### Phase equilibrium

For a system at phase equilibrium, the chemical potential of a species is uniform across all phases, even though the phases may be different states of matter or be at different pressures, gravitational potentials, or electric potentials. The natural variables for amounts of species thus count all phases.

### Chemical reaction systems

The amounts of components are natural variables for a closed reaction system.

With conservation matrix $\mathbf{A}$, the amounts of components are: $$\mathbf{n}_c = \mathbf{A} \mathbf{n}$$

With initial amounts of species $\mathbf{n}_0$, stoichiometric numbers of species $\boldsymbol{\mu}$, and extent of reactions $\boldsymbol{\xi}$, the amounts of species at any given time are: $$\mathbf{n} = \mathbf{n}_0 + \boldsymbol{\nu} \boldsymbol{\xi}$$

The fundamental equation for a multi-reaction system at constant temperature and pressure can be written as: $$(\mathrm{d} G)_{T,P} = \boldsymbol{\mu} \mathrm{d} \mathbf{n} = \boldsymbol{\mu}_c \mathrm{d} \mathbf{n}_c = \boldsymbol{\mu} \boldsymbol{\nu} \mathrm{d} \boldsymbol{\xi}$$

The equilibrium condition at constant temperature and pressure is then $\boldsymbol{\mu} \boldsymbol{\nu} = \mathbf{0}$.

The conservation matrix for a system is related to the stoichiometric number matrix by $\mathbf{A} \boldsymbol{\mu} = \mathbf{0}$, which means $\boldsymbol{\mu}$ is in the null space of $\mathbf{A}$ and $\mathbf{A}^\mathrm{T}$ is in the null space of $\boldsymbol{\mu}^\mathrm{T}$.

Isomers have the same chemical potential at equilibrium. When the chemical potential of B is specified, species that differ only in the number of B molecules that they contain become pseudo-isomers, and they have the same transformed chemical potential at equilibrium. The amount of species in a pseudo-isomer group $n'_i$. The number of pseudo-isomer groups $N′$.

### Other work terms

When introducing work terms other than volumetric or chemical work, their extensive variables may be proportional to the amounts of species, such as gravitational work. In these cases, the new work terms are not independent of the chemical work terms and do not introduce new natural variables to internal energy.