Lecture on the topic: “Thermodynamic potentials”

Plan:

Group of potentials “E F G H”, having the dimension of energy.

Dependence of thermodynamic potentials on the number of particles. Entropy as thermodynamic potential.

Thermodynamic potentials of multicomponent systems.

Practical implementation of the method of thermodynamic potentials (using the example of a chemical equilibrium problem).

One of the main methods of modern thermodynamics is the method of thermodynamic potentials. This method arose, largely, thanks to the use of potentials in classical mechanics, where its change was associated with the work performed, and the potential itself is an energy characteristic of a thermodynamic system. Historically, the originally introduced thermodynamic potentials also had the dimension of energy, which determined their name.

The mentioned group includes the following systems:

Internal energy;

Free energy or Helmholtz potential;

Thermodynamic Gibbs potential;

Enthalpy.

The potential of internal energy was shown in the previous topic. The potentiality of the remaining quantities follows from it.

The thermodynamic potential differentials take the form:

From relations (3.1) it is clear that the corresponding thermodynamic potentials characterize the same thermodynamic system in different ways.... descriptions (methods of specifying the state of a thermodynamic system). Thus, for an adiabatically isolated system described in variables, it is convenient to use internal energy as a thermodynamic potential. Then the parameters of the system, thermodynamically conjugate to the potentials, are determined from the relations:

, , , (3.2)

If the “system in a thermostat” defined by the variables is used as a description method, it is most convenient to use free energy as the potential . Accordingly, for the system parameters we obtain:

, , , (3.3)

Next, we will choose the “system under the piston” model as a description method. In these cases, the state functions form a set (), and the Gibbs potential G is used as the thermodynamic potential. Then the system parameters are determined from the expressions:

, , , (3.4)

And in the case of an “adiabatic system over a piston”, defined by state functions, the role of thermodynamic potential is played by enthalpy H. Then the system parameters take the form:

, , , (3.5)

Since relations (3.1) define the total differentials of thermodynamic potentials, we can equate their second derivatives.

For example, Considering that

we get

(3.6a)

Similarly, for the remaining parameters of the system related to the thermodynamic potential, we write:

(3.6b-e)

Similar identities can be written for other sets of parameters of the thermodynamic state of the system based on the potentiality of the corresponding thermodynamic functions.

So, for a “system in a thermostat” with potential , we have:

For a system “above the piston” with a Gibbs potential, the following equalities will be valid:

And finally, for a system with an adiabatic piston with potential H, we obtain:

Equalities of the form (3.6) – (3.9) are called thermodynamic identities and in a number of cases turn out to be convenient for practical calculations.

The use of thermodynamic potentials makes it possible to quite simply determine the operation of the system and the thermal effect.

Thus, from relations (3.1) it follows:

From the first part of the equality follows the well-known proposition that the work of a thermally insulated system ( ) is produced due to the decrease in its internal energy. The second equality means that free energy is that part of the internal energy that, during an isothermal process, is completely converted into work (accordingly, the “remaining” part of the internal energy is sometimes called bound energy).

The amount of heat can be represented as:

From the last equality it is clear why enthalpy is also called heat content. During combustion and other chemical reactions occurring at constant pressure (), the amount of heat released is equal to the change in enthalpy.

Expression (3.11), taking into account the second law of thermodynamics (2.7), allows us to determine the heat capacity:

All thermodynamic potentials of the energy type have the property of additivity. Therefore we can write:

It is easy to see that the Gibbs potential contains only one additive parameter, i.e. the specific Gibbs potential does not depend on. Then from (3.4) it follows:

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A physical quantity whose elementary change during the transition of a system from one state to another is equal to the amount of heat received or given divided by the temperature at which this transition occurred is called entropy.

For an infinitesimal change in the state of the system:

When a system transitions from one state to another, the change in entropy can be calculated as follows:

Based on the first law of thermodynamics, we can obtain

dS=dQ/T=C V dT/T+RdV/V, and

In an isothermal process T=const, i.e. T 1 = T 2:

DS=R×ln(V 2 /V 1).

In an isobaric process p=const, i.e. V 2 /V 1 =T 2 /T 1:

DS=(C V +R)×ln(T 2 /T 1)=C p ×ln(T 2 /T 1)=C p ×ln(V 2 /V 1).

For an isochoric process, V=const, i.e. V 1 = V 2:

DS=C V ×ln(T 2 /T 1).

In an adiabatic process dQ=0, i.e. DS=0:

S 1 =S 2 =const.

Changes in the entropy of a system performing a Carnot cycle:

DS=-(Q 1 /T 1 +Q 2 /T 2).

The entropy of a closed system performing a reversible Carnot cycle does not change:

dS=0 or S=const.

If the system undergoes an irreversible cycle, then dS>0.

Thus, the entropy of a closed (isolated) system cannot decrease during any processes occurring in it:

where the equal sign is valid for reversible processes, and the inequality sign is valid for irreversible ones.

The second law of thermodynamics: “In an isolated system, only such processes are possible in which the entropy of the system increases.” That is

dS³0 or dS³dQ/T.

The second law of thermodynamics determines the direction of thermodynamic processes and indicates the physical meaning of entropy: entropy is a measure of energy dissipation, i.e. characterizes that part of energy that cannot be converted into work.

Thermodynamic potentials are certain functions of volume V, pressure p, temperature T, entropy S, number of particles of the system N and other macroscopic parameters x that characterize the state of the thermodynamic system. These include: internal energy U=U(S,V,N,x), enthalpy H=H(S,p,N,x); free energy – F=F(V,T,N,x), Gibbs energy G=G(p,T,N,x).

The change in the internal energy of a system in any process is defined as the algebraic sum of the amount of heat Q that the system exchanges with the environment during the process, and the work A done by the system or performed on it. This reflects the first law of thermodynamics:

The change in U is determined only by the values ​​of the internal energy in the initial and final states:

For any closed process that returns the system to its original state, the change in internal energy is zero (U 1 =U 2 ; DU = 0; Q = A).

The change in the internal energy of the system in an adiabatic process (at Q = 0) is equal to the work done on the system or done by the system DU = A.

In the case of the simplest physical system with small intermolecular interactions (ideal gas), the change in internal energy is reduced to a change in the kinetic energy of the molecules:

where m is the gas mass;

c V – specific heat capacity at constant volume.

Enthalpy (heat content, Gibbs thermal function) – characterizes the state of a macroscopic system in thermodynamic equilibrium when choosing entropy S and pressure p – H(S,p,N,x) as the main independent variables.

Enthalpy is an additive function (i.e., the enthalpy of the entire system is equal to the sum of the enthalpies of its constituent parts). Enthalpy is related to the internal energy U of the system by the relation:

where V is the volume of the system.

The total enthalpy differential (with constant N and x) has the form:

From this formula we can determine the temperature T and volume V of the system:

T=(dH/dS), V=(dH/dp).

At constant pressure, the heat capacity of the system is

These properties of enthalpy at constant pressure are similar to the properties of internal energy at constant volume:

T=(dU/dS), p=-(dU/dV), c V =(dU/dT).

Free energy is one of the names for isochoric-isothermal thermodynamic potential or Helmholtz energy. It is defined as the difference between the internal energy of a thermodynamic system (U) and the product of its entropy (S) and temperature (T):

where TS is the bound energy.

Gibbs energy – isobaric-isothermal potential, free enthalpy, characteristic function of a thermodynamic system with independent parameters p, T and N – G. Determined through the enthalpy H, entropy S and temperature T by the equality

With free energy - the Helmholtz energy, the Gibbs energy is related by the relation:

The Gibbs energy is proportional to the number of particles N, per particle, called the chemical potential.

The work performed by a thermodynamic system in any process is determined by the decrease in the thermodynamic potential that meets the conditions of the process. Thus, with a constant number of particles (N=const) under thermal insulation conditions (adiabatic process, S=const), the elementary work dA is equal to the loss of internal energy:

For an isothermal process (T=const)

In this process, work is performed not only due to internal energy, but also due to the heat entering the system.

For systems in which the exchange of matter with the surrounding environment (change in N) is possible, processes are possible at constant p and T. In this case, the elementary work dA of all thermodynamic forces, except pressure forces, is equal to the decrease in the Gibbs thermodynamic potential (G), i.e.

According to Nernst's theorem, the change in entropy (DS) for any reversible isothermal processes occurring between two equilibrium states at temperatures approaching absolute zero tends to zero

Another equivalent formulation of Nernst's theorem is: "With the help of a sequence of thermodynamic processes it is impossible to achieve a temperature equal to absolute zero."

Lecture 14.

Basic inequality and basic equation of thermodynamics. The concept of thermodynamic potentials. Joule-Thompson effect. Le Chatelier-Brown principle. Introduction to thermodynamics of irreversible processes.

Basic inequality and fundamental equation of thermodynamics

For entropy, the relation is satisfied. Using the first law of thermodynamics, we get basic inequality of thermodynamics:

.

The equal sign corresponds to equilibrium processes . The basic equation of equilibrium (reversible) processes:

.

Method of thermodynamic potentials.

The application of the laws of thermodynamics makes it possible to describe many properties of macrosystems. For such a description, two ways have historically developed: the method of cycles and the method of thermodynamic functions. The first is based on the analysis of reversible cycles, and the second is based on the application of thermodynamic functions (potentials) introduced by Gibbs.

The starting point for deriving all thermodynamic potentials is the basic equation of thermodynamics:

,

connecting five quantities ( T, S, U, p, V), which can be state parameters or considered as functions of the system state.

To determine the state of the simplest thermodynamic system, it is enough to set the values ​​of two independent parameters. Therefore, to find the values ​​of the remaining three parameters, it is necessary to determine three more equations, one of which is the basic equation of thermodynamics, and the other two can be, for example, an equation of state and an additional equation arising from the properties of a specific state of the system:

;
;
.

In general, thermodynamic potentials can refer to any state function (for example, internal energy or entropy) if it is defined as an independent function of state parameters. Therefore, the number of thermodynamic functions is very large. Usually, those that have the following property are considered: the partial derivatives of the function with respect to the corresponding parameters are equal to one or another parameter of the state of the system.

Thermodynamic potentials ( thermodynamic functions ) – these are certain functions of volume, pressure, temperature, entropy, number of particles of the system and other macroscopic parameters characterizing the state of the system, which have the following property: if the thermodynamic potential is known, then by differentiating it according to the parameters noted above, all other parameters that determine the state of the system can be obtained.

Examples of thermodynamic potentials.

1) V and entropy S . Then it follows from the basic equation of thermodynamics:
. Where do we find
,
. Hence, internal energy
- potential.

The meaning of internal energy as potential : with V=const we get:
, i.e. the change in internal energy is equal to the amount of heat supplied to the system during an isochoric process.

If the process is irreversible, then
or
.

2) Let us choose pressure as independent parameters p and entropy S .

Subject to equality
and the basic equation of thermodynamics:
, we obtain that from the relation: it follows:
. Now let’s introduce the notation:
. Then
And
,
. Means, function
is a thermodynamic potential and is called enthalpy.

The meaning of enthalpy as a thermodynamic potential : at p=const we get that
, i.e. the change in enthalpy is equal to the amount of heat supplied during an isobaric process.

If the process is irreversible, then
or ,
.

3) Let us choose volume as independent parameters V and temperature T .

Let's rewrite the basic equation of thermodynamics
as:
and taking into account equality
we get: or . Now we introduce the notation:
, Then
,
,
. Thus, function
- thermodynamic potential, which is called free energy or Helmholtz thermodynamic potential.

The meaning of free energy as thermodynamic potential : with T=const we get: , i.e. the decrease in free energy is equal to the work done by the system in an isothermal process.

If the process is irreversible, then
or , i.e.

.

In an irreversible isothermal and isochoric process
- free energy decreases until the system reaches thermodynamic equilibrium - in this case, free energy takes on a minimum value.

Thermodynamic potentials, Pike, p.36

Thermodynamic potentials, Pike, p.36

For isolated systems, this relationship is equivalent to the classical formulation that entropy can never decrease. This conclusion was made by Nobel laureate I. R. Prigogine, analyzing open systems. He also put forward the principle that disequilibrium can serve as a source of order.

Third beginning thermodynamics describes the state of a system near absolute zero. In accordance with the third law of thermodynamics, it establishes the origin of entropy and fixes it for any system. At T 0 the coefficient of thermal expansion, the heat capacity of any process, goes to zero. This allows us to conclude that at absolute zero temperature, any changes in state occur without a change in entropy. This statement is called the theorem of Nobel laureate V. G. Nernst, or the third law of thermodynamics.

The third law of thermodynamics states :

absolute zero is fundamentally unattainable because when T = 0 And S = 0.

If there existed a body with a temperature equal to zero, then it would be possible to construct a perpetual motion machine of the second kind, which contradicts the second law of thermodynamics.

Modification of the third law of thermodynamics for calculating chemical equilibrium in a system formulated by Nobel Prize laureate M. Planck in this way.

Planck's postulate : at absolute zero temperature, entropy takes the value S 0 , independent of pressure, state of aggregation, and other characteristics of the substance. This value can be set equal to zero, orS 0 = 0.

In accordance with statistical theory, the value of entropy is expressed as S = ln, where  – Boltzmann constant,  – statistical weight, or thermodynamic probability of macrostates. It is also called -potential. By statistical weight we mean the number of microstates with the help of which a given macrostate is realized. Entropy of an ideal crystal at T = 0 K, provided  = 1, or in the case where a macrostate can be realized by a single microstate, is equal to zero. In all other cases, the entropy value at absolute zero must be greater than zero.

3.3. Thermodynamic potentials

Thermodynamic potentials are functions of certain sets of thermodynamic parameters, allowing one to find all the thermodynamic characteristics of the system as a function of these same parameters.

Thermodynamic potentials completely determine the thermodynamic state of the system, and by differentiation and integration any parameters of the system can be calculated.

The main thermodynamic potentials include the following functions .

1. Internal energy U, which is a function of independent variables:

entropy S,

volume V,

number of particles N,

generalized coordinates x i

or U = U(S, V, N,x i).

2. Helmholtz free energy F is a function of temperature T, volume V, number of particles N, generalized coordinates x i So F = F(T, V, N, x t).

3. Thermodynamic Gibbs potential G = G(T, p, N, x i).

4. Enthalpy H =H(S, P, N, x i).

5. Thermodynamic potential , for which the independent variables are temperature T, volume V, chemical potential x,  =  (T, V, N, x i).

There are classical relationships between thermodynamic potentials:

U = F + TS = H – PV,

F = U – TS = H – TS – PV,

H = U + PV = F + TS + PV,

G = U – TS + PV = F + PV = H – TS,

 = U – TS – V = F – N = H – TS – N, (3.12)

U = G + TS – PV =  + TS + N,

F = G – PV =  + N,

H = G + TS =  + TS + N,

G =  + PV + N,

 = G – PV – N.

The existence of thermodynamic potentials is a consequence of the first and second laws of thermodynamics and shows that the internal energy of the system U depends only on the state of the system. The internal energy of the system depends on the full set of macroscopic parameters, but does not depend on the method of achieving this state. Let us write the internal energy in differential form

dU = TdS– PdV– X i dx i + dN,

T = ( U/ S) V, N, x= const,

P = –( U/ V) S, N, x= const,

 = ( U/ N) S, N, x= const.

Similarly we can write

dF = – SDT–PdV – X t dx t + dN,

dH= TdS+VdP– X t dx t + dN,

dG= – SdT+VdP – X i dx i + dN,

d = – SDT–PdV – X t dx t – NdN,

S = – ( F/ T) V ; P = –( F/ V) T ; T = ( U/ S) V ; V = ( U/ P) T ;

S = – ( G/ T) P ; V = ( G/ P) S ; T = ( H/ S;); P = – ( U/ V) S

S = – ( F/ T); N = – ( F/);  = ( F/ N); X = – ( U/ x).

These equations take place for equilibrium processes. Let us pay attention to the thermodynamic isobaric-isothermal potential G, called Gibbs free energy,

G = U – TS + PV = H –TS, (3.13)

and isochoric-isothermal potential

F = U – T.S. (3.14)

which is called Helmholtz free energy.

In chemical reactions occurring at constant pressure and temperature,

G =  U – TS + PV = N, (3.15)

where  – chemical potential.

Under the chemical potential of some component of the system i we will understand the partial derivative of any of the thermodynamic potentials with respect to the amount of this component at constant values ​​of the remaining thermodynamic variables.

Chemical potential can also be defined as a quantity that determines the change in the energy of a system when one particle of a substance is added, for example,

 i = ( U/ N) S , V= cost , or G =  i N i .

From the last equation it follows that  = G/ N i , that is  represents the Gibbs energy per particle. Chemical potential is measured in J/mol.

Omega potential  is expressed in terms of a large statistical sum Z How

 = – Tln Z, (3.16)

Where [sum over N And k(N)]:

Z=   exp[( N – E k (N))/T].

thermodynamic potentials, thermodynamic potentials of elements

Thermodynamic potentials- internal energy, considered as a function of entropy and generalized coordinates (volume of the system, interface area, length of an elastic rod or spring, polarization of the dielectric, magnetization of the magnet, masses of the system components, etc.), and thermodynamic characteristic functions obtained by applying the Legendre transformation to internal energy

.

The purpose of introducing thermodynamic potentials is to use such a set of natural independent variables that describe the state of a thermodynamic system, which is most convenient in a particular situation, while maintaining the advantages that the use of characteristic functions with the dimension of energy gives. in particular, the decrease in thermodynamic potentials in equilibrium processes occurring at constant values ​​of the corresponding natural variables is equal to useful external work.

Thermodynamic potentials were introduced by W. Gibbs, who spoke of “fundamental equations”; The term thermodynamic potential belongs to Pierre Duhem.

The following thermodynamic potentials are distinguished:

• internal energy
• enthalpy
• Helmholtz free energy
• Gibbs potential
• high thermodynamic potential

• 1 Definitions (for systems with a constant number of particles)
  • 1.1 Internal energy
  • 1.2 Enthalpy
  • 1.3 Helmholtz free energy
  • 1.4 Gibbs potential
• 2 Thermodynamic potentials and maximum work
• 3 Canonical equation of state
• 4 Transition from one thermodynamic potential to another. Gibbs - Helmholtz formulas
• 5 Method of thermodynamic potentials. Maxwell's relations
• 6 Systems with a variable number of particles. Large thermodynamic potential
• 7 Potentials and thermodynamic equilibrium
• 8 Notes
• 9 Literature

Definitions (for systems with a constant number of particles)

Internal energy

Defined in accordance with the first law of thermodynamics, as the difference between the amount of heat imparted to the system and the work done by the system on external bodies:

.

Enthalpy

Defined as follows:

,

where is pressure and is volume.

Since the work is equal in an isobaric process, the increment in enthalpy in a quasi-static isobaric process is equal to the amount of heat received by the system.

Helmholtz free energy

Also often called simply free energy. Defined as follows:

,

where is temperature and is entropy.

Since in an isothermal process the amount of heat received by the system is equal, the loss of free energy in a quasi-static isothermal process is equal to the work done by the system on external bodies.

Gibbs potential

Also called Gibbs energy, thermodynamic potential, Gibbs free energy and even just free energy(which can lead to mixing of the Gibbs potential with the Helmholtz free energy):

.

Thermodynamic potentials and maximum work

Internal energy represents the total energy of the system. However, the second law of thermodynamics prohibits converting all internal energy into work.

It can be shown that the maximum total work (both on the environment and on external bodies) that can be obtained from a system in an isothermal process is equal to the decrease in Helmholtz free energy in this process:

,

where is the Helmholtz free energy.

In this sense, it represents free energy that can be converted into work. The remaining part of the internal energy can be called bound.

In some applications, you have to distinguish between complete and useful work. The latter represents the work of the system on external bodies, excluding the environment in which it is immersed. The maximum useful work of the system is

where is the Gibbs energy.

In this sense, Gibbs energy is also free.

Canonical equation of state

Specifying the thermodynamic potential of a certain system in a certain form is equivalent to specifying the equation of state of this system.

The corresponding thermodynamic potential differentials are:

• for internal energy

,

• for enthalpy

,

• for Helmholtz free energy

,

• for the Gibbs potential

.

These expressions can be considered mathematically as complete differentials of functions of two corresponding independent variables. Therefore, it is natural to consider thermodynamic potentials as functions:

, .

Specifying any of these four dependencies - that is, specifying the type of functions - allows you to obtain all the information about the properties of the system. So, for example, if we are given internal energy as a function of entropy and volume, the remaining parameters can be obtained by differentiation:

Here the indices mean the constancy of the second variable on which the function depends. These equalities become obvious if we consider that.

Setting one of the thermodynamic potentials as a function of the corresponding variables, as written above, represents the canonical equation of state of the system. Like other equations of state, it is valid only for states of thermodynamic equilibrium. In nonequilibrium states, these dependencies may not hold.

Transition from one thermodynamic potential to another. Gibbs - Helmholtz formulas

The values ​​of all thermodynamic potentials in certain variables can be expressed in terms of a potential whose differential is complete in these variables. For example, for simple systems in variables, thermodynamic potentials can be expressed in terms of Helmholtz free energy:

The first of these formulas is called the Gibbs-Helmholtz formula, but the term is sometimes applied to all similar formulas in which temperature is the only independent variable.

Method of thermodynamic potentials. Maxwell's relations

The method of thermodynamic potentials helps to transform expressions that include basic thermodynamic variables and thereby express such “hard-to-observe” quantities as the amount of heat, entropy, internal energy through measured quantities - temperature, pressure and volume and their derivatives.

Let us again consider the expression for the total differential of internal energy:

.

It is known that if mixed derivatives exist and are continuous, then they do not depend on the order of differentiation, that is

.

But also, therefore

.

Considering the expressions for other differentials, we obtain:

, .

These relations are called Maxwell's relations. Note that they are not satisfied in the case of discontinuity of mixed derivatives, which occurs during phase transitions of the 1st and 2nd order.

Systems with a variable number of particles. Large thermodynamic potential

The chemical potential () of a component is defined as the energy that must be expended in order to add an infinitesimal molar amount of this component to the system. Then the expressions for the differentials of thermodynamic potentials can be written as follows:

, .

Since thermodynamic potentials must be additive functions of the number of particles in the system, the canonical equations of state take the following form (taking into account that S and V are additive quantities, but T and P are not):

, .

And, since from the last expression it follows that

,

that is, the chemical potential is the specific Gibbs potential (per particle).

For a large canonical ensemble (that is, for a statistical ensemble of states of a system with a variable number of particles and an equilibrium chemical potential), a large thermodynamic potential can be defined that relates the free energy to the chemical potential:

;

It is easy to verify that the so-called bound energy is a thermodynamic potential for a system given with constants.

Potentials and thermodynamic equilibrium

In a state of equilibrium, the dependence of thermodynamic potentials on the corresponding variables is determined by the canonical equation of state of this system. However, in states other than equilibrium, these relationships lose their validity. However, thermodynamic potentials also exist for nonequilibrium states.

Thus, with fixed values ​​of its variables, the potential can take on different values, one of which corresponds to the state of thermodynamic equilibrium.

It can be shown that in a state of thermodynamic equilibrium the corresponding potential value is minimal. Therefore, the equilibrium is stable.

The table below shows the minimum of which potential corresponds to the state of stable equilibrium of a system with given fixed parameters.

Notes

1. Krichevsky I.R., Concepts and fundamentals of thermodynamics, 1970, p. 226–227.
2. Sychev V.V., Complex thermodynamic systems, 1970.
3. Kubo R., Thermodynamics, 1970, p. 146.
4. Munster A., ​​Chemical thermodynamics, 1971, p. 85–89.
5. Gibbs J. W., The Collected Works, Vol. 1, 1928.
6. Gibbs J.W., Thermodynamics. Statistical Mechanics, 1982.
7. Duhem P., Le potentiel thermodynamique, 1886.
8. Gukhman A. A., On the foundations of thermodynamics, 2010, p. 93.

Literature

• Duhem P. Le potentiel thermodynamique et ses applications à la mécanique chimique et à l "étude des phénomènes électriques. - Paris: A. Hermann, 1886. - XI + 247 pp.
• Gibbs J. Willard. The Collected Works. - N. Y. - London - Toronto: Longmans, Green and Co., 1928. - T. 1. - XXVIII + 434 p.
• Bazarov I. P. Thermodynamics. - M.: Higher School, 1991. 376 p.
• Bazarov I. P. Misconceptions and errors in thermodynamics. Ed. 2nd rev. - M.: Editorial URSS, 2003. 120 p.
• Gibbs J. W. Thermodynamics. Statistical mechanics. - M.: Nauka, 1982. - 584 p. - (Classics of science).
• Gukhman A. A. On the foundations of thermodynamics. - 2nd ed., corrected. - M.: Publishing house LKI, 2010. - 384 p. - ISBN 978-5-382-01105-9.
• Zubarev D.N. Nonequilibrium statistical thermodynamics. M.: Nauka, 1971. 416 p.
• Kvasnikov I. A. Thermodynamics and statistical physics. Equilibrium Systems Theory, vol. 1. - M.: Moscow State University Publishing House, 1991. (2nd ed., revised and supplemented. M.: URSS, 2002. 240 pp.)
• Krichevsky I. R. Concepts and fundamentals of thermodynamics. - 2nd ed., revision. and additional - M.: Chemistry, 1970. - 440 p.
• Kubo R. Thermodynamics. - M.: Mir, 1970. - 304 p.
• Landau, L. D., Lifshits, E. M. Statistical physics. Part 1. - 3rd edition, supplemented. - M.: Nauka, 1976. - 584 p. - (“Theoretical Physics”, Volume V).
• Mayer J., Geppert-Mayer M. Statistical mechanics. M.: Mir, 1980.
• Munster A. Chemical thermodynamics. - M.: Mir, 1971. - 296 p.
• Sivukhin D.V. General course in physics. - M.: Nauka, 1975. - T. II. Thermodynamics and molecular physics. - 519 p.
• Sychev V.V. Complex thermodynamic systems. - 4th ed., revised. and additional.. - M: Energoatomizdat, 1986. - 208 p.
• Thermodynamics. Basic concepts. Terminology. Letter designations of quantities. Collection of definitions, vol. 103/ Committee of Scientific and Technical Terminology of the USSR Academy of Sciences. M.: Nauka, 1984

thermodynamic potentials, thermodynamic potentials of elements, thermodynamic potentials