thermodynamic potentials, thermodynamic potentials of elements

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

The following thermodynamic potentials are distinguished:

• internal energy
• enthalpy
• Helmholtz free energy
• Gibbs potential
• great 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

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

Helmholtz free energy

Also often referred to simply free energy. Defined as follows:

,

where is the temperature and is the 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

The internal energy is the total energy of the system. However, the second law of thermodynamics prohibits the conversion of all internal energy into work.

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

,

where is the Helmholtz free energy.

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

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

where is the Gibbs energy.

In this sense, the Gibbs energy is also free.

Canonical equation of state

Setting the thermodynamic potential of a certain system in a certain form is equivalent to setting the equation of state for this system.

The corresponding differentials of thermodynamic potentials are:

• for internal energy

,

• for enthalpy

,

• for the Helmholtz free energy

,

• for the Gibbs potential

.

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

, .

Setting any of these four dependencies - that is, specifying the type of functions - allows you to get 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 and 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, is 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 the Helmholtz free energy:

The first of these formulas is called the Gibbs-Helmholtz formula, but this term is sometimes applied to all such 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 the main 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.

Consider again 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 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 takes place 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, while T and P are not):

, .

And since it follows from the last expression that

,

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

For the grand canonical ensemble (that is, for the 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 the 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 relations lose their force. However, for non-equilibrium states, thermodynamic potentials also exist.

Thus, for 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 the state of thermodynamic equilibrium the corresponding value of the potential is minimal. Therefore, the equilibrium is stable.

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

Notes

1. Krichevsky I. R., Concepts and foundations 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 JW, 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 p.
• Gibbs J. Willard. The Collected Works. - N. Y. - London - Toronto: Longmans, Green and Co., 1928. - V. 1. - XXVIII + 434 p.
• Bazarov I.P. Thermodynamics. - M.: Higher school, 1991. 376 p.
• Bazarov IP Delusions 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.: Izd-vo LKI, 2010. - 384 p. - ISBN 978-5-382-01105-9.
• Zubarev D.N. Non-equilibrium statistical thermodynamics. M.: Nauka, 1971. 416 p.
• Kvasnikov IA Thermodynamics and statistical physics. Theory of equilibrium systems, vol. 1. - M.: Publishing House of Moscow State University, 1991. (2nd ed., Rev. and add. M.: URSS, 2002. 240 p.)
• 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., Lifshitz, E. M. Statistical Physics. Part 1. - Edition 3, 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 DV General course of physics. - M.: Nauka, 1975. - T. II. Thermodynamics and molecular physics. - 519 p.
• Sychev VV 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 Academy of Sciences of the USSR. Moscow: Nauka, 1984

thermodynamic potentials, thermodynamic potentials of elements, thermodynamic potentials

The method of thermodynamic potentials or the method of characteristic functions was developed by Gibbs. This is an analytical method based on the use of the basic equation of thermodynamics for quasi-static processes.

The idea of ​​the method is that the basic equation of thermodynamics allows for a system under various conditions to introduce some state functions, called thermodynamic potentials, the change of which with a change in state is a total differential; using this, one can compose the equations necessary for the analysis of a particular phenomenon.

Consider simple systems. In this case, for quasi-static processes, the basic TD equation has the form for a closed system.

How will this equation change if the number of particles changes? Internal energy and entropy are proportional to the number of particles in the system: ~, ~, therefore ~, ~, and the equation will look like for an open system, where
- the chemical potential will be the generalized force for the independent variable of the number of particles in the system.

This equation relates five quantities, two of which are state functions: . The state of a simple system itself is determined by two parameters. Therefore, choosing two of the five named quantities as independent variables, we find that the main equation contains three more unknown functions. To determine them, it is necessary to add two more equations to the main equation, which can be the thermal and caloric equations of state: , , if , are chosen as independent parameters.

However, the definition of these three unknown quantities is simplified with the introduction of thermodynamic potentials.

We express from the main equation : for a closed system
or for an open system

We see that the increase in internal energy is completely determined by the increase in entropy and the increase in volume, i.e. if we choose or for an open system as independent variables, then to determine the other three variables, we need to know only one equation for the internal energy as a function or as a function of .

So, knowing the dependence , it is possible, using the basic TD identity, by simple differentiation (taking the first derivatives) to determine both other thermal variables:

If we take the second derivatives of , then we can determine the caloric properties of the system: and - the adiabatic modulus of elasticity of the system (determines the change in pressure \ elasticity \ per unit change in volume and is the reciprocal of the compressibility coefficient):

Taking into account that is the total differential, and equating the mixed derivatives , we find the relationship between the two properties of the system - the change in temperature during its adiabatic expansion and the change in pressure during isochoric heat transfer to the system:

Thus, the internal energy as a function of the variables , is a characteristic function. Its first derivatives determine the thermal properties of the system, the second - the caloric properties of the system, mixed - the relationship between other properties of the system. The establishment of such connections is the content of the method of TD potentials. A is one of the many TD potentials.

We can find an expression for TD potentials, its explicit, only for 2 systems, one of which is an ideal gas, the other is equilibrium radiation, since both the equations of state and the internal energy as a function of parameters are known for them. For all other TD systems, the potentials are found either from experience or by the methods of statistical physics, and then, using the obtained TD relations, the equations of state and other properties are determined. For gases, TD functions are most often calculated by methods of statistical physics; for liquids and solids, they are usually found experimentally using caloric definitions of heat capacity.

We obtain an expression for the internal energy of an ideal gas as a TD potential, i.e. as functions :

For an ideal gas, the internal energy only depends on ,
on the other hand, the entropy of an ideal gas depends on: . Express from the second equation and substitute into the first equation:

Let's take a logarithm

We take into account that

Transforming the second factor, we get:

We substitute the resulting expression into the first equation and obtain the TD potential internal energy: .

From a practical point of view, internal energy as a TD potential is inconvenient because one of its independent variables, entropy, cannot be measured directly, like the quantities .

Consider other TD potentials, transform the basic thermodynamic identity so that it includes differentials and .

We see that the TD function enthalpy is a TD potential with independent variables , since the derivatives of this function give the rest of the characteristics of the system.

Caloric and adiabatic modulus of elasticity;

give second derivatives.

The relationship between the two properties of the system, namely, the adiabatic change in temperature with a change in pressure and the isobaric change in volume when heat is imparted to the system, will be obtained by calculating the mixed derivatives:

Consider the TD potential, in independent variables, convenient for measurement. Let us transform the main TD identity so that it includes the differentials and .

We see that the TD free energy function or the Helmholtz function is a TD potential with independent variables , since the derivatives of this function give the remaining characteristics of the system.

Thermal, give the first derivatives.

Caloric heat capacity and compressibility coefficient - second derivatives:

This implies ;

This implies .

Mixed derivatives establish a relationship between two properties of a system - a change in entropy during its isothermal expansion and a change in pressure during isochoric heating:

Consider another function, with a different set of variables that are convenient for measurement. Let us transform the main TD identity so that it includes the differentials and .

The TD function is called the Gibbs potential, the Gibbs free energy is the TD potential with independent variables, since the derivatives of this function give the remaining characteristics of the system.

Thermal , , allowing, knowing the explicit form of the function, to find the thermal equation of the state of the system.

Caloric heat capacity and compressibility factor:

This implies ;

This implies .

Mixed derivatives establish a relationship between two properties of a system −

the change in entropy during its isothermal change in pressure and the change in volume during isobaric heating:

As you can see, in the general case, thermodynamic potentials are functions of three variables for open one-component systems and functions of only two variables for closed systems. Each TD potential contains completely all the characteristics of the system. and; from and expressions we obtain for .

The method of TD potentials and the method of cycles are two methods used in TD to study physical phenomena.

The change in entropy uniquely determines the direction and limit of the spontaneous flow of the process only for the simplest systems - isolated ones. In practice, for the most part, one has to deal with systems that interact with the environment. To characterize the processes occurring in closed systems, new thermodynamic state functions were introduced: isobaric-isothermal potential (Gibbs free energy) and isochoric-isothermal potential (Helmholtz free energy).

The behavior of any thermodynamic system in the general case is determined by the simultaneous action of two factors - enthalpy, which reflects the system's desire to minimize thermal energy, and entropy, which reflects the opposite trend - the system's desire for maximum disorder. If for isolated systems (ΔH = 0) the direction and limit of the spontaneous flow of the process is uniquely determined by the magnitude of the change in the entropy of the system ΔS, and for systems at temperatures close to absolute zero (S = 0 or S = const), the criterion for the direction of the spontaneous process is the change enthalpy ΔH, then for closed systems at temperatures not equal to zero, it is necessary to simultaneously take into account both factors. The direction and limit of the spontaneous flow of the process in any systems is determined by the more general principle of the minimum free energy:

Only those processes that lead to a decrease in the free energy of the system can proceed spontaneously; the system comes to a state of equilibrium when the free energy reaches its minimum value.

For closed systems that are in isobaric-isothermal or isochoric-isothermal conditions, the free energy takes the form of isobaric-isothermal or isochoric-isothermal potentials (the so-called Gibbs and Helmholtz free energy, respectively). These functions are sometimes called simply thermodynamic potentials, which is not quite strict, since internal energy (isochorically isentropic) and enthalpy (isobaric isentropic potential) are also thermodynamic potentials.

Consider a closed system in which an equilibrium process is carried out at constant temperature and volume. We express the work of this process, which we denote by A max (since the work of the process carried out in equilibrium is maximum), from equations (I.53, I.54):

(I.69)

We transform expression (I.69) by grouping terms with the same indices:

Entering the designation:

we get:

(I.72) (I.73)

The function is an isochoric-isothermal potential (Helmholtz free energy), which determines the direction and limit of the spontaneous flow of the process in a closed system under isochoric-isothermal conditions.

A closed system under isobaric-isothermal conditions is characterized by the isobaric-isothermal potential G:

(1.75)

(I.74)

Since –ΔF = A max , we can write:

The value A "max is called maximum useful work(maximum work minus expansion work). Based on the principle of minimum free energy, it is possible to formulate the conditions for the spontaneous flow of the process in closed systems.

Conditions for spontaneous processes in closed systems:

Isobaric-isothermal(P=const, T=const):

ΔG<0.dG<0

Isochoric-isothermal(V=const, T=const):

ΔF<0.dF< 0

Processes that are accompanied by an increase in thermodynamic potentials occur only when work is performed on the system from the outside. In chemistry, the isobaric-isothermal potential is most commonly used, since most chemical (and biological) processes occur at constant pressure. For chemical processes, the value of ΔG can be calculated, knowing the ΔH and ΔS of the process, according to equation (I.75), or using tables of standard thermodynamic potentials for the formation of substances ΔG ° arr; in this case, ΔG° of the reaction is calculated similarly to ΔH° according to equation (I.77):

The value of the standard change in the isobaric-isothermal potential in the course of any chemical reaction ΔG° 298 is a measure of the chemical affinity of the starting substances. Based on equation (I.75), it is possible to estimate the contribution of enthalpy and entropy factors to the value of ΔG and make some generalizing conclusions about the possibility of spontaneous occurrence of chemical processes, based on the sign of ΔН and ΔS.

1. exothermic reactions; ΔH<0.

a) If ΔS > 0, then ΔG is always negative; exothermic reactions accompanied by an increase in entropy always proceed spontaneously.

b) If ΔS< 0, реакция будет идти самопроизвольно при ΔН >TΔS (low temperatures).

2. Endothermic reactions; ΔH >0.

a) If ΔS > 0, the process will be spontaneous at ΔН< TΔS (высокие температуры).

b) If ΔS< 0, то ΔG всегда положительно; самопроизвольное протекание эндотермических реакций, сопровождающихся уменьшением энтропии, невозможно.

CHEMICAL EQUILIBRIUM

As shown above, the occurrence of a spontaneous process in a thermodynamic system is accompanied by a decrease in the free energy of the system (dG< 0, dF < 0). Очевидно, что рано или поздно (напомним, что понятие "время" в термодинамике отсутствует) система достигнет минимума свободной энергии. Условием минимума некоторой функции Y = f(x) является равенство нулю первой производной и положительный знак второй производной: dY = 0; d 2 Y >0. Thus, the condition for thermodynamic equilibrium in a closed system is the minimum value of the corresponding thermodynamic potential:

Isobaric-isothermal(P=const, T=const):

ΔG=0dG=0, d 2 G>0

Isochoric-isothermal(V=const, T=const):

ΔF=0dF=0, d 2 F>0

The state of the system with the minimum free energy is the state of thermodynamic equilibrium:

Thermodynamic equilibrium is such a thermodynamic state of a system that, under constant external conditions, does not change in time, and this invariability is not due to any external process.

The doctrine of equilibrium states is one of the branches of thermodynamics. Next, we will consider a special case of a thermodynamic equilibrium state - chemical equilibrium. As is known, many chemical reactions are reversible, i.e. can flow simultaneously in both directions - forward and reverse. If a reversible reaction is carried out in a closed system, then after a while the system will come to a state of chemical equilibrium - the concentrations of all reactants will cease to change with time. It should be noted that the achievement of a state of equilibrium by the system does not mean the termination of the process; chemical equilibrium is dynamic, i.e. corresponds to the simultaneous flow of the process in opposite directions at the same speed. The chemical equilibrium is mobile– any infinitely small external influence on the equilibrium system causes an infinitely small change in the state of the system; upon termination of the external influence, the system returns to its original state. Another important property of chemical equilibrium is that the system can spontaneously come to a state of equilibrium from two opposite sides. In other words, any state adjacent to the equilibrium one is less stable, and the transition to it from the equilibrium state is always associated with the need to expend work from outside.

The quantitative characteristic of chemical equilibrium is the equilibrium constant, which can be expressed in terms of equilibrium concentrations C, partial pressures P, or mole fractions X of the reactants. For some reaction

the corresponding equilibrium constants are expressed as follows:

(I.78) (I.79) (I.80)

The equilibrium constant is a characteristic quantity for every reversible chemical reaction; the value of the equilibrium constant depends only on the nature of the reacting substances and temperature. The expression for the equilibrium constant for an elementary reversible reaction can be derived from kinetic concepts.

Consider the process of establishing equilibrium in a system in which at the initial moment of time there are only initial substances A and B. The rate of the direct reaction V 1 at this moment is maximum, and the rate of the reverse reaction V 2 is equal to zero:

(I.81)

(I.82)

As the concentration of the starting substances decreases, the concentration of the reaction products increases; accordingly, the rate of the forward reaction decreases, the rate of the reverse reaction increases. Obviously, after some time, the rates of the forward and reverse reactions will become equal, after which the concentrations of the reactants will stop changing, i.e. chemical equilibrium is established.

Assuming that V 1 \u003d V 2, we can write:

(I.84)

Thus, the equilibrium constant is the ratio of the rate constants of the forward and reverse reactions. This implies the physical meaning of the equilibrium constant: it shows how many times the rate of the forward reaction is greater than the rate of the reverse at a given temperature and concentrations of all reactants equal to 1 mol / l.

Now consider (with some simplifications) a more rigorous thermodynamic derivation of the expression for the equilibrium constant. For this, it is necessary to introduce the concept chemical potential. Obviously, the value of the free energy of the system will depend both on external conditions (T, P or V), and on the nature and amount of substances that make up the system. If the composition of the system changes with time (i.e., a chemical reaction occurs in the system), it is necessary to take into account the effect of the composition change on the value of the free energy of the system. Let us introduce into some system an infinitely small number dn i moles of the i-th component; this will cause an infinitesimal change in the thermodynamic potential of the system. The ratio of an infinitesimal change in the value of the free energy of the system to an infinitesimal amount of a component introduced into the system is the chemical potential μ i of this component in the system:

(I.85) (I.86)

The chemical potential of a component is related to its partial pressure or concentration by the following relationships:

(I.87) (I.88)

Here μ ° i is the standard chemical potential of the component (P i = 1 atm., С i = 1 mol/l.). Obviously, the change in the free energy of the system can be related to the change in the composition of the system as follows:

Since the equilibrium condition is the minimum free energy of the system (dG = 0, dF = 0), we can write:

In a closed system, a change in the number of moles of one component is accompanied by an equivalent change in the number of moles of the remaining components; i.e., for the above chemical reaction, the following relation holds: If the system is in a state of chemical equilibrium, then the change in the thermodynamic potential is zero; we get:

(I.98) (I.99)

Here with i and p i – equilibrium concentrations and partial pressures of the initial substances and reaction products (in contrast to non-equilibrium С i and Р i in equations I.96 - I.97).

Since for each chemical reaction the standard change in the thermodynamic potential ΔF° and ΔG° is a strictly defined value, the product of equilibrium partial pressures (concentrations) raised to a power equal to the stoichiometric coefficient for a given substance in the chemical reaction equation (stoichiometric coefficients for starting substances are considered to be negative) there is a certain constant called the equilibrium constant. Equations (I.98, I.99) show the relationship between the equilibrium constant and the standard change in free energy during a reaction. The equation of the isotherm of a chemical reaction relates the values ​​of the real concentrations (pressures) of the reactants in the system, the standard change in the thermodynamic potential during the reaction, and the change in the thermodynamic potential during the transition from a given state of the system to equilibrium. The sign of ΔG (ΔF) determines the possibility of spontaneous flow of the process in the system. In this case, ΔG° (ΔF°) is equal to the change in the free energy of the system during the transition from the standard state (P i = 1 atm., С i = 1 mol/l) to the equilibrium state. The equation of the isotherm of a chemical reaction makes it possible to calculate the value of ΔG (ΔF) during the transition from any state of the system to equilibrium, i.e. answer the question whether the chemical reaction will proceed spontaneously at given concentrations C i (pressures P i) of the reagents:

If the change in the thermodynamic potential is less than zero, the process under these conditions will proceed spontaneously.

Similar information.

$S$ and generalized coordinates $x\_1,x\_2,...$(system volume, phase interface area, length of an elastic rod or spring, dielectric polarization, magnetization of a magnet, masses of system components, etc.), and thermodynamic characteristic functions obtained by applying the Legendre transformation to internal energy

$U=U(S,x\_1,x\_2,...)$.

The purpose of introducing thermodynamic potentials is to use such a set of natural independent variables describing the state of a thermodynamic system that is most convenient in a particular situation, while maintaining the advantages that the use of characteristic functions with energy dimension 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"; term thermodynamic potential owned by Pierre Duhem.

The following thermodynamic potentials are distinguished:

Definitions (for systems with a constant number of particles)

Internal energy

It is defined in accordance with the first law of thermodynamics, as the difference between the amount of heat communicated to the system and the work done by the system above external bodies:

$U=Q\; -\; A$.

Enthalpy

Defined as follows:

$H=U\; +\; PV$,

Since in an isothermal process the amount of heat received by the system is $T\backslash Delta\; S$, then decline free energy in a quasi-static isothermal process is equal to the work done by the system above 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):

$G\; =\; H\; -\; TS\; =\; F\; +\; PV\; =\; U+PV-TS$.

Thermodynamic potentials and maximum work

The internal energy is the total energy of the system. However, the second law of thermodynamics prohibits the conversion of all internal energy into work.

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

$A^f\_(max)=-\backslash Delta\; F$,

where $F$ is the Helmholtz free energy.

In this sense $F$ represents free energy that can be converted into work. The rest of the internal energy can be called related.

In some applications it is necessary to distinguish complete and useful work. The latter is the work of the system on external bodies, excluding the environment in which it is immersed. Maximum useful system work is equal to

$A^u\_(max)=-\backslash Delta\; G$

where $G$ is the Gibbs energy.

In this sense, the Gibbs energy is also free.

Canonical equation of state

Setting the thermodynamic potential of a certain system in a certain form is equivalent to setting the equation of state for this system.

The corresponding differentials of thermodynamic potentials are:

• for internal energy

$dU=\; \backslash delta\; Q\; -\; \backslash delta\; A=T\; dS\; -\; P\; dV$,

• for enthalpy

$dH\; =\; dU\; +\; d(PV)\; =\; T\; dS\; -\; P\; dV\; +\; P\; dV\; +\; V\; dP\; =\; T\; dS\; +\; V\; dP$,

• for the Helmholtz free energy

$dF\; =\; dU\; -\; d(TS)\; =\; T\; dS\; -\; P\; dV\; -\; T\; dS\; -\; S\; dT\; =\; -P\; dV\; -\; S\; dT$,

• for the Gibbs potential

$dG\; =\; dH\; -\; d(TS)\; =\; T\; dS\; +\; V\; dP\; -\; T\; dS\; -\; S\; dT\; =\; V\; dP\; -\; S\; dT$.

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

$U\; =\; U(S,V)$, $H\; =\; H(S,P)$, $F\; =\; F(T,V)$, $G\; =\; G(T,P)$.

Setting any of these four dependencies - that is, specifying the type of functions $U(S,V)$, $H(S,P)$, $F(T,V)$, $G(T,P)$- allows you to get all the information about the properties of the system. So, for example, if we are given the internal energy $U$ as a function of entropy $S$ and volume $V$, the remaining parameters can be obtained by differentiation:

$T=(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; S)\backslash right))\_V$ $P=-(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; V)\backslash right))\_S$

Here the indices $V$ and $S$ mean the constancy of the second variable on which the function depends. These equalities become clear when we consider that $dU\; =\; T\; dS\; -\; P\; dV$.

Specifying one of the thermodynamic potentials as a function of the corresponding variables, as written above, is canonical equation of state systems. Like other equations of state, it is valid only for states of thermodynamic equilibrium. In nonequilibrium states, these dependences may not be satisfied.

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 $V$, $T$ thermodynamic potentials can be expressed in terms of the Helmholtz free energy:

$U\; =\; -\; T^2\; \backslash left(\backslash frac(\backslash partial)(\backslash partial\; T\; )\backslash frac(F)(T)\; \backslash right)\_(V)$,

$H\; =\; -\; T^2\; \backslash left(\backslash frac(\backslash partial)(\backslash partial\; T\; )\backslash frac(F)(T)\; \backslash right)\_(V)\; -\; V\; \backslash left(\backslash frac(\backslash partial\; F)(\backslash partial\; V)\backslash right)\_(T)$,

$G=\; F-\; V\; \backslash left(\backslash frac(\backslash partial\; F)(\backslash partial\; V)\backslash right)\_(T)$.

The first of these formulas is called Gibbs-Helmholtz formula, but sometimes this term is applied to all such 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 the main 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.

Consider again the expression for the total differential of internal energy:

$dU\; =\; T\; dS\; -\; P\; dV$.

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

$\backslash frac(\backslash partial^2\; U)(\backslash partial\; V\; \backslash partial\; S)=\backslash frac(\backslash partial^2\; U)(\backslash partial\; S\; \backslash partial\; V)$.

But $(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; V)\backslash right))\_S=-P$ and $(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; S)\backslash right))\_V=T$, that's why

$(\backslash left(\backslash frac(\backslash partial\; P)(\backslash partial\; S)\backslash right))\_V=-(\backslash left(\backslash frac(\backslash partial\; T)(\backslash partial\; V)\backslash right))\_S$.

Considering expressions for other differentials, we obtain:

$(\backslash left(\backslash frac(\backslash partial\; T)(\backslash partial\; P)\backslash right))\_S=(\backslash left(\backslash frac(\backslash partial\; V)(\backslash partial\; S)\backslash right))\_P$, $(\backslash left(\backslash frac(\backslash partial\; S)(\backslash partial\; V)\backslash right))\_T=(\backslash left(\backslash frac(\backslash partial\; P)(\backslash partial\; T)\backslash right))\_V$, $(\backslash left(\backslash frac(\backslash partial\; S)(\backslash partial\; P)\backslash right))\_T=-(\backslash left(\backslash frac(\backslash partial\; V)(\backslash partial\; T)\backslash right))\_P$.

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

Systems with a variable number of particles. Large thermodynamic potential

Chemical potential ( $\backslash mu$ ) of a component is defined as the energy required to add an infinitesimal molar amount of that component to the system. Then the expressions for the differentials of thermodynamic potentials can be written as follows:

$dU\; =\; T\; dS\; -\; P\; dV\; +\; \backslash mu\; dN$, $dH\; =\; T\; dS\; +\; V\; dP\; +\; \backslash mu\; dN$, $dF\; =\; -S\; dT\; -\; P\; dV\; +\; \backslash mu\; dN$, $dG\; =\; -S\; dT\; +\; V\; dP\; +\; \backslash mu\; dN$.

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 the fact that $S$ and $V$ are additive quantities, and $T$ and $P$- No):

$U\; =\; U(S,V,N)\; =\; N\; f\; \backslash left(\backslash frac(S)(N),\backslash frac(V)(N)\backslash right)$, $H\; =\; H(S,P,N)\; =\; N\; f\; \backslash left(\backslash frac(S)(N),P\backslash right)$, $F\; =\; F(T,V,N)\; =\; N\; f\; \backslash left(T,\backslash frac(V)(N)\backslash right)$, $G\; =\; G(T,P,N)\; =\; N\; f\; \backslash left(T,P\backslash right)$.

And because $\backslash frac(d\; G)(dN)=\backslash mu$, it follows from the last expression that

$G\; =\; \backslash mu\; N$,

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

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

$\backslash Omega\; =\; F\; -\; \backslash mu\; N\; =\; -\; P\; V$; $d\; \backslash Omega\; =\; -S\; dT\; -\; N\; d\; \backslash mu\; -\; P\; dV$

It is easy to verify that the so-called bound energy $T\; S$ is the thermodynamic potential for a system given with constants $S\; P\; \backslash mu$.

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 relations lose their force. However, for non-equilibrium states, thermodynamic potentials also exist.

Thus, for 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 the state of thermodynamic equilibrium the corresponding value of the potential is minimal. Therefore, the equilibrium is stable.

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

fixed parameters	thermodynamic potential
S,V,N	internal energy
S,P,N	enthalpy
T,V,N	Helmholtz free energy
T,P,N	Gibbs potential
T,V, $\backslash mu$	Large thermodynamic potential
S,P, $\backslash mu$	bound energy

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Notes

Literature

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An excerpt characterizing Thermodynamic potentials

She looked to where she knew he was; but she could not see him otherwise than as he was here. She saw him again the same as he was in Mytishchi, at Trinity, in Yaroslavl.
She saw his face, heard his voice, and repeated his words and her own words spoken to him, and sometimes invented new words for herself and for him, which could then be said.
Here he is lying on an armchair in his velvet coat, leaning his head on a thin, pale arm. His chest is terribly low and his shoulders are raised. The lips are firmly compressed, the eyes shine, and a wrinkle jumps up and disappears on the pale forehead. One of his legs is trembling slightly. Natasha knows that he is struggling with excruciating pain. “What is this pain? Why pain? What does he feel? How it hurts!” Natasha thinks. He noticed her attention, raised his eyes and, without smiling, began to speak.
“One terrible thing,” he said, “is to bind oneself forever with a suffering person. It's eternal torment." And with a searching look—Natasha saw that look now—he looked at her. Natasha, as always, answered then before she had time to think about what she was answering; she said, "It can't go on like this, it won't happen, you'll be healthy - completely."
She now first saw him and now experienced everything that she felt then. She remembered the long, sad, stern look he gave at these words, and she understood the meaning of the reproach and despair of that long look.
“I agreed,” Natasha said to herself now, “that it would be terrible if he remained always suffering. I said it then only because it would be terrible for him, but he understood it differently. He thought it would be terrible for me. He then still wanted to live - he was afraid of death. And I told him so rudely, stupidly. I didn't think this. I thought something completely different. If I said what I thought, I would say: let him die, die all the time before my eyes, I would be happy in comparison with what I am now. Now... Nothing, no one. Did he know it? No. Didn't know and never will know. And now you can never, never fix it.” And again he spoke the same words to her, but now in her imagination Natasha answered him differently. She stopped him and said: “Terrible for you, but not for me. You know that without you there is nothing in my life, and suffering with you is the best happiness for me. And he took her hand and shook it the way he had squeezed it that terrible evening, four days before his death. And in her imagination she spoke to him still other tender, loving speeches, which she could have said then, which she spoke now. “I love you… you… love, love…” she said, clutching her hands convulsively, clenching her teeth with a fierce effort.
And sweet sorrow seized her, and tears were already coming into her eyes, but suddenly she asked herself: to whom is she saying this? Where is he and who is he now? And again everything was shrouded in dry, hard bewilderment, and again, tightly knitting her eyebrows, she peered at where he was. And now, now, it seemed to her, she was penetrating the secret ... But at that moment, when the incomprehensible, it seemed, was revealed to her, the loud knock of the handle of the door lock painfully struck her hearing. Quickly and carelessly, with a frightened, unoccupied expression on her face, the maid Dunyasha entered the room.
“Come to your father, quickly,” said Dunyasha with a special and lively expression. “A misfortune, about Pyotr Ilyich ... a letter,” she said with a sob.

In addition to the general feeling of alienation from all people, Natasha at that time experienced a special feeling of alienation from the faces of her family. All her own: father, mother, Sonya, were so close to her, familiar, so everyday that all their words, feelings seemed to her an insult to the world in which she had lived lately, and she was not only indifferent, but looked at them with hostility. . She heard Dunyasha's words about Pyotr Ilyich, about the misfortune, but did not understand them.
“What is their misfortune, what misfortune can there be? They have everything of their own, old, familiar and calm, ”Natasha mentally told herself.
When she entered the hall, her father quickly left the countess's room. His face was wrinkled and wet with tears. He must have run out of that room to let loose the sobs that were choking him. Seeing Natasha, he frantically waved his hands and burst into painfully convulsive sobs that distorted his round, soft face.
“Don’t… Petya… Go, go, she… she… is calling…” And he, sobbing like a child, quickly shuffling with his weakened legs, went up to a chair and almost fell on it, covering his face with his hands.
Suddenly, like an electric current, ran through Natasha's entire being. Something terribly hurt her in the heart. She felt a terrible pain; it seemed to her that something was coming off in her and that she was dying. But following the pain, she felt an instant release from the prohibition of life that lay on her. Seeing her father and hearing her mother's terrible, rude cry from behind the door, she instantly forgot herself and her grief. She ran up to her father, but he, waving his hand helplessly, pointed to her mother's door. Princess Mary, pale, with a trembling lower jaw, came out of the door and took Natasha by the hand, saying something to her. Natasha did not see or hear her. She walked quickly through the door, stopped for a moment, as if in a struggle with herself, and ran to her mother.
The countess was lying on an armchair, strangely awkwardly stretching herself, and banging her head against the wall. Sonya and the girls held her hands.
“Natasha, Natasha!” shouted the countess. - Not true, not true ... He is lying ... Natasha! she screamed, pushing away those around her. - Go away, everyone, it's not true! Killed! .. ha ha ha ha! .. not true!
Natasha knelt on an armchair, bent over her mother, embraced her, lifted her up with unexpected force, turned her face towards her, and clung to her.
- Mommy! .. my dear! .. I'm here, my friend. Mom, she whispered to her, not stopping for a second.
She did not let her mother out, tenderly wrestled with her, demanded a pillow, water, unbuttoned and tore her mother's dress.
“My friend, my dear ... mother, darling,” she whispered incessantly, kissing her head, hands, face and feeling how uncontrollably, in streams, tickling her nose and cheeks, her tears flowed.
The Countess squeezed her daughter's hand, closed her eyes, and fell silent for a moment. Suddenly she got up with unusual rapidity, looked around senselessly, and, seeing Natasha, began to squeeze her head with all her might. Then she turned her face, wrinkled with pain, to look at him for a long time.
“Natasha, you love me,” she said in a low, trusting whisper. - Natasha, you will not deceive me? Will you tell me the whole truth?
Natasha looked at her with tear-filled eyes, and in her face there was only a plea for forgiveness and love.
“My friend, mother,” she repeated, straining all the forces of her love to somehow remove from her the excess of grief that crushed her.
And again, in a powerless struggle with reality, the mother, refusing to believe that she could live when her beloved boy, blooming with life, was killed, fled from reality in a world of madness.
Natasha did not remember how that day, night, next day, next night went. She did not sleep and did not leave her mother. Natasha's love, stubborn, patient, not as an explanation, not as a consolation, but as a call to life, every second seemed to embrace the countess from all sides. On the third night, the Countess was quiet for a few minutes, and Natasha closed her eyes, leaning her head on the arm of the chair. The bed creaked. Natasha opened her eyes. The Countess sat on the bed and spoke softly.
- I'm glad you came. Are you tired, do you want some tea? Natasha walked over to her. “You have grown prettier and matured,” the countess continued, taking her daughter by the hand.
“Mommy, what are you talking about!”
- Natasha, he is gone, no more! And, embracing her daughter, for the first time the countess began to cry.

Princess Mary postponed her departure. Sonya and the count tried to replace Natasha, but they could not. They saw that she alone could keep her mother from insane despair. For three weeks Natasha lived hopelessly with her mother, slept on an armchair in her room, gave her water, fed her and talked to her without ceasing - she spoke, because one gentle, caressing voice calmed the countess.
The emotional wound of the mother could not heal. Petya's death tore off half of her life. A month after the news of Petya's death, which found her a fresh and vigorous fifty-year-old woman, she left her room half dead and not taking part in life - an old woman. But the same wound that half killed the Countess, this new wound called Natasha to life.
A spiritual wound resulting from a rupture of the spiritual body, just like a physical wound, however strange it may seem, after a deep wound has healed and seems to have come together, a spiritual wound, like a physical wound, heals only from within by the protruding force of life.
Natasha's wound also healed. She thought her life was over. But suddenly love for her mother showed her that the essence of her life - love - was still alive in her. Love has awakened, and life has awakened.
The last days of Prince Andrei connected Natasha with Princess Mary. A new misfortune brought them even closer. Princess Marya postponed her departure and for the last three weeks, as if she were a sick child, she looked after Natasha. The last weeks spent by Natasha in her mother's room had sapped her physical strength.
Once, in the middle of the day, Princess Mary, noticing that Natasha was trembling in a feverish chill, took her to her and laid her on her bed. Natasha lay down, but when Princess Mary, having lowered the blinds, wanted to go out, Natasha called her to her.
- I don't want to sleep. Marie, sit with me.
- You're tired - try to sleep.
- No no. Why did you take me away? She will ask.
- She's much better. She spoke so well today,” said Princess Marya.
Natasha was lying in bed and in the semi-darkness of the room she examined the face of Princess Marya.
"Does she look like him? thought Natasha. Yes, similar and not similar. But it is special, alien, completely new, unknown. And she loves me. What's on her mind? Everything is good. But how? What does she think? How does she look at me? Yes, she's beautiful."
“Masha,” she said, timidly pulling her hand to her. Masha, don't think I'm stupid. Not? Masha, dove. I love you so much. Let's be really, really friends.
And Natasha, embracing, began to kiss the hands and face of Princess Marya. Princess Mary was ashamed and rejoiced at this expression of Natasha's feelings.
From that day on, that passionate and tender friendship was established between Princess Mary and Natasha, which happens only between women. They kissed incessantly, spoke tender words to each other, and spent most of their time together. If one went out, the other was restless and hurried to join her. Together they felt a greater harmony with each other than separately, each with himself. A feeling stronger than friendship was established between them: it was an exceptional feeling of the possibility of life only in the presence of each other.
Sometimes they were silent for whole hours; sometimes, already lying in their beds, they began to talk and talked until the morning. They talked mostly about the distant past. Princess Marya talked about her childhood, about her mother, about her father, about her dreams; and Natasha, who previously with calm incomprehension turned away from this life, devotion, humility, from the poetry of Christian self-denial, now, feeling bound by love with Princess Marya, fell in love with Princess Marya’s past and understood the previously incomprehensible side of life to her. She did not think of applying humility and self-sacrifice to her life, because she was used to looking for other joys, but she understood and fell in love with another this previously incomprehensible virtue. For Princess Mary, who listened to stories about Natasha's childhood and early youth, a previously incomprehensible side of life was also revealed, faith in life, in the pleasures of life.
They still never talked about him in the same way, so as not to violate with words, as it seemed to them, that height of feeling that was in them, and this silence about him made them forget him little by little, not believing this.
Natasha lost weight, turned pale, and physically became so weak that everyone constantly talked about her health, and she was pleased with it. But sometimes not only the fear of death, but the fear of illness, weakness, loss of beauty suddenly came over her, and involuntarily she sometimes carefully examined her bare hand, surprised at its thinness, or looked in the mirror in the morning at her stretched out, miserable, as it seemed to her. , face. It seemed to her that it should be so, and at the same time she became frightened and sad.
Once she soon went upstairs and was out of breath. Immediately, involuntarily, she thought up a business for herself below, and from there she ran upstairs again, trying her strength and watching herself.
Another time she called Dunyasha, and her voice trembled. She called to her once more, in spite of the fact that she heard her footsteps - she called in that chesty voice with which she sang, and listened to him.
She didn’t know this, she wouldn’t have believed it, but under the impenetrable layer of silt that seemed to her that covered her soul, thin, tender young needles of grass were already breaking through, which were supposed to take root and so cover the grief that crushed her with their vital shoots that it would soon be invisible and not noticeable. The wound healed from within. At the end of January, Princess Marya left for Moscow, and the count insisted that Natasha go with her in order to consult with the doctors.

1. The group of potentials “E F G H” having the dimension of energy.

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

3. Thermodynamic potentials of multicomponent systems.

4. Practical implementation of the method of thermodynamic potentials (on the example of the problem of chemical equilibrium).

One of the main methods of modern thermodynamics is the method of thermodynamic potentials. This method arose largely due 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 initially 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;

Gibbs thermodynamic potential;

Enthalpy.

The potentiality of internal energy was shown in the previous topic. It implies the potentiality of the remaining quantities.

The differentials of thermodynamic potentials take the form:

From relations (3.1) it can be seen that the corresponding thermodynamic potentials characterize the same thermodynamic system with different methods .... descriptions (methods of setting the state of a thermodynamic system). So, 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:

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

Next, we will choose the “system under the piston” model as a way of describing it. 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:

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

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

For example, given that

we get

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

Similar identities can also 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 a potential, we have:

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

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

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 quite easy to determine the operation of the system and the thermal effect.

Thus, relations (3.1) imply:

From the first part of the equality follows the well-known provision that the work of a thermally insulated system () is carried out due to the decrease in its internal energy. The second equality means that the free energy is that part of the internal energy, which in the isothermal process is completely transformed into work (respectively, 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. Gibbs specific potential does not depend on. Then from (3.4) it follows:

That is, the chemical potential is the specific Gibbs potential, and the equality takes place

The thermodynamic potentials (3.1) are interconnected by direct relations, which make it possible to make a transition from one potential to another. For example, let's express all thermodynamic potentials in terms of internal energy.

In doing so, we obtained all thermodynamic potentials as functions of (). In order to express them in other variables, use the procedure re….

Let pressure be given in variables ():

Let us write the last expression as an equation of state, i.e. find the form

It is easy to see that if the state is given in variables (), then the thermodynamic potential is the internal energy. By virtue of (3.2), we find

Considering (3.18) as an equation for S, we find its solution:

Substituting (3.19) into (3.17) we get

That is, from variables () we moved to variables ().

The second group of thermodynamic potentials arises if, in addition to those considered above, the chemical potential is included as thermodynamic variables. The potentials of the second group also have the dimension of energy and can be related to the potentials of the first group by the relations:

Accordingly, the potential differentials (3.21) have the form:

As well as for the thermodynamic potentials of the first group, for the potentials (3.21) one can construct thermodynamic identities, find expressions for the parameters of the thermodynamic system, and so on.

Let us consider the characteristic relations for the “omega potential”, which expresses the quasi-free energy and is used in practice most often among the other potentials of the group (3.22).

The potential is given in variables () describing the thermodynamic system with imaginary walls. The system parameters in this case are determined from the relations:

The thermodynamic identities following from potentiality have the form:

Quite interesting are the additive properties of the thermodynamic potentials of the second group. Since in this case the number of particles is not among the parameters of the system, the volume is used as an additive parameter. Then for the potential we get:

Here - specific potential per 1. Taking into account (3.23), we obtain:

Accordingly, (3.26)

The validity of (3.26) can also be proved on the basis of (3.15):

The potential can also be used to convert thermodynamic functions written in form to form. For this, relation (3.23) for N:

permitted regarding:

Not only the energy characteristics of the system, but also any other quantities included in relation (3.1) can act as thermodynamic potentials. As an important example, consider entropy as a thermodynamic potential. The initial differential relation for entropy follows from the generalized notation of the I and II principles of thermodynamics:

Thus, entropy is the thermodynamic potential for a system given by parameters. Other system parameters look like:

By resolving the first of relations (3.28), the passage from variables to variables is relatively possible.

The additive properties of entropy lead to the known relations:

Let us proceed to the determination of thermodynamic potentials on the basis of given macroscopic states of a thermodynamic system. To simplify calculations, we assume the absence of external fields (). This does not reduce the generality of the results, since additional systems simply appear in the resulting expressions for .

As an example, let us find expressions for free energy, using the equation of state, the caloric equation of state, and the behavior of the system at as initial ones. Taking into account (3.3) and (3.12), we find:

Let us integrate the second equation of system (3.30) taking into account the boundary condition at:

Then system (3.30) takes the form:

The solution of system (3.31) makes it possible to find the specific free energy in the form

The origin of the specific free energy can also be found from the conditions at:

Then (3.32) takes the form:

and the expression for the entire free energy of the system, up to an additive constant, takes the form:

Then the reaction of the system to the inclusion of an external field is given by an additional equation of state, which, depending on the set of state variables, has the form:

Then the change in the corresponding thermodynamic potential associated with the inclusion of zero from zero to is determined from the expressions:

Thus, setting the thermodynamic potential in the macroscopic theory is possible only on the basis of using the given equations of the thermodynamic state, which, in turn, are themselves obtained on the basis of setting the thermodynamic potentials. This “vicious circle” can be broken only on the basis of a microscopic theory, in which the state of the system is set on the basis of distribution functions, taking into account statistical features.

Let us generalize the obtained results to the case of multicomponent systems. This generalization is carried out by replacing the parameter with a set. Let's take a look at specific examples.

Let us assume that the thermodynamic state of the system is given by parameters, i.e. we consider a system in a thermostat, consisting of several components, the number of particles in which is equal to The free energy, which in this description is the thermodynamic potential, has the form:

The additive parameter in (3.37) is not the number of particles, but the volume of the system V. Then the density of the system is denoted by . The function is a non-additive function of non-additive arguments. This is quite convenient, since when the system is divided into parts, the function does not change for each part.

Then, for the parameters of the thermodynamic system, we can write:

Considering that we have

For the chemical potential of an individual component, we write:

There are other ways to take into account the additive properties of free energy. Let us introduce the relative densities of the number of particles of each of the components:

independent of the volume of the system V. Here, is the total number of particles in the system. Then

The expression for the chemical potential in this case takes a more complex form:

Calculate the derivatives of and and substitute them into the last expression:

The expression for pressure, on the contrary, will be simplified:

Similar relations can also be obtained for the Gibbs potential. So, if the volume is given as an additive parameter, then, taking into account (3.37) and (3.38), we write:

the same expression can be obtained from (3.yu), which in the case of many particles takes the form:

Substituting expression (3.39) into (3.45), we find:

which completely coincides with (3.44).

In order to switch to the traditional Gibbs potential recording (through state variables ()) it is necessary to solve the equation (3.38):

Regarding the volume V and substitute the result in (3.44) or (3.45):

If the total number of particles in the system N is given as an additive parameter, then the Gibbs potential, taking into account (3.42), takes the following form:

Knowing the type of specific values: , we get:

In the last expression, summation over j replace by summation over i. Then the second and third terms add up to zero. Then for the Gibbs potential we finally obtain:

The same relation can be obtained in another way (from (3.41) and (3.43)):

Then for the chemical potential of each of the components we obtain:

In the derivation of (3.48), transformations similar to those used in the derivation of (3.42) were performed using imaginary walls. The system state parameters form a set ().

The role of the thermodynamic potential is played by the potential, which takes the form:

As can be seen from (3.49), the only additive parameter in this case is the volume of the system V.

Let us determine some thermodynamic parameters of such a system. The number of particles in this case is determined from the relation:

For free energy F and the Gibbs potential G can be written:

Thus, the relations for thermodynamic potentials and parameters in the case of multicomponent systems are modified only due to the need to take into account the number of particles (or chemical potentials) of each component. At the same time, the very idea of ​​the method of thermodynamic potentials and calculations based on it remains unchanged.

As an example of using the method of thermodynamic potentials, consider the problem of chemical equilibrium. Let us find the conditions of chemical equilibrium in a mixture of three substances entering into a reaction. Additionally, we assume that the initial reaction products are rarefied gases (this allows us to ignore intermolecular mutual production), and constant temperature and pressure are maintained in the system (this process is the easiest to implement in practice, therefore the condition of constant pressure and temperature is created in industrial installations for a chemical reaction) .

The equilibrium condition of a thermodynamic system, depending on the way it is described, is determined by the maximum entropy of the system or the minimum energy of the system (for more details, see Bazarov Thermodynamics). Then we can obtain the following equilibrium conditions for the system:

1. The state of equilibrium of an adiabatically isolated thermodynamic system, given by the parameters (), is characterized by a maximum of entropy:

The second expression in (3.53a) characterizes the stability of the equilibrium state.

2. The state of equilibrium of an isochoric-isothermal system, given by the parameters (), is characterized by a minimum of free energy. The equilibrium condition in this case takes the form:

3. The equilibrium of the isobaric-isothermal system, given by the parameters (), is characterized by the conditions:

4. For a system in a thermostat with a variable number of particles, defined by the parameters (), the equilibrium conditions are characterized by potential minima:

Let us turn to the use of chemical equilibrium in our case.

In the general case, the equation of a chemical reaction is written as:

Here - the symbols of chemicals - the so-called stoichiometric numbers. So for the reaction

Since pressure and temperature are chosen as parameters of the system, which are assumed to be constant. It is convenient to consider the Gibbs potential as a state of the thermodynamic potential G. Then the equilibrium condition for the system will consist in the requirement of the constancy of the potential G:

Since we are considering a three-component system, we set In addition, taking into account (3.54), we can write the balance equation for the number of particles ():

Introducing the chemical potentials for each of the components: and taking into account the assumptions made, we find:

Equation (3.57) was first obtained by Gibbs in 1876. and is the desired chemical equilibrium equation. It is easy to see, comparing (3.57) and (3.54), that the equation of chemical equilibrium is obtained from the equation of a chemical reaction by simply replacing the symbols of the reacting substances with their chemical potentials. This technique can also be used when writing the chemical equilibrium equation for an arbitrary reaction.

In the general case, the solution of equation (3.57), even for three components, is sufficiently loaded. This is due, firstly, to the fact that it is very difficult to obtain explicit expressions for the chemical potential even for a one-component system. Second, the relative concentrations and are not small quantities. That is, it is impossible to perform series expansion on them. This further complicates the problem of solving the equation of chemical equilibrium.

Physically noted difficulties are explained by the need to take into account the rearrangement of the electron shells of the atoms entering into the reaction. This leads to certain difficulties in the microscopic description, which also affects the macroscopic approach.

Since we agreed to confine ourselves to the study of gas rarefaction, we can use the ideal gas model. We assume that all reacting components are ideal gases that fill the total volume and create pressure p. In this case, any interaction (except chemical reactions) between the components of the gas mixture can be neglected. This allows us to assume that the chemical potential i-th component depends only on the parameters of the same component.

Here is the partial pressure i-th component, and:

Taking into account (3.58), the equilibrium condition for the three-component system (3.57) takes the form:

For further analysis, we use the equation of state of an ideal gas, which we write in the form:

Here, as before, we denote the thermodynamic temperature. Then the record known from the school takes the form: , which is written in (3.60).

Then for each component of the mixture we get:

Let us determine the form of the expression for the chemical potential of an ideal gas. As follows from (2.22), the chemical potential has the form:

Taking into account equation (3.60), which can be written in the form, the problem of determining the chemical potential is reduced to determining the specific entropy and specific internal energy.

The system of equations for the specific entropy follows from the thermodynamic identities (3.8) and the heat capacity expression (3.12):

Taking into account the equation of state (3.60) and passing to the specific characteristics, we have:

Solution (3.63) has the form:

The system of equations for the specific internal energy of an ideal gas follows from (2.23):

The solution to this system can be written as:

Substituting (3.64) - (3.65) into (3.66) and taking into account the equation of state for an ideal gas, we obtain:

For a mixture of ideal gases, expression (3.66) takes the form:

Substituting (3.67) into (3.59), we get:

Performing transformations, we write:

Performing potentiation in the last expression, we have:

Relation (3.68) is called the law of mass action. The value is a function of temperature only and is called the component of a chemical reaction.

Thus, the chemical equilibrium and the direction of a chemical reaction is determined by the magnitude of pressure and temperature.