Theory of thermodynamic equilibrium. Local thermodynamic equilibrium What is called the point of thermodynamic equilibrium

Equilibrium conditions in heat transfer processes (thermal equilibrium) are determined by relatively simple measurements of the temperatures of the contacting phases. Mechanical equilibrium (when momentum is transferred) is determined by the equality of directly measured pressures in adjacent phases. It is much more difficult to determine the equilibrium condition of a system in mass transfer processes. Therefore, the main attention will be paid to this type of equilibrium here.

The process of mass transfer from one phase to another in an isolated closed system consisting of two or more phases occurs spontaneously and proceeds until a mobile phase equilibrium is established between the phases under given conditions (temperature and pressure). It is characterized by the fact that in a unit of time as many molecules of the component pass from the first phase to the second as from the second to the first (i.e., there is no preferential transition of a substance from one phase to another). Having reached a state of equilibrium, the system can stay in it without quantitative and qualitative changes for an arbitrarily long time, until some external influence takes it out of this state. Thus, the state of an isolated system at equilibrium is determined only by internal conditions. Therefore, the gradients of the intensive parameters and the fluxes corresponding to them must be equal to zero:

dT = 0; dP = 0; dm i = 0

where T is temperature; P - pressure; m i - chemical potential of the i-th component.

These expressions are called the conditions of thermal, mechanical and chemical (material) equilibrium, respectively. All spontaneous processes proceed in the direction of achieving equilibrium. The more the state of the system deviates from equilibrium, the higher the rate of the transfer of substances between phases due to the increase in the driving force that determines this process. Therefore, in order to carry out the processes of transfer of substances, it is necessary to prevent the establishment of an equilibrium state, for which matter or energy is supplied to the system. In practice, in open systems, this condition is usually implemented by creating a relative phase movement in apparatuses with counterflow, cocurrent, or other flow patterns.

It follows from the second law of thermodynamics that in spontaneous processes the entropy S of the system increases and reaches its maximum value in the equilibrium condition, i.e. in this case dS = 0.

This condition, as well as the three previous ones, determine the equilibrium condition of the system.

The chemical potential dm i is defined as the increase in the internal energy U of the system when an infinitely small number of moles of the i-th component is added to the system, referred to this amount of substance, at constant volume V, entropy S and the number of moles of each of the remaining components n j (where n = l , 2, 3, …, j).

In the general case, the chemical potential can be defined as the increment of any of the thermodynamic potentials of the system at various constant parameters: the Gibbs energy G– at constant pressure Р, temperature Т and n j ; enthalpy H - at constant S, P and n j .

In this way

The chemical potential depends not only on the concentration of a given component, but also on the type and concentration of other components of the system. For a mixture of ideal gases, m i depends only on the concentration of the considered component and temperature:

where is the value of m i in the standard state (usually at Р i = 0.1 MPa), depends only on temperature; P i - partial pressure of the i-th component of the mixture; is the pressure of the i-th component in the standard state.

The chemical potential characterizes the ability of the considered component to leave this phase (by evaporation, crystallization, etc.). In a system consisting of two or more phases, the transition of a given component can occur spontaneously only from a phase in which its chemical potential is greater to a phase with a lower chemical potential. Under equilibrium conditions, the chemical potential of the component in both phases is the same.

In general, the chemical potential can be written as:

where a i is the activity of the i-th component of the mixture; x i and g i, respectively, the mole fraction and activity coefficient of the i-th component.

The activity coefficient g i is a quantitative measure of the imperfection of the behavior of the i-th component in the mixture. For g i > 1, the deviation from the ideal behavior is called positive, for g i< 1 - отрицательным. Для отдельных систем g i £ 1. Тогда а i = х i ‚ и уравнение принимает вид:

For ideal systems, the chemical potential can also be expressed in terms of the volatility f i of the component:

Where is the volatility of the i-th component under standard conditions. The values of a i and f i are found in the reference literature.

When carrying out technological processes, working media (gas, steam, liquid) are in a non-equilibrium state, which cannot be described by thermodynamic parameters. To describe the non-equilibrium state of systems, additional non-equilibrium, or dissipative, parameters are introduced, which are used as gradients of intense thermodynamic quantities - temperature, pressure, chemical potential and density of the corresponding dissipative flows associated with the transfer of energy, mass and momentum.

Phase rule

The existence of a given phase in a system or the equilibrium of phases is possible only under certain conditions. When these conditions change, the equilibrium of the system is disturbed, a phase shift or a transition of a substance from one phase to another occurs. The possible existence of a given phase in equilibrium with others is determined by the phase rule, or Gibbs phase equilibrium law:

C + F = k + n

where C is the number of degrees of freedom (pressure, temperature, concentration) - the minimum number of parameters that can be changed independently of each other without disturbing the equilibrium of this system; Ф - number of phases of the system; k is the number of independent system components; n is the number of external factors affecting the equilibrium position in the given system.

For mass transfer processes, n = 2, since the external factors in this case are temperature and pressure. Then the expression takes the form

C + F = k + 2

From here C \u003d k - F + 2.

Thus, the phase rule makes it possible to determine the number of parameters that can be changed without violating the phase equilibrium of the system. For example, for a one-component equilibrium system "liquid - vapor", the number of degrees of freedom will be:

C = 1 - 2 + 2 = 1

That is, in this case, only one parameter can be arbitrarily set - pressure or temperature. Thus, for a one-component system, there is an unambiguous relationship between temperature and pressure under equilibrium conditions. An example is the widely used reference data - the relationship between temperature and pressure of saturated vapors of water.

For a one-component equilibrium system consisting of three phases "solid - liquid - vapor", the number of degrees of freedom is zero: С = 1 - 3 + 2 = 0.

For example, the "water - ice - water vapor" system is in equilibrium at a pressure of 610.6 Pa and a temperature of 0.0076 °C.

For a two-component equilibrium system "liquid - vapor", the number of degrees of freedom C \u003d 2 - 2 + 2 \u003d 2. In this case, one of the variables (for example, pressure) is set and an unambiguous relationship between temperature and concentration or (at constant temperature) between pressure and concentration. The relationship between the parameters (temperature - concentration, pressure - concentration) is built in flat coordinates. Such diagrams are usually called phase diagrams.

Thus, the phase rule determines the possibility of coexistence of phases, but does not indicate the quantitative dependences of the transfer of matter between phases.

Thermodynamic condition of chemical equilibrium

The thermodynamic condition for the equilibrium of a process occurring under isobaric-isothermal conditions is the equality to zero of the change in the Gibbs energy (D rG(T)=0). When the reaction proceeds n a A+n b B=n with C+n d D

the change in the standard Gibbs energy is

D rG 0 T=(n c×D f G 0 C+ n d×D f G 0 D)–(n a×D f G 0 A+ n b×D f G 0 B).

This expression corresponds to an ideal process in which the concentrations of the reactants are equal to unity and remain unchanged during the reaction. In the course of real processes, the concentrations of the reactants change; the concentration of the starting substances decreases, while the concentration of the reaction products increases. Taking into account the concentration dependence of the Gibbs energy (see Sec. 1 . 3. 4) its change during the reaction is

D r G T=–

– =

=(n c×D f G 0 C+ n d×D f G 0 D)–(n a×D f G 0 A+ n b×D f G 0 B) +

+ R× T×(n c×ln C + n d×ln C D–n a×ln C A–n b×ln C B)

D r G T=D rG 0 T+R× T× ,

where is the dimensionless concentration i-th substance; X i– mole fraction i-th substance; pi- partial pressure i-th substance; R 0 \u003d \u003d 1.013 × 10 5 Pa - standard pressure; with i– molar concentration i-th substance; With 0 \u003d 1 mol / l - standard concentration.

In a state of equilibrium

D rG 0 T+R×T× = 0,

Value To 0 is called the standard (thermodynamic) equilibrium constant of the reaction. So at a certain temperature T as a result of the flow of direct and reverse reactions, equilibrium is established in the system at certain concentrations of reactants - equilibrium concentrations (C i) R . The values of equilibrium concentrations are determined by the value of the equilibrium constant, which is a function of temperature and depends on the enthalpy (D r N 0) and entropy (D rS 0) reactions˸

D rG 0 T+R× T×ln K 0 = 0,

because D rG 0 T=D r N 0 T - T×D rS 0 T,

If the enthalpy values (D r N 0 T) and entropy (D rS 0 T) or D rG 0 T reaction, then you can calculate the value of the standard equilibrium constant.

The reaction equilibrium constant characterizes ideal gas mixtures and solutions. Intermolecular interactions in real gases and solutions lead to a deviation of the calculated values of the equilibrium constants from the real ones. To take this into account, instead of the partial pressures of the components of gas mixtures, their fugacity is used, and instead of the concentration of substances in solutions, their activity. fugacity i th component is related to ᴇᴦο partial pressure by the relation fi=g i× pi, where g i- coefficient of fugacity. The activity and concentration of the component are related by the ratio a i=g i× C i, where g i– activity coefficient.

It should be noted that in a fairly wide range of pressures and temperatures, gas mixtures can be considered ideal and calculations of the equilibrium composition of the gas mixture can be carried out, considering the fugacity coefficient g i@ 1. In the case of liquid solutions, especially electrolyte solutions, the activity coefficients of their components can differ significantly from unity (g i¹ 1) and activities should be used to calculate the equilibrium composition.

Thermodynamic condition of chemical equilibrium - concept and types. Classification and features of the category "Thermodynamic condition of chemical equilibrium" 2015, 2017-2018.

Macroscopic systems often have a "memory", as if they remember their history. For example, if using a spoon to organize the movement of water in a cup, then this movement will continue for some time but inertia. Steel acquires special properties after machining. However, memory fades over time. The movement of water in the cup stops, the internal stresses in the steel weaken due to plastic deformation, and concentration inhomogeneities decrease due to diffusion. It can be argued that systems tend to achieve relatively simple states that are independent of the system's prior history. In some cases this state is reached quickly, in others slowly. However, all systems tend to states in which their properties are determined by internal factors, and not by previous perturbations. Such simple, limiting states are, by definition, time-independent. These states are called equilibrium. Situations are possible when the state of the system is unchanged, but mass or energy flows take place in it. In this case, we are not talking about an equilibrium, but about a stationary state.

The state of a thermodynamic system, characterized under constant external conditions by the invariability of parameters over time and the absence of flows in the system, is called equilibrium.

equilibrium state- the limiting state, to which the thermodynamic system, isolated from external influences, tends. The condition of isolation should be understood in the sense that the rate of processes of establishing equilibrium in the system is much higher than the rate of change in conditions at the boundaries of the system. An example is the process of fuel combustion in the combustion chamber of a rocket engine. The residence time of the fuel element in the chamber is very short (10 _3 - 1 (N s), however, the equilibrium time is approximately 10 ~ 5 s. Another example is that geochemical processes in the earth's crust proceed very slowly, but the lifetime of thermodynamic systems of this kind is calculated in millions of years, therefore, in this case, the thermodynamic equilibrium model is also applicable.

Using the introduced concept, we can formulate the following postulate: there are special states of simple systems - those that are fully characterized by macroscopic values of internal energy U, volume V and mole numbers n and n 2 > i, chemical components. If the system under consideration has more complex mechanical and electrical properties, then the number of parameters required to characterize the equilibrium state increases (it is necessary to take into account the presence of surface tension forces, gravitational and electromagnetic fields, etc.).

From a practical point of view, the experimenter must always establish whether the system under study is in equilibrium. For this, the absence of visible changes in the system is not enough! For example, two bars of steel may have the same chemical composition, but completely different properties due to mechanical processing (forging, pressing), heat treatment, etc. one of them. If the properties of the system under study cannot be described using the mathematical apparatus of thermodynamics, this maybe mean that the system is not in equilibrium.

In reality, only very few systems reach an absolutely equilibrium state. In particular, in this state, all radioactive materials must be in a stable form.

It can be argued that the system is in equilibrium if its properties are adequately described using the apparatus of thermodynamics.

It is useful to recall that in mechanics, the equilibrium of a mechanical system is the state of a mechanical system under the influence of forces, in which all its points are at rest with respect to the reference frame under consideration.

Let us consider two examples explaining the concept of equilibrium in thermodynamics. If contact is established between the thermodynamic system and the environment, then in the general case a process will begin, which will be accompanied by a change in some parameters of the system. In this case, some of the parameters will not change. Let the system consist of a cylinder containing a piston (Fig. 1.9). At the initial moment of time, the piston is fixed. To the right and left of it is gas. The pressure to the left of the piston is R Ah right - R in, and R A > R B If you remove the fastener, the piston will be released and will start moving to the right, while the volume of the subsystem BUT will begin to increase, and the right - to decrease (-D VB = D VA). Subsystem BUT loses energy, subsystem AT acquires it, pressure r A drops, pressure r in increases until the pressures to the left and right of the piston are equal. In this case, the gas masses of the subsystems to the left and right of the piston do not change. Thus, in the considered process, energy is transferred from one subsystem to another due to changes in pressure and volume. The independent variables in the considered process are pressure and volume. These state parameters some time after the release of the piston will take on constant values and will remain unchanged until the system is influenced from the outside. The state reached is equilibrium.

Balance state - it is the final state of the process of interaction of one or more systems with their environment.

As is clear from the above example, the parameters of a system in a state of equilibrium depend on the initial state of the system (its subsystems) and the environment. It should be noted that the indicated interrelation of the initial and final states is one-sided and does not allow restoring the initial nonequilibrium state on the basis of information about the parameters of the equilibrium state.

Rice. 1.9.

A thermodynamic system is in equilibrium if all state parameters do not change after the system is isolated from other systems and the environment.

The driving force of the considered process of establishing equilibrium was the pressure difference to the left and right of the piston, i.e. difference in intensity parameters. At the initial moment Ar \u003d r l -r in*0, at the end moment Ap \u003d 0, p "A \u003d Pv-

As another example, consider the system shown in Fig. 1.10.

Rice. 1.10.

System shells BUT and AT - non-deformable and heat-resistant (adiabatic). At the initial time, the gas in the system AT is at room temperature, water in the system BUT heated. System pressure AT measured with a manometer. At some point in time, the heat-insulating layer between BUT and AT removed (in this case, the wall remains undeformed, but becomes heat-permeable (diathermic)). System pressure AT begins to grow, it is obvious that the energy is transferred from A in B at the same time, no visible changes in the systems are observed, there are no mechanical movements. Looking ahead, let's say that this energy transfer mechanism can be justified using the second law of thermodynamics. In the previous example, in the process of establishing equilibrium, two coordinates changed - pressure and volume. It can be assumed that in the second example, two coordinates should also change, one of which is pressure; the change of the second we could not observe.

Experience shows that after a certain period of time, the states of systems Aw B cease to change, a state of equilibrium is established.

Thermodynamics deals with equilibrium states. The term "equilibrium" implies that the action of all forces on the system and within the system is balanced. In this case, the driving forces are equal to zero, and there are no flows. The state of an equilibrium system does not change if the system is isolated from the environment.

It is possible to consider separate types of equilibrium: thermal (thermal), mechanical, phase and chemical.

In a system in the state thermal equilibrium, the temperature is the same at any point and does not change with time. In a system in the state mechanical equilibrium, pressure is constant, although the magnitude of pressure can vary from point to point (column of water, air). phase equilibrium - equilibrium between two or more phases of a substance (vapor - liquid; ice - water). If the system has reached the state chemical equilibrium, it cannot detect changes in the concentrations of chemicals.

If a thermodynamic system is in equilibrium, it is assumed that it has reached equilibrium of all kinds (thermal, mechanical, phase and chemical). Otherwise, the system is non-equilibrium.

Characteristic signs of an equilibrium state:

1) does not depend on time (stationarity);
2) characterized by the absence of flows (in particular, heat and mass);
3) does not depend on the “history” of the system development (the system “does not remember” how it got into this state);
4) stable with respect to fluctuations;
5) in the absence of fields does not depend on the position in the system within the phase.

1. Extreme properties of thermodynamic potentials.

2. Conditions for equilibrium and stability of a spatially homogeneous system.

3. General conditions for phase equilibrium in thermodynamic systems.

4. Phase transitions of the first kind.

5. Phase transitions of the second kind.

6. Generalization of semi-phenomenological theory.

Questions of stability of thermodynamic systems were considered in the previous topic in relation to the problem of chemical equilibrium. Let us pose the problem of theoretical substantiation of the previously formulated conditions (3.53) on the basis of the II law of thermodynamics, using the properties of thermodynamic potentials.

Consider a macroscopic infinitesimal change in the state of the system: 1 -2, in which all its parameters are related to an infinitesimal value:

Respectively:

Then, in the case of a quasi-static transition, from the generalized formulations of the I and II principles of thermodynamics (2.16) it follows:

If 1-2 is non-quasi-static, then the following inequalities hold:

In expression (4.3), the values with a prime correspond to a non-quasi-static process, and the quantities without a prime correspond to a quasi-static one. The first inequality of system (4.3) characterizes the principle of maximum heat absorption obtained on the basis of generalization of numerous experimental data, and the second characterizes the principle of maximum work.

Writing the work for a non-quasistatic process in the form and introducing the parameters and in a similar way, we get:

Expression (4.4) is absolutely equivalent to the Clausius inequality.

Let us consider the main consequences of (4.4) for various ways of describing thermodynamic systems:

1. Adiabatically isolated system: (). Respectively. Then:

This means that if the state variables of the system are fixed, then, due to (4.5), its entropy will arise until the equilibrium state occurs in the system, according to the zero law of thermodynamics. That is, the equilibrium state corresponds to the maximum entropy:

Variations in (4.6) are made according to those parameters that, for the specified fixed parameters of the system, can take on nonequilibrium values. It may be the concentration P, pressure R, temperature, etc.

2. The system in the thermostat (). Accordingly, which allows us to rewrite (4.4) in the form:

Taking into account the form of the expression for free energy: and equality, we obtain:

Thus, the course of nonequilibrium processes for a system placed in a thermostat is accompanied by a decrease in its free energy. And the equilibrium value corresponds to its minimum:

3. The system under the piston (), i.e. .In this case, relation (4.4) takes the form:

Thus, equilibrium in the system under the piston occurs when the minimum value of the Gibbs potential is reached:

4. System with imaginary walls (). Then. Then

which allows you to write

Accordingly, in a system with imaginary walls, non-equilibrium processes are directed towards a decrease in potential, and equilibrium is achieved under the condition:

The condition determines the equilibrium state of the system itself and is widely used in the study of multicomponent or multiphase systems. The minimum or maximum conditions determine the criteria for the stability of these equilibrium states with respect to spontaneous or artificially created perturbations of the system.

In addition, the presence of extreme properties of thermodynamic potentials allows one to use variational methods for their study by analogy with the variational principles of mechanics. However, this requires the use of a statistical approach.

Let us consider the conditions of equilibrium and stability of thermodynamic systems using the example of a gas placed in a cylinder above a piston. In addition, to simplify the analysis, we neglect the external fields, assuming Then the state variables are ().

It was noted earlier that a thermodynamic system can be influenced either by doing work on it or by imparting some heat to it. Therefore, it is necessary to analyze the balance and stability in relation to each of the noted influences.

The mechanical action is associated with the displacement of the loose piston. In this case, the work done on the system is

As an internal parameter that can change and for which variation should be carried out, we choose the volume.

Representing the Gibbs potential through free energy

and varying, we write:

From the last equality follows:

Expression (4.13) should be considered as an equation for the equilibrium value of the volume for given system parameters ().

The equilibrium state stability conditions have the form:

Taking into account (4.13), the last condition can be rewritten as:

Condition (4.14) imposes certain requirements on the equation of state. So, ideal gas isotherms

everywhere satisfy the stability condition. At the same time, the van der Waals equation

or the Dieteriga equation

have sections on which the stability conditions are not met, and which do not correspond to real equilibrium states, i.e. experimentally implemented.

If, however, there are isotherms at some point, then special methods of mathematical analysis are used to check stability, i.e. check if the following conditions are met:

Similarly, the stability requirements imposed on the equation of state can be formulated for other system parameters. Consider, as an example, the dependence of the chemical potential. Let us introduce the density of the number of particles. Then the chemical potential can be represented as

Let's calculate the differential depending on the state variables:

When writing the last expression, it was taken into account that the thermodynamic identity (3.8) was used. Then

That is, the stability condition for the chemical potential takes the form

At the critical point in the presence of deflection, we have:

Let us turn to the analysis of the stability of the system to thermal action associated with the transfer of a certain amount of heat. Then, as a variational parameter, we consider the entropy of the system S. To take into account exactly the thermal effect, we fix the mechanical parameters. Then it is convenient to choose a set as variables of the thermodynamic state, and free energy as the thermodynamic potential.

By varying, we find:

From the equilibrium condition we obtain

Equations (4.21) should be considered as an equation for the equilibrium value of entropy. From the positiveness of the second variation of the free energy:

Since the temperature always takes positive values, it follows from (4.22):

Expression (4.23) is the desired condition for the stability of a thermodynamic system with respect to heating. Some authors consider the positive heat capacity as one of the manifestations of the Le Chatelier-Brown principle. When informing the thermodynamic system of the amount of heat:

Its temperature arises, which, in accordance with the second law of thermodynamics in the formulation of Clausius (1850), leads to a decrease in the amount of heat entering the system. In other words, in response to external influences - the message of the amount of heat - the thermodynamic parameters of the system (temperature) change in such a way that external influences are weakened.

Consider first a one-component system in a two-phase state. Hereinafter, by a phase we mean a homogeneous substance in chemical and physical terms.

Thus, each phase will be considered as a homogeneous and thermodynamically stable subsystem characterized by a common pressure value (in accordance with the requirement of the absence of heat fluxes). Let us study the equilibrium condition of a two-phase system with respect to the change in the number of particles and located in each of the phases.

Taking into account the assumptions made, it is most convenient to use the description of the system under the piston with fixed parameters (). Here, is the total number of particles in both phases. Also, for simplicity, we “switch off” the external fields ( a=0).

In accordance with the chosen method of description, the equilibrium condition is the condition (4.10) for the minimum of the Gibbs potential:

which is supplemented by the condition of constancy of the number of particles N:

By varying in (4.24a), taking into account (4.24b), we find:

Thus, the general criterion for the equilibrium of a two-phase system is the equality of their chemical potentials.

If the expressions of chemical potentials are known and then the solution of equation (4.25) will be some curve

called the curve of phase equilibrium or discrete phase equilibrium.

Knowing the expressions for chemical potentials, from equality (2.yu):

we can find the specific volumes for each of the phases:

That is, (4.26) can be rewritten as equations of state for each of the phases:

Let us generalize the obtained results to the case n phases and k chemically unreactive components. For arbitrary i th component, equation (4.25) takes the form:

It is easy to see that expression (4.28) represents the system ( n- 1) independent equations. Accordingly, from the equilibrium conditions for k component we get k(n-1) independent equations ( k(n-1) connections).

The state of the thermodynamic system in this case is given by temperature, pressure p and k-1 values of the relative concentrations of the components in each phase. Thus, the state of the system as a whole is set by a parameter.

Taking into account the superimposed constraints, we find the number of independent parameters of the system (degree freedom).

Equality (4.29) is called the Gibbs phase rule.

For a one-component system () in the case of two phases () there is one degree of freedom, i.e. we can arbitrarily change only one parameter. In the case of three phases (), there are no degrees of freedom (), that is, the coexistence of three phases in a one-component system is possible only at one point, called the triple point. For water, the triple point corresponds to the following values: .

If the system is not one-component, more complicated cases are possible. So, a two-phase () two-component system () has two degrees of freedom. In this case, instead of the phase equilibrium curve, we obtain a region in the form of a strip, the boundaries of which correspond to the phase diagrams for each of the pure components, and the internal regions correspond to different values of the relative concentration of the components. One degree of freedom in this case corresponds to the curve of coexistence of three phases, and corresponds to the fourth point of coexistence of four phases.

As discussed above, the chemical potential can be represented as:

Accordingly, the first derivatives of the chemical potential are equal to the specific values of the entropy, taken with the opposite sign, and the volume:

If at points satisfying phase equilibrium:

the first derivatives of the chemical potential for different phases experience a discontinuity:

the thermodynamic system is said to undergo a phase transition of the first order.

Phase transitions of the first kind are characterized by the presence of a latent heat of the phase transition, which is different from zero, and a jump in the specific volumes of the system. The latent specific heat of the phase transition is determined from the relationship:

and the jump in specific volume is:

Examples of phase transitions of the first kind are the processes of boiling and evaporation of liquids. Melting of solids, transformation of the crystal structure, etc.

Consider two nearby points on the phase equilibrium curve () and (), whose parameters differ by infinitesimal values. Then equation (4.25) is also valid for differentials of chemical potentials:

this implies:

Performing transformations in (4.34), we obtain:

Expression (4.35) is called the Clausius-Clapeyron equation. This equation makes it possible to obtain the form of a phase equilibrium curve from the values of the phase transition heat and phase volumes known from the experiment and without involving the concept of chemical potential, which is rather difficult to determine both theoretically and experimentally.

The so-called metastable states are of great practical interest. In these states, one phase continues to exist in the stability region of the other phase:

Examples of sufficiently stable metastable states are diamonds, amorphous glass (along with crystalline rock crystal), etc. Metastable states of water are widely known in nature and industrial installations: superheated liquid and supercooled vapor, as well as supercooled liquid.

An important circumstance is that the condition for the experimental realization of these states is the absence of a new phase, impurities, impurities, etc. in the system, i.e. the absence of a center of condensation, vaporization and crystallization. In all these cases, the new phase appears initially in small quantities (drops, bubbles or crystals). Therefore, surface effects that are commensurate with volume ones become significant.

For simplicity, we confine ourselves to considering the simplest case of the coexistence of two spatially disordered phase states - liquid and vapor. Consider a liquid containing a small bubble of saturated vapor. In this case, the surface tension force acts along the interface. To take it into account, we introduce the parameters:

Here is the surface area of the film,

Surface tension coefficient. The “-” sign in the second equation (4.36) corresponds to the fact that the film is contracted and the work of the external force is directed to increase the surface:

Then, taking into account the surface tension, the Gibbs potential will change by the value:

Introducing the model of the system under the piston and taking into account the equality, we write the expression for the Gibbs potential in the form

Here and are the specific values of free energy, and are the specific volumes of each of the phases. At fixed values of (), the value (4.39) reaches a minimum. In this case, the Gibbs potential can be varied with. These quantities are related using the relation:

where R can be expressed in terms of: Let us choose quantities as independent parameters, then the Gibbs potential (4.39) can be rewritten as:

(here taken into account)

Performing variation (4.40), we write:

Taking into account the independence of quantities, we reduce (4.41) to the system

Let us analyze the obtained equality. From (4.42a) it follows:

Its meaning is that the pressure in phase 1 is equal to the external pressure.

Introducing expressions for the chemical potentials of each of the phases and taking into account

we write (4.42b) in the form:

Here is the pressure in phase II. The difference between equation (4.44) and the phase equilibrium condition (4.25) is that the pressure in (4.44) in each of the phases can be different.

From equality (4.42c) it follows:

Comparing the resulting equality with (4.44) and the expression for the chemical potential, we obtain the formula for the gas pressure inside a spherical bubble:

Equation (4.45) is the Laplace formula known from the course of general physics. Generalizing (4.44) and (4.45), we write the equilibrium conditions between a liquid and a vapor bubble in the form:

In the case of studying the problem of the liquid-solid phase transition, the situation becomes much more complicated due to the need to take into account the geometric features of crystals and the anisotropy of the direction of predominant crystal growth.

Phase transitions are also observed in more complex cases, in which only the second derivatives of the chemical potential with respect to temperature and pressure suffer a discontinuity. In this case, the phase equilibrium curve is determined not by one, but by three conditions:

Phase transitions satisfying equations (4.47) are called phase transitions of the second kind. Obviously, the latent heat of the phase transition and the change in specific volume in this case is equal to zero:

To obtain the differential equation of the phase equilibrium curve, the Clausius-Clapeyron equation (4.35) cannot be used, because when directly substituting the values (4.48) into the expression (4.35), an uncertainty is obtained. Let us take into account that the condition u is preserved when moving along the phase equilibrium curve. Then:

Let us calculate the derivatives in (4.49)

Substituting the obtained expressions into (4.49), we find:

The system of linear equations (4.51) written with respect to and is homogeneous. Therefore, its non-trivial solution exists only if the determinant composed of the coefficients is equal to zero. Therefore, we write

Taking into account the obtained condition and choosing any equation from system (4.51), we obtain:

Equations (4.52) for the phase equilibrium curve in the case of a second-order phase transition are called the Ehrenfest equations. In this case, the phase equilibrium curve can be determined from the known characteristics of heat capacity jumps, thermal expansion coefficient, elasticity coefficient.

Second-order phase transitions occur much earlier than first-order phase transitions. This is obvious even from condition (4.47), which is much more rigid than the equation of the phase equilibrium curve (4.10) with conditions (4.31). An example of such phase transitions is the transition of a conductor from the superconducting state to the normal state in the absence of a magnetic field.

In addition, there are phase transitions with a latent heat equal to zero, for which, during the transition, the presence of a singularity in the caloric equation is observed (the heat capacity suffers a discontinuity of the second kind). This type of phase transition is called a type phase transition. Examples of such transitions are the transition of liquid helium from the superfluid state to the normal state, the transition at the Curie point for ferromagnets, the transition from the inelastic state to the elastic state for alloys, and so on.

) in conditions of isolation from the environment. In general, these values are not constant, they only fluctuate (fluctuate) around their average values. If an equilibrium system corresponds to several states, in each of which the system can be indefinitely long, then the system is said to be in a metastable equilibrium. In a state of equilibrium in the system, there are no flows of matter or energy, non-equilibrium potentials (or driving forces), changes in the number of phases present. There are thermal, mechanical, radiation (radiant) and chemical equilibrium. In practice, the condition of isolation means that the processes of establishing equilibrium proceed much faster than changes occur at the boundaries of the system (that is, changes in conditions external to the system), and the system exchanges matter and energy with the environment. In other words, thermodynamic equilibrium is achieved if the rate of relaxation processes is sufficiently high (as a rule, this is characteristic of high-temperature processes) or the time to reach equilibrium is long (this case occurs in geological processes).

In real processes, incomplete equilibrium is often realized, but the degree of this incompleteness can be significant and insignificant. In this case, three options are possible:

equilibrium is achieved in some part (or parts) of a relatively large system - local equilibrium,
incomplete equilibrium is achieved due to the difference in the rates of relaxation processes occurring in the system - partial equilibrium,
both local and partial equilibrium take place.

In non-equilibrium systems, changes occur in the flows of matter or energy, or, for example, phases.

Stability of thermodynamic equilibrium

The state of thermodynamic equilibrium is called stable if in this state there is no change in the macroscopic parameters of the system.

Criteria for thermodynamic stability of various systems:

Isolated (absolutely not interacting with the environment) system is the maximum entropy.
Closed (exchanges only heat with the thermostat) system- minimum free energy.
Fixed temperature and pressure system is the minimum of the Gibbs potential.
System with fixed entropy and volume- minimum internal energy.
System with fixed entropy and pressure- minimum enthalpy.