The phase equilibrium is characterized. Phase balance. Equilibrium states at phase transitions

The method of determining the equilibrium constants according to the NGAA atlas issued by the American Gasoline Production Association has received the greatest distribution both in our country and abroad. The atlas contains equilibrium constants from methane to decane inclusive, as well as nitrogen and carbon dioxide. The equilibrium constant at a given convergence pressure is determined from an atlas based on pressure and temperature data. When calculating on a computer, it is difficult to use graphs of equilibrium constants, because one has to repeatedly remove the values ​​of the constants from the graphs and enter them into the computer memory. In this regard, the NGAA atlases are translated into tables:

Recommendations for automating the choice of equilibrium constants of hydrocarbon systems on a computer. Tables of equilibrium constants. All-Russian Research Institute of Gas, Moscow, 1972. A partial atlas of equilibrium constants is presented in the reference book: Katz, "Gas Production and Transportation".

Phase equilibrium equations.

For calculations, the following initial data are needed:

Initial reservoir pressure

Initial formation temperature

Composition of the hydrocarbon system

Equilibrium constants of individual constituent components, K=y/x, where

y is the molar fraction of the component in the gas phase,

x-molar fraction of a component in the liquid phase in equilibrium with the gas phase.

Each component of the mixture at a given temperature and pressure has its own equilibrium constants. They are determined experimentally. Dependences of the equilibrium constant are expressed in terms of the given parameters:

, , .

In the case of a multicomponent mixture, the critical pressure is called the convergence pressure. Since there is no difference between liquid and vapor (gas) in the critical and supercritical regions, then for a mixture:

The convergence pressure corresponds to a certain composition of the mixture and is a numerical characteristic of the composition.

The atlas of constants was compiled for a wide range of changes in P cx.

For a multicomponent mixture, an equation of phase concentrations is compiled. N is the mass of all components in a certain volume V. N G is the mass of components in a gas, N W is the mass of components in a liquid N = N G +N L.

If we divide by the sum of the molecular weights of all components contained in the volume V, we get n M = n MG + n MF is the number of moles of the components in the gas and liquid phases.

Mole fraction of components in gas "y i" and in liquid "x i":

N G i - mass of the i-th component in the gas phase

N w i is the mass of the i-th component in the liquid phase

M i - molecular weight of the i-th component

The molar fraction of the i-th component in the volume as a whole is expressed:

N i is the mass of the i-th component in the volume V.

From the above expressions it follows: (*)

Denote:

n MG / n M = Y is the mole fraction of all components in the gas phase.

n MF / n M \u003d X is the mole fraction of all components in the liquid phase

Considering y i = K i x i ? 1=X+Y, substitute into equality (*)

, Equations of phase concentrations

When determining the phase state, various problems can be solved.

For example, given: ν i (composition), P,T,Y → then determine x i and y i from ur-th concentrations. More often there is a problem of finding Y,X according to the known composition ν i , P,T. Then the equality is used . The equation is solved by an iterative method. The original equation is solved in the form: .

It is obvious that for Y=0 f(Y)=0.

The maximum value of the proportion of the gas phase Y=1. hence the solution is sought in the interval 0 .

Taking Y 0 =0.5 as the initial approximation and applying sequentially the iterative formulas of the Newton method - the method of chords, a solution is found with a small number of iterations:

Two-phase filtration.

In connection with the design and analysis of the development of oil and gas fields, it is necessary to study the joint flow in a porous medium of several liquids, most often water, oil and gas, which are separate phases that do not mix with each other.

The formation of deposits occurs as a result of the displacement of the water that was originally there. Therefore, along with oil and gas in the reservoirs there is a certain amount (10-30%) of buried water. In addition, many deposits are filled with oil and gas only in the upper dome part, while the underlying zones are filled with water with its original content that was not pushed aside during the formation process. The uppermost parts of the reservoir contain gas that may be present or collected during development. A two- or three-phase flow occurs during the development of oil deposits, oil and gas, gas condensate and simply gas deposits in the presence of water underlying the gas cap, i.e. almost always, except for dry gas traps.

When filtering two liquids (oil-water), or liquid and gas (oil-gas, water-gas), Darcy's law has a different form than with a single-phase function:

,

Here K 1 (S)? K 2 (S) - relatively phase permeability, depending on S - saturation of the 2nd phase, usually water, 1st phase - oil and gas.

In hydrodynamic calculations, it is often convenient to use empirical dependences of relative phase permeability on saturation obtained from experimental data. Let's consider the empirical formulas obtained by Chen-Zhong-Xiang, which can be used in estimating calculations.

1. For water and oil (s-water saturation):

2. for gas and water (s-gas saturation):

The behavior of relative phase permeabilities is described by graphs of the form:

Dependencies have two characteristic points S st, S *

At the point S \u003d Sv relative water permeability \u003d 0 \u003d K 2 (S)

At point S=S * relative permeability of oil (gas) = ​​0

At these points, the phase with zero permeability is dispersed and occupies isolated dead ends in a porous medium, and therefore is not mobile. Simultaneous filtration of 3 phases has been studied less than two-phase. Use this approach. S n + S in \u003d S well, considering 2 phases - liquid and gas S G + S well \u003d 1 two-phase system.

two-phase system, K n (S), K in (S)

K G (S f), K f (S f)

All relative permeabilities are determined from two-phase diagrams, (S G, S W) and

Then the relative permeability for oil is K f (S f) K n (S)

for water - K w (S w) K in (S)

for gas - K G (S well)

For thick seams, or sloping seams, where gravity must be taken into account, if the Z axis, then the vertical component of the two-three-phase filtration rate instead contains:

P are the same pressures in the phases.

P * = P + ρgZ reduced pressures.

We have considered expressions for the filtration rate for two, three-phase filtration flow. If two or three immiscible phases (oil, gas, water) are moving, then the same type of equations will be written for each separately:

I=1, 2, 3 where 1 is gas

2 - oil

3 - water or:

The difference from the equation of continuity of a single-phase liquid - the equation includes - the saturation of the phases. On the left side, S i affect the phase permeabilities. On the right side, when compiling the mass balance for an element, we must take into account for a separate phase not the entire pore volume, but its share occupied by the i-th phase.

If we substitute expressions for velocities as functions of pressure and saturation, as well as expressions for phase densities as functions of pressure, into the system of equations for the continuity of phases, then for a system of 3 equations we have unknowns 4-P, S 1, S 2 , S 3.

The system is closed by the relation: S 1 + S 2 + S 3 = 1.

In this case, it was assumed that the pressures in the phases are the same Р.

capillary forces. Interfacial (capillary) pressure forces arise in pore channels, for example, in a two-phase flow.

R g - R f = R k (S f)

Since the functions P k (S w) have been studied experimentally, the input cap. forces in the equation does not add the number of unknowns.

7. Multi-phase multi-component filtration. Three-phase - two-three-dimensional filtration.

We consider a system from " n l" phases, for example:

1st phase - oil wetting by gas, non-wetting by water;

2nd phase - water, wetting;

3rd phase - gas, non-wetting.

In general, the system consists of nc"chemical components. When moving, changing the pressure, temperature of the mixture, individual hydrocarbon components can transfer from the oil phase to the gas phase and vice versa. The transition of water into a gaseous state is not excluded, during thermal action on the formation. Mass transfer of various components occurs between the phases. In this case, the material balance when deriving the continuity equation is written for each component separately, and as a result we have " nc"continuity equations.

Fraction of the pore space of the mesh element occupied by " 1 "th phase - S 1 ;

C e j - concentration j- th component in 1 is the phase in the grid volume under consideration.

Then the change in mass j- th component in the grid element must be considered as the sum of its changes in each phase, taking into account ( S 1 C e j) – fraction of the pore volume of the element occupied j- component in 1 – oh phase;

S l C lj ρ l- mass fraction j- 1 – oh phase;

Total mass fraction j- th component in the pore volume of the grid element 1 – oh phase;

- mass fraction j- th component in the pore volume of the grid element;

Ω el is the volume of the grid element.

Mass change j- th component in a short period of time Δt, in the case of a Cartesian coordinate system, we write the form:

Let be the density of the source (sink) 1 – oh phase, - concentration j- th component in the source 1 - oh phase.

Then - the total density of the source according to j- oh component.

The flow terms in the continuity equation, in contrast to the single-phase motion, contain

Bulk speed j- th component in the stream 1 - oh phase.

In the material balance equation j- th component, the flows of the considered component over all phases are summed up.

As a result, the continuity equation for j- th component looks like:

The number of system equations is determined by the number of moving components j=1,2,…,n c .

Three-phase model of an oil reservoir.

In the design of oil reservoirs, the non-volatile oil model (beta model) has been widely used. The hydrocarbon system is approximated by two components: non-volatile (oil) and volatile (gas) soluble in oil. It is assumed that three separate phases coexist in a porous medium: oil, gas, and water.

Water and oil do not mix, do not exchange masses and do not change phases.

The gas is soluble in oil, insoluble in water.

Fluids are assumed to be in thermodynamic equilibrium at constant temperature.

Consider the movement of a three-phase fluid: oil, gas, water (g, o, c):

the gas phase consists of one component - free gas;

water phase - from one water component;

oil phase - 2-component, oil and gas dissolved in it.

Let us determine the concentrations of the components in the phase; 1,2 - oil, gas.

C H1 is the concentration of oil in the oil phase;

C H2 - gas concentration in the oil phase;

C B1 \u003d C B2 \u003d 0, i.e. the water phase does not contain a component of gas and oil;

C G2 = 1, i.e. the gas phase contains only gas;

FROM lj– mass concentration j- th component in 1 - oh phase.

Consider the left side of the equation for the continuity of a multiphase fluid.

(sums for all phases of flows j- th component).

Let us introduce the concept of volumetric phase coefficients: the ratio of the phase volume in reservoir conditions to the volume in standard conditions.

For the gas phase:

For the aqueous phase:

For the oil phase:

here - the volume of oil in reservoir conditions, taking into account the gas dissolved in it;

.

Main literature

additional literature

1. Lysenko VD Innovative development of oil fields. - M.: Nedra-Business Center, 2000. - 516s. - Bibliography: pp.513-514

2. Zakirov, S. N. Development of gas, gas condensate and oil and gas condensate fields / S.N. Zakirov. - M. : Struna, 1998. - 626 p. - Bibliography: p. 597-620. - ISBN 5-85926-011-3

3. Zheltov, Yu. P. Development of oil fields: a textbook for universities / Yu. P. Zheltov. - 2nd ed., revised. and additional - M. : Nedra, 1998. - 365 p. : ill. - Bibliography. from. 359. - ISBN 5-247-03806-1

4. Kanevskaya R.D. Mathematical modeling of hydrodynamic processes in the development of hydrocarbon deposits. - M. - Izhevsk: Institute of Computer Research, 2002. - 140 p.

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1 Method for constructing phase equilibrium constants of multicomponent solutions EV Koldoba New analytical formulas for phase equilibrium constants are proposed in this work, taking into account the influence of fluid composition and more accurately conveying the phase behavior of multicomponent solutions. The approach makes it possible to build a thermodynamically consistent model of multicomponent filtration convenient for numerical simulation: the required computing resources are reduced, and the reliability of calculations is increased. Key words: phase equilibrium constants, EOS, phase transition. Introduction To predict the development of oil and gas fields, numerical modeling methods using three-dimensional hydrodynamic models are widely used. Calculations can take from several minutes to several months, depending on the complexity and accuracy of the model and computer performance. Oil and gas condensate contain hundreds of components, and even small concentrations of one of them can change the phase state of the mixture, so one of the urgent problems is to take into account as many components as possible in order to more accurately describe the complex phase 269

2 E. V. Koldoba of system behavior. Phase transitions in solutions are accompanied not only by the formation of new phases, but also by a continuous change in the component composition of the phases. When distributing the components between the gas and liquid phases, phase equilibrium constants (distribution coefficients) are used. In the Russian-language literature, the terms "constants" or "coefficients" are traditionally used to designate these quantities, although for the class of problems under consideration, these are complex functions of pressure, temperature, and composition of solutions. To simulate phase transitions in multicomponent solutions, modern hydraulic simulators use compositional and thermal models. thermal model, multidimensional phase diagrams (if there are N components in the solution, then the phase diagram is N-dimensional) are not calculated, the distribution of components over phases is carried out using phase equilibrium constants, so the correct setting of these functions can significantly increase the accuracy of calculations. takes the calculation of phase equilibria of multicomponent solutions (flash). Iterative methods calculate "the exact values ​​of the phase equilibrium constants, but the stability of these complex and time-consuming calculations depends largely on the accuracy of their initial approximations, which for given by formulas. It is possible to increase the stability of numerical calculations of phase equilibrium in the compositional model and significantly reduce its time if the formulas specifying the phase equilibrium constants are more accurate. There are many explicit and implicit (iterative) methods for calculating phase equilibrium constants. In iterative methods, equations of state for solutions (EOS) are used to calculate phase equilibrium. The accuracy of which can be improved by adjusting the pair interaction constants . Explicit methods for calculating phase equilibrium constants use reference data on the properties of each component and some characteristics of the mixture known from laboratory tests. 270

3 Method for constructing phase equilibrium constants for multicomponent solutions For an N-component solution with a total molar concentration z, the phase equilibrium constants are functions of pressure, temperature, and composition K = K (p, T, z 1, z 2,..., z N). K are defined as the ratio of the concentrations of the -th component in the gas and liquid phases: K = y /x (1) where is the component number. For weak solutions, K is determined, according to Raoult's laws, as the ratio of the partial pressure of the saturated vapor of the th component p s to the total pressure in the system: K = p s (T) / p (2) The most well-known methods of calculating by formulas are listed below. The formula proposed by Wilson: K = p c p exp (5.31(1 + ω) (1 T c T)) where p is the pressure in the solution, T is the temperature. The following reference data are used to calculate the -th component: p c - critical pressure, T c - critical temperature, ω - acentric factor. Thor and Witson's modification: K = (pc p) A 1 (ps p A = 1) (exp 5.31(1 + ω)(1 T) c T) () 0.6 p 14.7 pk 14.7 in (4) there are several approaches. Praza, for example, proposed the following relation: p k = (MW γ) C7+ (3) (4) 271

4 E. V. Koldoba + 3 [ a (MW γ)c7+)/(T 460) ] ln p + a 5 p 2 + a 6 p where a 1, a 2, a 3, a 4, a 5, a 6 are constants known for each component. Modified by Reid et al., used in modern thermal simulators, has the following form: () a1 K = p + a 2 + a 3 p exp [ a 4 /(T a 5)] (6) where a 1, a 2, a 3, a 4, a 5, a 6 are constants known for each component. The listed models (2-5) do not take into account the composition of real solutions and the features of the dissolution of the components, which sometimes leads to significant errors. For example, in a two-component C 4 H 10 C 10 H 22 solution, the C 4 H 10 component is lighter and its concentration in the gas phase is higher than in the liquid phase, i.e. by definition (1): K C4 H 10 >> 1. However, according to formulas (2-5) it turns out K C4 H 10< 1. В работах - был предложен способ построения констант фазового равновесия с настраиваемыми параметрами, учитывающими поведение конкретного реального раствора в заданном диапазоне давлений и температур: (5) K = A (p + p) α p β (7) где A, p, α, β - настраиваемые параметры, учитывающие свойства реального флюида. В работе была продемонстрирована эффективность и быстродействие такого подхода при моделировании многокомпонентной фильтрации с фазовыми переходами. 272

5 Method for constructing phase equilibrium constants for multicomponent solutions In this paper, we propose a further development of this approach. The assessment of the accuracy of the proposed model and its adjustment was carried out on a more complex and resource-intensive compositional model using the Peng-Robinson equation of state (PR) and the corresponding procedure for calculating phase equilibrium. The accuracy is compared with other methods for calculating K. Problem Statement Let an N-component solution with a total molar concentration z be in a two-phase state, separating into a gas with a concentration y and a liquid with a concentration x. Indices G and L here and below will denote quantities related to the gas and liquid phases, respectively. For the mixture and each phase separately, the following normalization conditions must be met: N z = 1, N x = 1, N y = 1 The gas phase is a non-ideal gas that can be in a supercritical state. To describe the pvt properties, we propose to use the hyperbolic equation of state (EOS): V G = βrt p + b (8) where V G is the molar volume of the gas phase, T is temperature, p is pressure, β, b are adjustable parameters. The hyperbola (8) has two asymptotes: 1) at V p = 0 2) p V G = b (curve 2 in Fig.1). The liquid phase can be either incompressible or compressible, and near the critical point the EOS of the liquid must change into the EOS for the gas, so the EOS of the liquid phase is written 273

6 E. V. Koldoba Fig. 1. Hyperbolic equations of state: 1 - for liquid, 2 for gas, ABCD - isotherm of the Peng-Robinson equation. also in hyperbolic form: VL = αrt +b p + p (9) where VL is the molar volume of the liquid phase, α, p, b are adjustable parameters. The hyperbola (9) has two asymptotes: 1) for V p = p 2) p VG = b. Let us construct the Gibbs molar potential for the gas phase: X X gg = βrt ln p + b p + RT y ln B y + y χ (10) The first and second terms of equation (10) are obtained by integrating the EOS of the gas. The remaining terms of the expression are added according to the mixing rules of the physics of solutions and describe the processes of dissolution. Correction coefficients B are introduced in the third term, taking into account the imperfection of the dissolution of the components. χ are functions that depend only on temperature and characterize the pure th component; these functions are the same both in the gas and in the liquid phase. 274

7 Method for constructing phase equilibrium constants of multicomponent solutions After transformations, the Gibbs molar potential for the gas phase has the form: g G = RT ln (p β exp(bp/rt)) + RT y ln B y + y χ (11) It is known that the potential Gibbs is a first order homogeneous function of the number of moles of the components, so the Gibbs molar potential is a first order homogeneous function of the component concentrations. For this property to hold, it is necessary to multiply the first term of equation (11) by the sum of concentrations N y = 1, after which the chemical potentials of the components in the gas phase are calculated: or () gg µ,g = yp = RT ln (p β exp(bp/rt ) + RT y ln B y + χ) µ,g = RT ln (B yp β exp(bp/rt) + χ) (12) The chemical potentials of the components for the liquid phase are calculated similarly: µ,l = RT ln (A x ( p + p) α exp(bp/rt)) + χ (13) where A are some correction factors characterizing the imperfection of the processes of component dissolution in the liquid. From the equality of the chemical potentials of the components µ,l = µ,g in the phases (one of the phase equilibrium conditions) we have: From the equality we obtain an expression for y /x, i.e. an expression for the phase equilibrium constants K: exp(bp/rt) B p β exp(bp/rt) = C (p + p) α p β exp((bb)p/rt) (14) 275

8 EV Koldoba where C = A /B are integral correction factors that simultaneously characterize the imperfection of dissolution processes in both gas and liquid. In this model, the equilibrium constant of the th component does not explicitly depend on the concentrations and characteristics of other components. However, by adjusting the parameters C, β, b, α, p, b, we thereby take into account the properties of a real solution in the considered range of pressures and temperatures. The parameters C are determined from the calculated or measured values ​​of the concentrations y 0 and x 0 at pressure p 0, we calculate the values ​​K 0 = y 0 /x 0 and find C. Finally, we get: K = K 0 (p + p) α p β p β 0 (p 0 + p) α exp((bb)(pp 0)/RT) (15) Calculation of model parameters Parameters K 0, β, b, α, p, b are calculated at some reference pressure p 0 from experimental data or data , obtained from more complex and costly models, let's call them "exact" models: α, p, b - are determined from the approximation of the "exact" urs of the liquid by the model; β, b - are determined from the approximation of the "exact" gas EOS by the model one, K 0 - from the values ​​of the concentrations y 0 and x 0 at the equilibrium node, corresponding to the total fluid concentration z. Compositional model

9 Method for constructing phase equilibrium constants of multicomponent solutions del, using the iterative method and the Peng-Robinson equation of state, which is given as follows: where p = b = RT V ba V (V + b) + b(v + b) pb =1 a = N =1 b = RT c, P c, N jaaj (1 kj) j=1 a = R2 T 2 c, P c, [ T 1 + m (1) T c, ] 2 (16) f ω m = ω ω 2 f ω > m = ω ω ω 3 , then c = x, if gas, then c = y) The parameter b in the Peng-Robinson EOS has the physical meaning of the volume of molecules, therefore V > b is naturally always satisfied, and, moreover, V = b is the pole of the function. The same parameter b is used in the model hyperbolic EOS of a fluid (9) and in it V = b is the asymptote. Thus, the parameter b in the EOS equation (9) is calculated in the same way as in the Peng-Robinson model. The ABCD curve in Fig.1 is the Peng-Robinson EOS isotherm, the AB branch describes the liquid state of the fluid, CD is the gaseous state. Hyperbola 1 given by equation (9) approximates the liquid branch of the EOS, hyperbola 2 given by equation (8) approximates the gas branch. 277

10 EV Koldoba In the compositional model, the Peng-Robinson EOS is used to calculate the chemical potentials of components in gas and liquid. The system of equations for chemical potentials is solved by iterative methods: µ,g = µ,l, = 1, 2,...N 0. To solve system (17), a computer program was used that simulates the calculation of phase equilibrium in a compositional model (flash), a description of such calculations can be found in . Substituting the concentrations in the gas and the pressure into equation (16), we find the gas root (the largest of the roots, moreover, V > b). Substituting the concentrations in the liquid into equation (16), we find the liquid root (the smallest of the roots, moreover, V > b). We find the derivatives p for the Peng-Robinson EOS: V p V = RT (V b) + 2a(V + b) 2 (V 2 + 2bV b 2) 2 We calculate the derivatives at pressure p 0. On the other hand, we calculate the derivatives of the equations ( 8-9): () p = αrt 2 () p = βrt 2 VLV b VGV b Equating the values ​​of the EOS and their derivatives for the gas and liquid phases, we find the values ​​of the parameters β, b, α, p. For a C 1 H 4 C 10 H 22 solution, the results obtained by formulas (3-6) were compared using the proposed method and the "exact" compositional model. The comparison results are shown in Fig. 2. In the pressure range under consideration (atm), a good agreement between the new model and the "exact" solution was obtained. 278

11 Method for constructing phase equilibrium constants for multicomponent solutions Fig. 2. Phase equilibrium constants for methane in solution C1 H4 C10 H22: K - "exact" solution, W - according to the Wilson formula, R - according to the Reid formula, Kol - according to the new formula References Habbalah W.A., Startzman R.a., Barrafet M.A. use of neural networks for prediction of vapour/lqud equlbrum K-values ​​for lght hydrocarbon mxture, SPE Reservor Engneerng, May Wlson G.M. A modfed Redlch-Kwong EOS. Applcaton to General Physcal Data Calculatons. Paper 15c presented at the 1969 AlChE Natl.Meetng, Cleveland, Oho. Whtson C.H. and Torp S.B. Evaluatng Constant Volume Depleton Data. JPT (March 1983), Trans., AIME,

12 E. V. Koldoba Red R. C., Prausntz J. M. and Sherwood T.K. The properties of Gases and Lquds, 3rd edton, McGraw-Hll, New York, Koldoba A.V., Koldoba E.V. Model Equation of State and Gibbs Potential for Numerical Calculation of Multicomponent Filtration Problems with Phase Transitions. - Geochemistry, 2004, N 5, c Koldoba A.V., Koldoba E.V. Thermodynamically consistent model of a multicomponent mixture with phase transitions. Mathematical modeling, 2010, v.22, N 4, with Koldoba A.V., Koldoba E.V., Myasnikov A.V. Efficient thermodynamically consistent approach for numerical modeling of oil displacement processes, - Mathematical Modeling, 2009, N 10, with Sivukhin D.V. Thermodynamics and molecular physics. 5th ed., rev. - M.: FIZMATLIT, 2005, 544 p. Brusilovsky A.I. Phase transformations in the development of oil and gas. Publishing house "Grail Moscow 2002, 575 p. 280

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Heterogeneous equilibria associated with the transition of a substance from one phase to another without changing the chemical composition are called phase.

These include equilibria in the processes of evaporation, melting, condensation, etc. Phase equilibria, like chemical ones, are characterized by the minimum value of the Gibbs energy of the system (G 0 T = 0) and the equality of the rates of direct and reverse processes. Thus, the equilibrium in the “water-ice” system H 2 O (l.)  H 2 O (cr.) is characterized by the equality of the rates of ice melting and water crystallization.

Equilibrium in heterogeneous systems depends on pressure, temperature and concentration of components in the system. For phase equilibrium, as well as for chemical equilibrium, Le Chatelier's principle is valid.

Before formulating the Gibbs phase rule, let's define some concepts.

Phase (F)- a part of a thermodynamic system, homogeneous at all points in composition and properties and separated from other parts of the system by an interface.

Component (K) or integral part of the system- a substance that can be isolated from the system and exist outside it.

The smallest number of components in terms of which the composition of any phase is expressed is called the number of independent components of the system. When considering phase equilibria, when no chemical transformations occur in the system, the concepts of “component” and “independent component” coincide.

The number of degrees of freedom (C) or the variance of the system is the number of conditions (temperature, pressure, concentration) that can be changed arbitrarily without changing the number and type of system phases.

The ratio between the number of phases (F), components (K), and degrees of freedom (C) in a heterogeneous equilibrium system is determined by Gibbs phase rule:

C \u003d K - F +n, (7)

where n is the number of external factors affecting the equilibrium in the system.

These are usually temperature and pressure. Then n = 2 and equation (7) takes the form:

C \u003d K - F + 2. (8)

According to the number of degrees of freedom, systems are divided into:

invariant (C = 0),

monovariant (C = 1),

ivariant (C = 2),

polyvariant (C > 2).

When studying phase equilibria, the graphical method is widely used - the method of constructing state diagrams. The state diagram for any substance is built on the basis of experimental data. It makes it possible to judge: 1) the stability of any one phase of the system; 2) about the stability of equilibrium between two or three phases under given conditions. For example, consider the state diagram of a single-component system (K = 1) - water (Fig. 7.2).

Three curves AO, OB, and OS, intersecting at one point O, divide the diagram into three parts (fields, areas), each of which corresponds to one of the aggregate states of water - vapor, liquid, or ice. The curves correspond to the equilibrium between the corresponding phases.

AO curve expresses the dependence of saturated vapor pressure over ice on temperature and is called the sublimation (sublimation) curve. For the AO curve: K \u003d 1, Ф \u003d 2, n \u003d 2, then the number of degrees of freedom, C \u003d 1 - 2 + 2 \u003d 1. This means that You can arbitrarily change only the temperature (or only the pressure) - the system is monovariant.

Curve OS expresses the dependence of saturated vapor pressure over liquid water on temperature and is called the evaporation or condensation curve. For the OS curve: K= 1, F= 2, n = 2, then the number of degrees of freedom C = 1–2 + 2 = 1, i.e. the system is monovariant.

RH curve expresses the dependence of the melting point of ice (or freezing of liquid water) on pressure and is called the melting or crystallization curve. For the OB curve: K= 1, F= 2, n= 2, then the number of degrees of freedom C = 1 - 2 + 2 = 1, i.e. the system is monovariant.

All considered curves (AO, OB, OS) correspond to the transition of water from one phase state to another, i.e. determine the balance of two phases . The regions bounded by these curves correspond to the conditions for the existence of a single phase. For each of these areas: K = 1, F = 1, n = 2, then the number of degrees of freedom C = 1 - 2 + 2 = 2 - the system is bivariant, those. within certain limits, it is possible to change the values ​​of p and T independently of each other without changing the number of phases.

All curves intersect at point O- triple point - it corresponds to the equilibrium of all three phases:

Ice  Liquid water  Steam

For the triple point: K = 1, Ф = 3, n = 2, then the number of degrees of freedom C = 1 - 3 + 2 = 0 - the system is invariant, those. the equilibrium conditions (temperature and pressure) are strictly defined and none of them can be changed: T = 273.1 K, P = 610 Pa (4.58 mmHg).

Heterogeneous equilibria associated with the transition of a substance from one phase to another without changing the chemical composition are called phase.

These include equilibria in the processes of evaporation, melting, condensation, etc. Phase equilibria, like chemical ones, are characterized by the minimum value of the Gibbs energy of the system (DG 0 T = 0) and the equality of the rates of the direct and reverse processes. Thus, the equilibrium in the “water-ice” system H 2 O (l.) Û H 2 O (cr.) is characterized by the equality of the rates of ice melting and water crystallization.

Equilibrium in heterogeneous systems depends on pressure, temperature and concentration of components in the system. For phase equilibrium, as well as for chemical equilibrium, Le Chatelier's principle is valid.

Before formulating the Gibbs phase rule, let's define some concepts.

Phase (F)- a part of a thermodynamic system, homogeneous at all points in composition and properties and separated from other parts of the system by an interface.

Component (K) or integral part of the system- a substance that can be isolated from the system and exist outside it.

The smallest number of components in terms of which the composition of any phase is expressed is called the number of independent components of the system. When considering phase equilibria, when no chemical transformations occur in the system, the concepts of “component” and “independent component” coincide.

The number of degrees of freedom (C) or the variance of the system is the number of conditions (temperature, pressure, concentration) that can be changed arbitrarily without changing the number and type of system phases.

The ratio between the number of phases (F), components (K), and degrees of freedom (C) in a heterogeneous equilibrium system is determined by Gibbs phase rule:

C \u003d K - F + n, (7)

where n is the number of external factors affecting the equilibrium in the system.

These are usually temperature and pressure. Then n = 2 and equation (7) takes the form:

C \u003d K - F + 2. (8)

According to the number of degrees of freedom, systems are divided into invariant (C = 0), monovariant (C = 1), bivariant (C = 2) and polyvariant (C > 2).

When studying phase equilibria, the graphical method is widely used - the method of constructing state diagrams. The state diagram for any substance is built on the basis of experimental data. It makes it possible to judge: 1) the stability of any one phase of the system; 2) about the stability of equilibrium between two or three phases under given conditions. For example, consider the state diagram of a single-component system (K = 1) - water (Fig. 7.2).

Three curves AO, OB, and OS, intersecting at one point O, divide the diagram into three parts (fields, areas), each of which corresponds to one of the aggregate states of water - vapor, liquid, or ice. The curves correspond to the equilibrium between the corresponding phases. The AO curve expresses the dependence of saturated vapor pressure over ice on temperature and is called the sublimation (sublimation) curve. For the AO curve: K = 1, F = 2, n = 2, then the number of degrees of freedom, C = 1 - 2 + 2 = 1. This means that You can arbitrarily change only the temperature (or only the pressure) - the system is monovariant.

The OS curve expresses the dependence of the saturation vapor pressure over liquid water on temperature and is called the evaporation or condensation curve. For the OS curve: K= 1, F= 2, n = 2, then the number of degrees of freedom C = 1–2 + 2 = 1, i.e. the system is monovariant.

The RH curve expresses the dependence of the melting temperature of ice (or freezing of liquid water) on pressure and is called the melting or crystallization curve. For the OB curve: K= 1, F= 2, n= 2, then the number of degrees of freedom C = 1 - 2 + 2 = 1, i.e. the system is monovariant.

All considered curves (AO, OB, OS) correspond to the transition of water from one phase state to another, i.e. determine the balance of two phases . The regions bounded by these curves correspond to the conditions for the existence of a single phase. For each of these areas: K = 1, F = 1, n = 2, then the number of degrees of freedom C = 1 - 2 + 2 = 2 - the system is bivariant, those. within certain limits, it is possible to change the values ​​of p and T independently of each other without changing the number of phases.

All curves intersect at point O - the triple point - it corresponds to the equilibrium of all three phases:

Ice Û Liquid water Û Steam.

For the triple point: K = 1, Ф = 3, n = 2, then the number of degrees of freedom C = 1 - 3 + 2 = 0 - the system is invariant, those. the equilibrium conditions (temperature and pressure) are strictly defined and none of them can be changed: T = 273.1 K, P = 610 Pa (4.58 mmHg).

If Avogadro's law is applicable only for gases, then Avogadro's number has a universal character for any aggregate state of matter.

All the considered gas laws are strictly observed at very low pressures, at ordinary low pressures they are observed approximately, and at high pressures large deviations from these laws are observed.

1 See section 2.8 for an idea of ​​the electron families of elements and their valence electrons.

1 The valence electrons of an atom are electrons that can participate in the formation of chemical bonds in molecules, ions, etc.

1 The effective radii found experimentally for the metallic state of simple substances are given.

2 In this case, the f-compression is called lanthanide.

1 The number of electrons in the outer energy level of the ion is given in parentheses.

The reaction takes place in a melt of substances.

1 The reaction takes place in a melt of substances.

2 The reaction takes place in the melt of substances.

3 The reaction takes place in solution.

Like chemical equilibrium, phase equilibrium is dynamic (the rates of forward and reverse processes are equal). As for chemical equilibrium, the thermodynamic condition for phase equilibrium is zero Gibbs energy:D G P, T = 0. Like the chemical equilibrium, the phase equilibrium obeys law of mass action And Le Chatelier-Brown principle(see above).

Let us define some basic concepts used in the theory of phase equilibrium. Component (part of the system ) - each of the chemically homogeneous substances contained in the system, which can be isolated from it and can exist in an isolated form for a long time. For example, an aqueous solution of NaCl and KCl consists of three components: H 2 O, NaCl and KCl.

Number of independent components K- the smallest number of components sufficient both for the formation of the entire system and for the formation of any of its phases. TO is equal to the total number of components of the equilibrium system minus the number of equations relating their concentrations at equilibrium (chemical or phase). For example, in a heterogeneous system in a state of chemical equilibrium:

C (gr.) + CO 2 (g.) « 2CO (g.)

3 components (C (gr.), CO 2 (g.), CO (g.)) and 2 independent components, since there is one equation that relates the concentrations of the system components in a state of chemical equilibrium - the equation of the chemical equilibrium constant ( TO= 3 – 1 = 2). Indeed, two components are sufficient for the formation of this entire system and any of its phases: C (g) and CO 2 (g).

Number of degrees of freedom (variability ) FROM is the number of external conditions (T, R, concentration of components, etc.), which can be arbitrarily changed within certain limits independently of each other without changing the number and type of equilibrium coexisting phases.

One of the most general laws of heterogeneous equilibrium (chemical and phase) is phase rule , bred in 1876 by J.W. Gibbs and used for the first time in practice in 1889 by H. Rosebaum. According to the phase rule, in a thermodynamically equilibrium system, the number of degrees of freedom FROM, phases F, independent components TO and external conditions n, affecting the equilibrium, are related by the relation:

For a one-component system ( TO= 1) the phase rule has the form:

C = 3 – F.

(3.36)

To illustrate the practical application of the phase rule, consider the simplest case - the state diagram of a one-component system - water (Fig. 3.5). This diagram is a combination of the following curves describing the equilibrium state of two phases in the coordinates temperature T - pressure R:

1. OS – melting curve . Characterizes the dependence of the melting temperature of ice on external pressure.

2. OA – sublimation curve (sublimations ). It characterizes the dependence of the pressure of saturated ice vapor on temperature.

3. OK – evaporation curve . It characterizes the dependence of the saturation vapor pressure of liquid water on temperature.

4. OV (depicted by dotted line). It characterizes the dependence of saturated vapor pressure over supercooled water on temperature. OV describes the behavior of water in metastable state (i.e., such a state when there are all signs of phase equilibrium, but D G P, T ¹ 0). For example, water can be cooled to -72°C at atmospheric pressure without causing crystallization. The system can be in this state (metastable) for an indefinitely long time. However, it is worth adding ice crystals to the water or mixing, rapid crystallization will begin with intense heat release and an increase in temperature to 0 ° C. The system goes into an equilibrium state.

The regions between the curves are single-phase regions (ice, liquid, vapor). The following points can be noted on the diagram:

1. X is the melting point of ice at atmospheric pressure.

2. Y is the boiling point of water at atmospheric pressure.

3. ABOUT – triple point . At this point, three phases (ice, liquid water, steam) are simultaneously in equilibrium.

4. TO – critical point . At this point, the distinction between the liquid and its vapor disappears. Above this point, water vapor cannot be converted to liquid water by any increase in pressure, and the water changes from vapor to gas (steam can be in equilibrium with the condensed phase, but gas cannot). Water (and any other substance) at a temperature above the critical point is also called supercritical fluid (dense fluid phase formed at temperatures and pressures above their critical values ​​[Atkins, 2007]).

For any point in a single-phase area, for example for a point Z(liquid), according to the formula (3.36) FROM= 3 – 1 = 2. The system is bivariant , i.e., within certain limits, independently of each other, you can change 2 external conditions (T and R

For any point corresponding to two equilibrium phases, for example, for a point X(ice/liquid), FROM= 3 – 2 = 1. The system is monovariant , i.e., only one external condition can be changed (either T or R) without changing the number and type of phases.

Finally, for the triple point ABOUT(ice/liquid/steam) FROM= 3 – 3 = 0. The system is invariant (non-variant ), i.e., without changing the number and type of phases, it is impossible to change any of the external conditions.

Shown in Fig. 3.5. the state diagram of water is valid for low pressures. At high pressures, ice can exist in several crystalline modifications. The diagram looks much more complicated [Kireev].

Rice. 3.5. Diagram of a water state diagram

[Ravdel], [Chemist's Handbook 1], [Eisenberg]