Gibbs phase rule states that the number of degrees of freedom WITH equilibrium thermodynamic system is equal to the difference between the number of components TO and the number of phases Ф, plus the number of factors P, affecting balance:
The phase rule allows, by a number of degrees of freedom, to predict the behavior of a system when one, two or more external conditions change and to calculate the maximum number of phases that can be in equilibrium under given conditions. Using the phase rule, one can predict the thermodynamic possibility of the existence of a system.
Usually the value P = 2, since only two factors are taken into account: temperature and pressure. Other factors (electrical, magnetic, gravitational) are taken into account as necessary. Then the number of degrees of freedom is equal to
If the temperature (or pressure) in the system remains constant, then the number of state parameters decreases by another unit
If the system maintains constant temperature and pressure (P = 0), then the number of degrees is equal to
The number of degrees of freedom for a one-component two-phase system (for example, crystal - liquid, crystal - vapor, liquid - vapor) is equal to
This means that each temperature corresponds to one single meaning pressure and, conversely, any pressure in a two-phase one-component system is realized only at a strictly defined temperature.
Consequently, heating of any two coexisting phases must be simultaneously accompanied by a strictly defined change in pressure, i.e. the temperature and pressure of the two phases are related by a functional relationship P=f (T).
Example 5.1. Determine the largest number of phases that can be in equilibrium in a system consisting of water and sodium chloride.
Solution. In this system the number of components (TO) equals two. Hence, C = = 4 - F. The largest number of phases corresponds to the smallest number of degrees of freedom. Since the number of degrees of freedom cannot be negative, the smallest value WITH equals zero. Therefore, the largest number of phases is four. A given system satisfies this condition when a solution of sodium chloride in water is in equilibrium simultaneously with ice, solid salt and water vapor. In this state, the system is variantless (invariant), i.e. this state is achieved only at strictly defined temperature, pressure and concentration of the solution.
One-component systems
At TO = 1 the equation of the phase rule takes the form
If one phase is in equilibrium, then WITH = 2. In this case they say that the system bivariant ;
two phases - C = 1, system monovariant;
three phases - WITH = 0, system invariant.
A diagram expressing the dependence of the state of a system on external conditions or on the composition of the system is called phase diagram. The relationship between pressure ( R ), temperature (7) and volume (V) of the phase can be represented by a three-dimensional phase diagram. Each point (every point is called figurative point) such a diagram depicts some equilibrium state. It is usually more convenient to work with sections of this diagram using a plane p - T (at V = const) or plane p - V (at T = const). Let us examine in more detail the case of a section by a plane p - T (at V= const).
Let us consider as an example the phase diagram of a monocomponent system - water (Fig. 5.1).
Phase diagram of water in coordinates p - T shown in Fig. 5.1. It is made up of three phase fields - areas of various (p, T) values at which water exists in the form of a certain phase - ice, liquid water or steam (indicated by the letters L, F and P, respectively). For these single-phase regions, the number of degrees of freedom is two, the equilibrium is bivariant (C = 3 - 1 = 2). This means that to describe the system it is necessary two independent variables - temperature and pressure. These variables can change in these areas independently, and there will be no change in the type or number of phases.
The phase fields are separated by three boundary curves.
Rice. 5.1.
AB curve - evaporation curve , expresses dependence vapor pressure of liquid water from temperature (or represents the dependence of the boiling point of water on pressure). In other words, this line answers two-phase liquid water - steam equilibrium, and the number of degrees of freedom calculated according to the phase rule is C = 3 - 2 = 1. Such balance monovariate. This means that for full description it is enough to define the system only one variable - either temperature or pressure. The second variable is the dependent variable, it is determined by the shape of the curve LW. Thus, for a given temperature there is only one equilibrium pressure or for a given vapor pressure there is only one equilibrium temperature.
At pressures and temperatures corresponding to points below the line AB, the liquid will completely evaporate and this area is the vapor area.
At pressures and temperatures corresponding to points above the line AB , the vapor is completely condensed into liquid (C = 2). Upper limit of the evaporation curve AB is at the point IN, which is called critical point (for water 374°C and 218 atm). Above this temperature, the liquid and vapor phases become indistinguishable (the clear liquid/vapor phase boundary disappears), therefore Ф = 1.
Line AC is the sublimation curve of ice (sometimes called a line sublimation ), reflecting the dependence water vapor pressure above ice on temperature. This line corresponds monovariate ice-steam equilibrium (C = 1). Above the line AC lies a region of ice, below is a region of steam.
Line AD melting curve , expresses dependence ice melting temperature versus pressure and corresponds monovariate equilibrium between ice and liquid water. For most substances the line AD deviates from the vertical to the right, but the behavior of the water is abnormal: Liquid water takes up less volume than ice. Based on Le Chatelier's principle, it can be predicted that an increase in pressure will cause a shift in equilibrium towards the formation of liquid, i.e. the freezing point will decrease.
Research conducted by GT.-U. Bridgman to determine the course of the ice melting curve at high pressures, showed that there is seven different crystalline modifications of ice , each of which, with the exception of the first, denser than water. So the upper limit of the line AD- point D where ice I (ordinary ice), ice III and liquid water are in equilibrium. This point is at -22°C and 2450 atm.
Triple point of water (a point reflecting the equilibrium of three phases - liquid, ice and steam) in the absence of air is at 0.0100 ° C and 4.58 mm Hg. Art. Number of degrees of freedom C = 3 - 3 = 0, and such an equilibrium is called invariant. When any parameter changes, the system ceases to be three-phase.
In the presence of air, the three phases are in equilibrium at 760 mm Hg. Art. and 0°C. The decrease in the temperature of the triple point in air is caused by the following factors:
- 1) the solubility of gaseous components of air in liquid water at 1 atm, which leads to a decrease in the triple point by 0.0024 ° C;
- 2) an increase in pressure from 4.58 mm Hg. Art. up to 1 atm, which reduces the triple point by another 0.0075°C.
Crystalline sulfur exists in the form two modifications - rhombic (S p) and monoclinic (S M). Therefore, the existence of four phases is possible: orthorhombic, monoclinic, liquid and gaseous (Fig. 5.2).
Solid lines delineate four regions: vapor, liquid, and two crystalline modifications. The lines themselves correspond to monovariant equilibria of the two corresponding phases. Note that the equilibrium line
monoclinic sulfur - melt deviated from vertical to the right (compare with the phase diagram of water). This means that when sulfur crystallizes from the melt, reduction in volume. At points A, B And WITH three phases coexist in equilibrium (point A rhombic, monoclinic and steam, point IN - rhombic, monoclinic and liquid, point WITH - monoclinic, liquid and vapor). It is easy to notice that there is another point O, at which there is an equilibrium of three phases - superheated orthorhombic sulfur, supercooled liquid sulfur and steam, supersaturated relative to steam, in equilibrium with monoclinic sulfur. These three phases form metastable system , i.e. a system that is in a state relative stability. The kinetics of the transformation of metastable phases into a thermodynamically stable modification is extremely slow, however, with prolonged exposure or the introduction of seed crystals of monoclinic sulfur, all three phases still transform into monoclinic sulfur, which is thermodynamically stable under conditions corresponding to the point ABOUT. Equilibria to which the curves correspond OA, OV And OS (sublimation, melting and evaporation curves, respectively) are metastable.
Rice. 5.2.
Clausius-Clapeyron equation
Movement along the lines of two-phase equilibrium on the phase diagram (C = 1) means a consistent change in pressure and temperature, i.e. R = f(T). General form Such a function for single-component systems was established by Clapeyron.
Suppose we have a monovariant equilibrium water - ice (line AD in Fig. 5.1). The equilibrium condition will look like this: for any point with coordinates (R, D) belonging to the line A.D.
For a one-component system p = dG/dv, where G- Gibbs free energy, and v is the number of moles. We need to express Formula Δ G=
= Δ H - T Δ S not suitable for this purpose, since it was bred for r, T = const. According to equation (4.3)
According to the first law of thermodynamics 
and according to the second law of thermodynamics _, and then
Obviously, in equilibrium
since the amount of ice formed at equilibrium is equal to the amount of water formed). Then
Molar (i.e., divided by the number of moles) volumes of water and ice; S water, S ice - molar entropies of water and ice. Let's transform the resulting expression into
(5.2)
where ΔSф, ΔVф p - change in molar entropy and volume at phase transition (ice -> water in this case).
Because the following type of equation is more often used:
where ΔHф p is the change in enthalpy during the phase transition; ΔV p - change in molar volume during transition; ΔTf p is the temperature at which the transition occurs.
The Clapeyron equation allows, in particular, to answer the following question: what is the dependence of the phase transition temperature on pressure ? The pressure can be external or created due to the evaporation of a substance.
Example 5.2. It is known that ice has a larger molar volume than liquid water. Then, when water freezes, ΔVф „ = V |да - V water > 0, at the same time ДНф „ = = ДН К < 0, since crystallization is always accompanied by the release of heat. Therefore, DHf „ /(T ΔVf p)< 0 и, согласно уравнению Клапейрона, производная dp/dT< 0. This means that the line of monovariant equilibrium ice - water on the phase diagram of water should form an obtuse angle with the temperature axis.
Clausius simplified the Clapeyron equation in the case evaporation And sublimation , assuming that:
Let's substitute (from the Mendeleev-Clapey equation
ron) into the Clapeyron equation:
Separating the variables, we get
(5.4)
This equation can be integrated if the dependence of ΔH IS11 on T. For a small temperature range, we can take ΔH NSP constant, then
Where WITH - integration constant.
Dependency In R from /T should give a straight line, from the slope of which the heat of evaporation D# isp can be calculated.
Let's integrate the left side of equation (5.4) in the range from R ( before p 2, and the right - from G, to T 2> those. from one point (p, 7,) lying on the liquid-vapor equilibrium line, to another - (p 2, T 2):
We write the result of integration in the form
(5.6)
sometimes called Clausius-Clapeyron equation. It can be used to calculate the heat of vaporization or sublimation if the vapor pressures at two different temperatures are known.
Entropy of evaporation
Molar entropy of evaporation 
equal to the difference
Because it can be assumed
The next assumption is that steam is considered an ideal gas. This implies the approximate constancy of the molar entropy of evaporation of a liquid at the boiling point, called Trouton’s rule.
Trouton's rule: the molar entropy of evaporation of any liquid is of the order of 88 JDmol K).
If during the evaporation of different liquids there is no association or dissociation of molecules, then the entropy of evaporation will be approximately the same. For compounds that form hydrogen bonds (water, alcohols), the entropy of evaporation is greater than 88 JDmol K). Trouton's rule allows us to determine the enthalpy of evaporation of a liquid from a known boiling point, and then, using the Clausius-Clapeyron equation, determine the position of the monovariant liquid-vapor equilibrium line on the phase diagram.
Example 5.3. Estimate the vapor pressure over diethyl ether at 298 K, knowing its boiling point (308.6 K).
Solution. According to Trouton's rule AS.. rn = 88 JDmol K), on the other hand,
Let us apply the Clausius - Clapeyron equation (5.6), taking into account that at boiling (T = 308.6 K) the vapor pressure of the ether p = 1 atm. Then we have: In /; - In 1 = 27.16 x x 10 3 /8.31(1/308.6 - 1 /T), or In R = -3268/7" + 10.59 (and this is the equation of the line of monovariant equilibrium liquid - vapor on the phase diagram of the ether). Hence, at T = 298 K (25°C), R = 0.25 atm.
Entropy of melting is not as constant for different substances as the entropy of evaporation. This is due to the fact that disorder (of which entropy is a measure) does not increase as much during the transition from a solid to a liquid state as it does during the transition to a gaseous state.
The state of a one-component system is determined by two independent parameters (for example, P and T), and the volume of the system V = f(P,T). If we plot the pressure, temperature, and volume of the system along three coordinate axes, respectively, we get a spatial diagram that characterizes the dependence of the state of the system and phase equilibria in it from external conditions. Such a diagram is called a state diagram or phase diagram.
For one-component system two-dimensional the phase diagram shows the conditions of existence at different values of the parameters (T - temperature and P - pressure) of several phases; the simplest option is three phases: gas, liquid and solid.
Water diagram
An example of such a diagram for water is shown in Fig. 8.2.
The same water, over a wider range of pressures and temperatures, has several – phases – 9 in a solid state of aggregation (Fig. 8.3).
Diagram of the state of sulfur.
Let us consider the phase diagram of the state of sulfur.
Lines EA, AC and CN – temperature dependence of saturated vapor pressure over solid Srhomb., Smon. and Szh. respectively. Line AB is the temperature dependence of the external pressure of the phase transition Srhomb. Smon. Lines CB and OB – dependence of the melting temperature Smon on external pressure. and Sromb. respectively. Stability region Srhomb. limited by lines EA, AB, BD. Stability area Smon. limited by lines BC, SA, AB. The region of existence of the liquid phase is located to the right of the BC and BD lines and above the CN line. The stability region of vaporous sulfur lies below the EA, AC and CN lines.
Unlike the phase diagram of water with one triple point, the phase diagram of sulfur has three such points: A, B and C. In each of them three phases can exist simultaneously. At point A - solid rhombic, solid monoclinic and vaporous sulfur; at point B – solid rhombic, solid monoclinic and liquid sulfur; at point C – solid monoclinic and vaporous sulfur and liquid sulfur.
State diagrams of two-component systems. Thermal analysis
Physicochemical systems that contain two components are called two-component systems. The components can be both simple substances and chemical compounds. The relationship between components can significantly change the properties of the system. To determine the state of the system unambiguously, it is necessary to know the parameters: P, T and the concentrations of components C1 and C2. The equation of state of a two-component system has the form: f(P,T,C1,C2)=0
To construct phase diagrams, the method of thermal analysis is used.
Thermal analysis
The thermal analysis method is based on the analysis of cooling curves of mixtures of various compositions. The cooling curve is a temperature-time dependence of the cooling of the mixture, showing the points of phase transitions. In Fig. 8.5. The shape of the cooling lines is shown (the left part of the figure) on which the plateau corresponds to the crystallization of the pure component (curves A and B), the inflection point is the beginning of the crystallization of one of the components of the solution.
State diagrams of two-component systems with complete solubility of substances in the solid phase
An example of such a diagram is shown in Fig. 8.5. - right part. It is shown how this diagram is constructed from these cooling curves.
The crystal structure of a substance is determined not only by its chemical composition, but also by the conditions of formation. There are many examples in nature where, depending on the conditions of formation, substances can have different crystal structures, i.e. lattice type, and therefore different physical properties. This phenomenon is called polymorphism. The presence of one or another modification of a substance can thus characterize the conditions of its formation. Polymorphic modifications are denoted by the Greek letters , , , .
Two types of polymorphism are possible: enantiotropic and monotropic.
Enantiotropy characterized by a reversible spontaneous transition at certain P and T of one form to another. This transition is accompanied by a decrease in the Gibbs energy when thermodynamic conditions (P, T) change. In Fig. Figure 4.2 shows the enantiotropic transition from phase to phase at (P, T) ↔ . Up to the temperature and pressure of the phase transition (to the left of the point (P,T) ↔ ), the modification of the substance is more stable, because it has a smaller supply of free energy G. At the point of intersection of the curves G =G , both phases are in equilibrium. Beyond this point, i.e. at high P and T, the phase is more stable. Thus, the -phase is a low-temperature, and -high-temperature modification of the substance.
Rice. 4.2. Change in Gibbs function with enantiotropy
Enantiotropic transformations are characteristic of carbon, sulfur, silicon dioxide and many other substances.
If only one phase is stable in the entire range of P and T, then the phase transition is not associated with certain values P and T and is irreversible. This type of polymorphism is called monotropy (Fig.4.3).The more stable phase is the one whose Gibbs energy is lower. (in Fig. 4.3. phase ). Monotropic forms are less common in natural conditions; an example would be the system
Fe 2 O 3 Fe 2 O 3
maghemite hematite
Rice. 4.3. Change in Gibbs energy during monotropy
In the phase diagram, the polymorphism of a substance is characterized by additional lines that limit the regions of existence of individual polymorphic modifications.
4.5.3. Sulfur phase diagram
As an example, consider the phase diagram of sulfur, which can exist in the form of orthorhombic or monoclinic sulfur, i.e. she is dimorphic. On the phase diagram of sulfur ( rice. 4.4), in contrast to the water diagram, two fields of solid phases: the region of orthorhombic sulfur (to the left of the EABD line, field 1) and monoclinic (inside the ABC triangle, field 2). Field 3 is the region of molten sulfur, field 4 is vaporous sulfur.
BC - melting curve of monoclinic sulfur,
BD – melting curve of orthorhombic sulfur,
AB - polymorphic transformation curve: S rhombus ↔ S monocle
EA and AC are the sublimation curves of orthorhombic and monoclinic sulfur, respectively.
SC is the evaporation curve of liquid sulfur.
The dotted lines reflect the possibility of the existence of metastable phases, which can be observed with a sharp change in temperature:
A 
O: ↔ (S); CO: (S)↔ (S); VO: ↔ (S).
Rice. 4.4. Sulfur phase diagram
Triple points correspond to three-phase equilibria: A – rhombic, monoclinic and vaporous sulfur; Cmonoclinic, liquid and vapor; Rhombic, monoclinic and liquid sulfur. At point O (the point of intersection of the dotted lines inside the triangle) there is a metastable equilibrium of three phases: rhombic, liquid and vapor.
Thus, using a phase diagram, it is possible to determine the phase state of a substance under given conditions or, conversely, having discovered one or another polymorphic modification of a substance (for example, an alloy or mineral), to characterize the conditions of its formation.
In Sect. 3.2 stated that if a compound can exist in more than one crystalline form, then it is said to exhibit polymorphism. If any free element (simple substance) can exist in several crystalline forms, then this type of polymorphism is called allotropy. For example, sulfur can exist in two allotropic forms: the β-form, which has an orthorhombic crystal structure, and the β-form, which has a monoclinic crystal structure. The molecules in -sulfur are packed more densely than in -sulfur.
In Fig. Figure 6.7 shows the temperature dependence of the free energy (see Chapter 5) of two allotropic forms of sulfur, as well as its liquid form. Free energy of anyone
Rice. 6.7. Dependence of the free energy of sulfur on temperature at atmospheric pressure.
substances decreases with increasing temperature. In the case of sulfur, the α-allotrope has the lowest free energy at temperatures below 368.5 K and is therefore most stable at such temperatures. At temperatures from 368.5 (95.5 °C) to 393 K (120 °C) -allogrope is most stable. At bittern temperatures of 393 K, the liquid form of sulfur is most stable.
In cases where an element (a simple substance) can exist in two or more allotropic forms, each of which is stable over a certain range of changing conditions, it is considered to exhibit estiotropic Temperature at which two enantiotropes are in equilibrium with each other is called the transition temperature. The temperature of the enantiotropic transition of sulfur at a pressure of 1 atm is 368.5 K.
The effect of pressure on the transition temperature is shown by the AB curve on the sulfur phase diagram shown in Fig. 6.8. An increase in pressure leads to an increase in the transition temperature.
Sulfur has three triple points - A, B and C. At point A, for example, two solid and vapor phases are in equilibrium. These two solid phases are two enantiotropes of sulfur. Dashed curves correspond to metastable conditions; For example, the AD curve represents the vapor pressure curve of sulfur at temperatures above its transition temperature.
Enantiotropy of other elements
Sulfur is not the only element that exhibits enantiotropy. Tin, for example, has two enantiotropes - gray tin and white tin. The transition temperature between them at a pressure of 1 atm is 286.2 K (13.2 °C).
