Phase equilibrium pure substance

Whether or not the solid or liquid phases are pure substances or mixtures, the equilibrium constant can also be obtained from calorimetrically determined values of and A5 according to equation (15). 

Clapeyron-Clausius equation A thermodynamic equation applying to any two-phase equilibrium for a pure substance. The equation states ... 

The Clapeyron equation expresses the dynamic equilibrium existing between the vapor and the condensed phase of a pure substance ... 

The term "pliase" for a pure substance indicates a state of matter - that is, solid, liquid, or gas. For mi. tures, however, a more stringent connotation must be used, since a totally liquid or solid system may contain more dian one phase. A phase is characterized by uniformity or homogeneity die same composition and properties must c. ist tliroughout the pliase region. At most temperatures and pressures, a pure substance normally exists as a single phase. At certain temperatures mid pressures, two or perhaps even dmee phases can coe.xist in equilibrium. 

Vapor—Liquid Systems. The vapor-liquid region of a pure substance is contained within the phase or saturation envelope on a P-V diagram (see Figure 2-80), A vapor, whether it exists alone or in a mixture of gases, is said to be saturated if its partial pressure (P.) equals its equilibrium vapor pressure (P, ) at the system temperature T. This temperature is called the saturation temperature or dew point T ... 

For a pure substance, the melting point is identical to the freezing point It represents the temperature at which solid and liquid phases are in equilibrium. Melting points are usually measured in an open container, that is, at atmospheric pressure. For most substances, the melting point at 1 atm (the normal melting point) is virtually identical with the triple-point temperature. For water, the difference is only 0.01°C. 

Thus, the condition for equilibrium between two phases of a pure substance is given by... 

For a pure substance, having three phases in equilibrium results in a triple point that is invariant. When pure solid, liquid, and gaseous water are in equilibrium, the temperature is fixed at a value of 273.16 K, and the pressure of the gas is fixed at the vapor pressure value (0.6105 kPa). 

In this section we limit our discussion to the phase equilibria involved with pure substances. In this case, the condition for equilibrium between phases A, B, C,..., becomes... 

So far, we have described the effect of pressure and temperature on the phase equilibria of a pure substance. We now want to describe phase equilibrium for mixtures. Composition, usually expressed as mole fraction x or j, now becomes a variable, and the effect of composition on phase equilibrium in mixtures becomes of interest and importance. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

Where applications to industrial combustion systems involve a relatively limited set of fuels, fire seeks anything that can bum. With the exception of industrial incineration, the fuels for fire are nearly boundless. Let us first consider fire as combustion in the gas phase, excluding surface oxidation in the following. For liquids, we must first require evaporation to the gas phase and for solids we must have a similar phase transition. In the former, pure evaporation is the change of phase of the substance without changing its composition. Evaporation follows local thermodynamics equilibrium between the gas... 

The conditions that apply for the saturated liquid-vapor states can be illustrated with a typical p-v, or (1 /p), diagram for the liquid-vapor phase of a pure substance, as shown in Figure 6.5. The saturated liquid states and vapor states are given by the locus of the f and g curves respectively, with the critical point at the peak. A line of constant temperature T is sketched, and shows that the saturation temperature is a function of pressure only, Tsm (p) or psat(T). In the vapor regime, at near normal atmospheric pressures the perfect gas laws can be used as an acceptable approximation, pv = (R/M)T, where R/M is the specific gas constant for the gas of molecular weight M. Furthermore, for a mixture of perfect gases in equilibrium with the liquid fuel, the following holds for the partial pressure of the fuel vapor in the mixture ... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

Let us apply Equation (6.8) to the two-phase liquid-vapor equilibrium requirement for a pure substance, namely p = p T) only. This applies to the mixed-phase region under the dome in Figure 6.5. In that region along a p-constant line, we must also have T constant. Then for all state changes along this horizontal line, under the p—v dome, dg = 0 from Equation (6.8b). The pure end states must then have equal Gibbs functions ... 

Fig. 3.2. A stylized phase diagram for a simple pure substance. The dashed line represents 1 atm pressure and the intersection with the solid-liquid equilibrium line represents the normal boiling point and the intersection with the liquid-vapor equilibrium line represents the normal boiling point.

The principle of operation is illustrated in Figure 15.37 which shows the pressme-volume relationship. Curve a shows the phase change of a pure liquid as it is pressurised isother-mally. Crystallisation begins at point Ai and proceeds by compression without any pressure change until it is complete at point A2. Beyond this point, the solid phase is compressed resulting in a very sharp rise in pressure. If the liquid contains impurities, these nucleate at point Bi. As the crystallisation of the pure substance progresses, the impurities are concentrated in the liquid phase and a higher pressure is required to continue the crystallisation process. As a result, the equilibrium pressure of the liquid-solid system rises exponentially with increase of the solid fraction, as shown by curve b which finally approaches... 

For a system consisting of one pure substance only which may exist in two different phases in equilibrium the criterion is obtained in a simple manner in terms of free energy. In such a case, if a certain amount of the substance is transferred from one phase to the other, the molar free energy of one phase decreases while that of the other phase increases by an equal amount. Hence the net result is that there is no change in free energy. 

The KTTS depends upon an absolute zero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equilibrium, together with specification of an interpolation instrument and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

Fig. 2. PT diagram for a pure substance that expands on melting (not to scale). For a substance that contracts on melting, eg, water, the fusion curve, A, has a negative slope point /is a triple state point is the gas—liquid critical state (—) are phase boundaries representing states of two-phase equilibrium ...

Equation (1) may be applied to the equilibrium between vapor and liquid of a pure substance (X = vapor pressure) or to the equilibrium between an ideal dilute solution and the pure phase of a solute X = solubility) or to the equilibrium of a chemical reaction (X = equilibrium constant). 

First we inspect the normal melting points (Tm) of the compounds. Note that because Tm, Th and Tc already have a subscript denoting that they are compound specific parameters, we omit the subscript i. Tm is the temperature at which the solid and the liquid phase are in equilibrium at 1.013 bar (= 1 atm) total external pressure. At 1 bar total pressure, we would refer to Tm as standard melting point. As a first approximation, we assume that small changes in pressure do not have a significant impact on the melting point. Extending this, we also assume that Tm is equal to the triple point temperature (Tt). This triple point temperature occurs at only one set of pressure/temperature conditions under which the solid, liquid, and gas phase of a pure substance all simultaneously coexist in equilibrium. 

We generally distinguish between two methods when the determination of the composition of the equilibrium phases is taking place. In the first method, known amounts of the pure substances are introduced into the cell, so that the overall composition of the mixture contained in the cell is known. The compositions of the co-existing equilibrium phases may be recalculated by an iterative procedure from the predetermined overall composition, and equilibrium temperature and pressure data It is necessary to know the pressure volume temperature (PVT) behaviour, for all the phases present at the experimental conditions, as a function of the composition in the form of a mathematical model (EOS) with a sufficient accuracy. This is very difficult to achieve when dealing with systems at high pressures. Here, the need arises for additional experimentally determined information. One possibility involves the determination of the bubble- or dew point, either optically or by studying the pressure volume relationships of the system. The main problem associated with this method is the preparation of the mixture of known composition in the cell. 

Figure 15-2B gives the corresponding chemical potentials calculated as in Equation 15-1. A loop also appears on this figure. The loop is nonexistent physically but can be used analytically. The point of intersection, e, meets the requirements of equilibria for the gas and liquid of a pure substance. At point e, the pressure of the gas equals the pressure of the liquid, and the chemical potentials of the two phases are equal. Point f has the same pressure as points e but is not an equilibrium point because its chemical potential is higher than that of points e. 

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