Vapor-liquid equilibrium consistency relation

Select the criterion to be used for thermodynamic consistency. Deviations from thermodynamic consistency arise as a result of experimental errors. Impurities in the samples used for vapor-liquid equilibrium measurements are often the source of error. A complete set of vapor-liquid equilibrium data includes temperature T. pressure P. liquid composition x, and vapor composition y,. Usual practice is to convert these data into activity coefficients by the following equation, which is a rearranged form of the equation that rigorously defines K values (i.e., defines the ratio y, /x, under Related Calculations in Example 3.1) ... 

For a system consisting of C components, the phase rule indicates that, in the two-phase region, there are F=C-2 + 2 = C degrees of freedom. That is, it takes C independent variables to define the thermodynamic state of the system. The independent variables may be selected from a total of 2C intensive variables (i.e., variables that do not relate to the size of the system) that characterize the system the temperature, pressure, C - 1 vapor-component mole fractions, and C - 1 liquid-component mole fractions. The number of degrees of freedom is the number of intensive variables minus the number of equations that relate them to each other. These are the C vapor-liquid equilibrium relations, Yj = K,X, i=l,. .., C. The equilibrium distribution coefficients, AT, are themselves functions of the temperature, pressure, and vapor and liquid compositions. The number of degrees of freedom is, thus, 2C - C = C, which is the same as that determined by the phase rule. 

If the logarithms of the activity coefficients of a binary system are plotted against molar composition, Eq. (53) relates the slopes at any value of the x s. This is useful for testing the consistency of data. Such testing becomes easier if we have a systematic way to correlate and smooth data and extend them over the entire range of composition. Integrated forms of the Gibbs-Duhem relation allow us to do this. Theoretically, one reliable measurement of vapor-liquid equilibrium at any point can then be used to characterize a system. 

In the distillation process, it is assumed that the vapor formed within a short period is in thermodynamic equilibrium with the liquid. Hence, the vapor composition y is related to the liquid composition x by an equilibrium relation of the functional form y = f x). The exact relationship for a particular mixture may be obtained from a thermodynamic analysis and is also dependent upon temperature and pressure. Figure 4.1 shows an example equilibrium curve for a system consisting of CS2 and CCI4 at 1 atmosphere pressure. 

Solvent in Solution. We shall use the pure substance at the same temperature as the solution and at its equilibrium vapor pressure as the reference state for the component of a solution designated as the solvent. This choice of standard state is consistent with the limiting law for the activity of solvent given in Equation (16.2), where the limiting process leads to the solvent at its equilibrium vapor pressure. To relate the standard chemical potential of solvent in solution to the state that we defined for the pure liquid solvent, we need to use the relationship... 

When a system consists of saturated-liquid and saturated-vapor phases coexisting in equilibrium, the total value of any extensive property of the two-phase system is the sum of the total properties of the phases. Written for the volume, this relation is... 

The nonidealities of equilibrium mixtures result from various combinations of molecular interactions one such interaction that has been recently studied is molecular association. Molecules of fatty acids such as acetic acid typically form dimers and, to a lesser extent, trimers by hydrogen bonding in the vapor and liquid states. Failure of the equilibrium data of binary systems containing acetic acid to meet established criteria of consistency based upon the Gibbs-Duhem relation has been observed by Rius et al. (I), Campbell et al. (2), Herington (3), and... 

Vapor sorption measurements yield equilibrium composition and fugacity or chemical potential the isopiestic version (19) is used to determine the uptake of a pure vapor by a nonvolatile material. This technique determines equilibrium composition of a phase which cannot be separated quantitatively from the liquid phase in equilibrium with it. In our application, a nonvolatile crystalline surfactant specimen S is equilibrated with vapor of V, which is, in turn, at equilibrium with a system of S and V consisting of two phases, one rich in S, and one rich in V. At equilibrium, the Gibbs-Duhem relation guarantees. that the initial specimen of S takes up enough V from the vapor phase that the chemical potential of S, as well as of V, is the same as in the biphasic system, and so the composition of the phase formed by vapor sorption is the same as that of the S-rich phase. This composition is easily determined by weight measurement. If the temperature were a triple point, i.e. three phases at... 

This report is concerned with contact angle hysteresis and with a closely related quantity referred to as "critical line force (CLF)." More particularly, it is concerned with the relationship between contact angle hysteresis and the magnitude of the contact angle itself. Two sets of liquid-solid-vapor systems have been investigated to provide the experimental data. One set consists of Teflon [poly(tetrafluoroethylene), Du Pont] and a series of liquids forming various contact angles at the Teflon-air interface. The second set consists of polyethylene and a similar series of liquids. In neither case was the ratio of air to test liquid vapor at the boundary line controlled, but it can be assumed that the ambient vapor phase operative in all the systems was close to an equilibrium mixture. 

Knowing the equation of state for a multi-component system of interest, one can determine thermodynamic functions consistent with basic thermodynamic relations [5]. In view of applications to calculation of the liquid-vapor equilibrium, we now turn to considering the definition of chemical potentials of components. 

The phase rule is a relation among the number of independent components, the number of phases, and the variance of a system in equilibrium. The independent components (or, briefly, the components) of a system are the substances that must be added to realize the system. The phases have been defined earlier (Section 1-3). Thus a system containing ice, water, and water vapor consists of three phases but only one component (water-substance), since any two of the phases can be formed from the third. The variance of the system is the number of independent ways in which the system can be varied these ways may include varying the temperature and the pressure, and also varying the composition of any solutions (gaseous, liquid, or crystalline) that exist as phases in the system. 

Essentially, the thermodynamic modeling consists in a set of equations that relate process parameters through mass and heat balances and liquid-vapor equilibrium equations. Below are the critical equations ... 

The method basically relates the changes in the composition of the liquid phase to the composition of the released vapor. This information would not be of practical value if the evaporation took place in completely dry air or in vacuum the difference in the composition of the liquid phase obviously equals the composition of the escaping vapor. In addition, if the evaporation were to take place under conditions close to equilibrium, the partial vapor pressures could be utilized to calculate the activities of the components in the liquid phase. Evaporation however, usually takes place into an atmosphere at a certain level of humidity and this fact affects the interaction between the liquid phase and its vapor. This influence is conveniently attended to using the algebraic approach as illustrated by an example [17], The system to be discussed consists of water (W), cosurfactant (C), and surfactant (S) and is depicted in Figure 1.16. 

Membrane structure and external conditions determine water sorption and swelling. The resulting water distribution determines transport properties and operation. Water sorption and swelling are central in rationalizing physical properties and electrochemical performance of the PEM. The key variable that determines the thermodynamic state of the membrane is the water content k. The equilibrium water content depends on the balance of capillary, osmotic, and electrostatic forces. Relevant external conditions include the temperature, relative humidity, and pressure in adjacent reservoirs of liquid water or vapor. The theoretical challenge is to establish the equation of state of the PEM that relates these conditions to A.. A consistent treatment of water sorption phenomena, presented in the section A Model of Water Sorption, revokes many of the contentious issues in understanding PEM structure and function. 

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