Single-Component Systems Vapor Pressure

Note that while a substance can only exist as a gas above the critical temperature, its density is nevertheless high enough that it can also qualify as a quasi-liquid. This dual behavior has certain advantages, which are exploited in a process termed supercritical fluid extraction, which is taken up in Section 6.2.5. 

Equation 6.1a cannot be integrated in straightforward fashion since both AH and AV depend on temperature in a complex fashion. However, when one of the states is represented by the vapor phase, the exact Equation 6.1a can be simplified by introducing the following two approximations  

The molar volume of the vapor phase is much greater than that of the liquid or solid phase. Thus, 

At low pressures, it may be assumed that the vapor phase behaves ideally, so that 

On substituting these two simplifying relations into Equation 6.1a and rearranging, we obtain 

Note that while a substance can exist only as a gas above the critical temperature, its density is nevertheless high enough that it can also qualify 

Mass Transfer and Separation Processes Principles and Applications 

Recall from thermod5mamics that Equation 6.1a integrates to the celebrated Clausius-Clapeyron equation, of which we present two versions below  

Although these equations are used routinely in the plotting, computation, and extrapolation of vapor pressure, it should be kept in mind that they are based on the following three assumptions  

Variations of AH with temperature are often considered negligible over a wide range of T but are, in fact, a thermod5mamic requirement (see Practice Problem 6.2). All three assumptions tend to break down with increasing 

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Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

The upper limit of the vapor pressure line is the point A. This is known as the critical point and the temperature and pressure represented by this point are the critical temperature To and the critical pressure Pc, respectively. At this point the intensive properties of the liquid phase and the vapor phase become identical and they are no longer distinguishable. For a single-component system the critical temperature may also be defined as the temperature above which a vapor cannot be liquefied, regardless of the applied pressure. Similarly, the critical pressme of a single-component system may be... 

In our discussion of single-component systems, we considered the case of a liquid being heated in a container exposed to the atmosphere and observed that the liquid boils at a temperature at which the vapor pressure of the liquid equals the total pressure of the atmosphere above it. A similar phenomenon occurs for liquid mixtures. If a mixture is heated slowly in an open container, vapor bubbles will form at the heated surface and emerge into the gas phase when the vapor pressure of the liquid equals the pressure above the liquid. Some reflection should convince you that the temperature at which this occurs is the bubble point of the liquid at that pressure. For an ideal liquid solution, the boiling point may therefore be determined approximately from Equation 6.4-9. 

To find the vapor pressure using an equation of state (eos), we begin with vapor-liquid equilibrium for a single-component system ... 

Gas-solid equilibrium for a single-component system is commonly referred to as sublimation equilibrium. Sublimation pressure, the vapor pressure of a solid, is basic to the modeling of solid-gas equilibrium. Sublimation pressure changes with temperature by an equation similar to that of the vapor pressure of a liquid. Equation (4.453),... 

In single component systems bubble point and dew point are represented by the same line, called vapor pressure curve (Fig. 5.1-6). In multicomponent systems... 

For a single-component liquid-vapor interface, it is not possible to vary temperature and pressure independendy while maintaining the phases in equihbrium. Flowever, measurement of interfacial tension variation with pressure at constant temperature is possible in a binary system. Good (1976) has shown that such data can be used to obtain useful information about interfacial characteristics in this case. 

Therefore, the two shaded parts in Figure 2.12 have an equal area. This situation is the same as the constant-pressure line for the vapor-liquid coexistence in the isothermal process of a single-component system (Maxwell construction). We can find (f>i and < 2 from this equahty of the areas. It is, however, easier to find 4>i and < 2 from A/tp(2) and Afisi4>i) = although solving these two... 

The whole new field that opens up is where the gas is present in relatively large amounts, and now the nueleation and phase change represent not the ebullition of the liquid, which in this case mainly acts as a carrier solvent for the dissolved gas, but the appearance of bubbles of the previously dissolved gas itself, with a lesser contribution from the vapor pressure of the solvent. Of course, there is a continuum of behavior from pure boiling in a single component system consisting of the liquid, through the bubble evolution of essentially the pure gas from solution—and this eontinuity is reflected in the mathematical description of the process, with essentially one equation describing both extremes. 

Before closing this section it should be pointed out that the present review of the vaporization thermodynamics and structural trends of rare-earth halides reveals that several of the single-component systems should be reinvestigated with a view both to establishing the existence of dimer species in the equilibrium vapors, if they exist, and to refine the relevant vapor pressure data. Furthermore, the thermochemical properties of the vapors need to be studied more thoroughly in order to identify possible trends in the vaporization thermodynamics. Finally, a clear, unambiguous high-temperature and/or matrix isolation spectroscopic (Raman and IR) characterization of the vibrational properties of the rare-earth halide vapors still remains to be realized before their structural picture is fully settled and a reliable third-law treatment of the data is attempted. 

As mentioned earlier the ease or difficulty of separating two products depends on the difference in their vapor pressures or volatilities. There are situations in the refining industry in which it is desirable to recover a single valuable compound in high purity from a mixture with other hydrocarbons which have boiling points so close to the more valuable product that separation by conventional distillation is a practical impossibility. Two techniques which may be applied to these situations are azeotropic distillation and extractive distillation. Both methods depend upon the addition to the system of a third component which increases the relative volatility of the constituents to be separated. 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

Although the third component of these systems is usually a single inorganic salt, mixtures of two or more salts have been studied, and some research has been done with third components of low vapor pressure (18,19). Some qualitative studies have been done on salt effect in vapor-liquid equilibrium with salts which are either soluble in only one or both components, hygroscopic or non-hygro-scopic, etc. 

As an illustration consider a single-component, two-phase system in a potential field. Let the phases be a gas and a liquid phase. Such a system has one degree of freedom, and we then choose the temperature, volume, and moles of the component to be the variables that define the state of the system. The cross section of the container, A, is uniform. Let the bottom of the container be at the position rt in the field. Figure 14.2 illustrates this system where r0 is the position of the phase boundary. With knowledge of the volume of the system and the cross section of the container, (r2 — rt) is known. The unknown quantities are r0 and the moles of the component in each phase. The pressure at the phase boundary is the vapor pressure of the liquid at the chosen temperature. Then the number of moles of the component in the gas phase is... 

For a system containing only a single component, boundaries between the vapor and the liquid determine the state of the substance. For the substance to be in the liquid phase, the temperature must be less than the critical temperature and the pressure must be greater than the vapor pressure. To be in the gas phase, the pressure must be less than the critical pressure and the temperature must be less than the vapor pressure. If the pressure is less than the critical... 

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