The Fundamental Fact of Phase Equilibrium

The principal business of this chapter is to establish the thermodynamic relations obeyed by two or more phases that are at equilibrium with each other. A phase is a portion of a system (or an entire system) inside which intensive properties do not change abruptly as a function of position. The principal kinds of phases are solids, liquids, and gases, although plasmas (ionized gases), liquid crystals, and glasses are sometimes considered to be separate types of phases. Solid and liquid phases are called condensed phases and a gas phase is often called a vapor phase. Several elements such as carbon exhibit solid-phase allotropy. That is, there is more than one kind of solid phase of the element. For example, diamond and graphite are both solid carbon, but have different crystal structures and different physical properties. With compounds, this phenomenon is called polymorphism instead of allotropy. Most pure substances have only one liquid phase, but helium exhibits allotropy in the liquid phase. 

In a multicomponent system there can often be several solid phases or several liquid phases at equilibrium. For example, if one equilibrates mercury, a mineral oil, a methyl silicone oil, water, benzyl alcohol, and a perfluoro compound such as perfluoro (A-ethyl piperidine) at room temperature, one can obtain six coexisting liquid phases. Each of these phases consists of a large amount of one substance with small amounts of the other substances dissolved in it. Under ordinary conditions, a system can exhibit only a single gas phase. However, if certain gaseous mixtures are brought to supercritical temperatures and pressures, where the distinction between gas and liquid disappears, two fluid phases can form without first making a gas-liquid phase transition. 

The substances in the system whose amounts can be varied independently are called components. We denote the number of components by c. Since the system is closed, if component number i moves out of one phase it must move into the other phase  

For an infinitesimal change of state involving changes in T and P and a transfer of matter from one phase to the other, 

If T and P are constant, the fundamental criterion of equilibrium implies that dG must vanish for an infinitesimal change that maintains equilibrium  

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The properties of a system at equilibrium do not depend on how the system arrived at equilibrium. Therefore, Eq. (5.1-5) is valid for any system at equilibrium, not only for a system that arrived at equilibrium under conditions of constant T and P. We call it the fundamental fact of phase equilibrium In a multiphase system at equilibrium the chemical potential of any substance has the same value in all phases in which it occurs. 

Section 5.1 The Fundamental Fact of Phase Equilibrium 5.3 For water at equilibrium at 23.756 torr and 298.15 K,... 

The vapor pressure that we have discussed thus far is measured with no other substances present. We are often interested in the vapor pressure of a liquid that is open to the atmosphere. The other gases in the atmosphere exert an additional pressure on the liquid that modifies its vapor pressure. Small amounts of the other gases dissolve in the liquid, but we neglect these impurities in the liquid. Denote the vapor pressure corresponding to a total pressure of P by P. From the fundamental fact of phase equilibrium for a one-component system,... 

The fundamental fact of phase equilibrium is that at equilibrium... 

We now show that a component of an ideal solution obeys Raoult s law if the solution is at equilibrium with an ideal gas mixture. From the fundamental fact of phase equilibrium the chemical potential of component / has the same value in the solution and in the vapor ... 

Physical chemists always want to write a single equation that applies to as many different cases as possible. We would like to write equations similar to Eq. (6.1-8) for the chemical potential of every component of every solution. Consider a dilute solution in which the solvent and the solute are volatile. We equilibrate the solution with a vapor phase, which we assume to be an ideal gas mixture. Using Henry s law, Eq. (6.2-1), for the partial vapor pressure of substance number i (a solute) and using the fundamental fact of phase equilibrium ... 

Consider a solid solute that is soluble in a liquid solvent but insoluble in the solid solvent. Assume that the pure solid solvent (component number 1) is at equilibrium with a liquid solution containing the dilute solute. From the fundamental fact of phase equilibrium. 

Consider a volatile solvent (component 1) and a nonvolatile solute (component 2) in a solution that is at equilibrium with the gaseous solvent at a constant pressure. We assume that the gas phase is an ideal gas and that the solvent acts as though it were ideal. Our development closely parallels the derivation of the freezing point depression formula earlier in this section. The fundamental fact of phase equilibrium gives... 

Hydrochloric acid, HCl, is one of a half-dozen strong acids, which means that its acid ionization constant is too large to measure accurately. We must find a way to handle the activity of unionized species such as HCl in spite of their unmeasurably small concentrations. Since aqueous HCl has an appreciable vapor pressure we assume that aqueous unionized HCl in an aqueous solution of HCl is at equilibrium with gaseous HCl. From the fundamental fact of phase equilibrium... 

We call M/.chem the chemical part of the chemical potential. It is assumed to be independent of the electric potential and depends only on temperature, pressure, and the composition of the system. The chemical potential including the electric potential term is the true chemical potential that obeys the fundamental fact of phase equilibrium. Some electrochemists use the term electrochemical potential for the chemical potential in Eq. (8.1-7) and refer to the chemical part of the chemical potential as the chemical potential. We will use the term chemical potential for the tme chemical potential and the term chemical part of the chemical potential for /r,diein-... 

Solubility and kinetics methods for distinguishing adsorption from surface precipitation suffer from the fundamental weakness of being macroscopic approaches that do not involve a direct examination of the solid phase. Information about the composition of an aqueous solution phase is not sufficient to permit a clear inference of a sorption mechanism because the aqueous solution phase does not determine uniquely the nature of its contiguous solid phases, even at equilibrium (49). Perhaps more important is the fact that adsorption and surface precipitation are essentially molecular concepts on which strictly macroscopic approaches can provide no unambiguous data (12, 21). Molecular concepts can be studied only by molecular methods. 

Mercury electrodeposition is a model system for experimental studies of electrochemical phase formation. On the one hand, the product obtained is a liquid drop, corresponding very well with the liquid drop model of classical nucleation theory. Besides, electron transfer is fast [61] and therefore the growth of nuclei is controlled by mass transport to the electrode surface [44]. On the other hand, the properties of the mercuryjaqueous solution interface have been the object of study for over a century and hence are fairly well understood. The high overpotential for proton reduction onto both mercury and vitreous carbon favor the study of the process over a wide range of overpotentials. In spite of the complications introduced by the equilibrium between the Hg +, Hg2 " ", and Hg species, this system offers an excellent opportunity to verily the fundamental postulates of the electrochemical nucleation theory. In fact, the dependence of the nucleation rate on the oxidation state of the electrodepositing species is fiiUy consistent with theory critical nuclei appear with similar sizes and onto similar number densities of active sites... 

So far we have not touched on the fact that the important topic of solvation energy is not yet taken into account. The extent to which solvation influences gas-phase energy values can be considerable. As an example, gas-phase data for fundamental enolisation reactions are included in Table 1. Related aqueous solution phase data can be derived from equilibrium constants 31). The gas-phase heats of enolisation for acetone and propionaldehyde are 19.5 and 13 keal/mol, respectively. The corresponding free energies of enolisation in solution are 9.9 and 5.4 kcal/mol. (Whether the difference between gas and solution derives from enthalpy or entropy effects is irrelevant at this stage.) Despite this, our experience with gas-phase enthalpies calculated by the methods described in this chapter leads us to believe that even the current approach is most valuable for evaluation of reactivity. 

One of the most fundamental problems of chemical physics is the study of the forces between atoms and molecules. We have seen in many preceding chapters that these forces are essential to the explanation of equations of state, specific heats, the equilibrium of phases, chemical equilibrium, and in fact all the problems we have taken up. The exact evaluation of these forces from atomic theory is one of the most difficult branches of quantum theory and wave mechanics. The general principles on which the evaluation is based, however, are relatively simple, and in this chapter we shall learn what these general principles are, and see at least qualitatively the sort of results they lead to. 

Thus we obtain a picture of the phase equilibrium between solidlike and liquidlike clusters which differs in a fundamental way from bulk equilibrium because of the finite tempierature range over which the solidlike and liquidlike forms may coexist. Clearly the transformation between these phases cannot be the same as a first-order phase transition, although it becomes so in the limit of large N. In fact the transformation of a cluster between solidlike and liquidlike is simply not in any of the traditional categories of first, second, or higher order. 

When the discussion turns to removal of some component from a fluid stream by a high surface area porous solid, such as silica gel, which is found in many consumer products (often in a small packet and sometimes in the product itself), then the term "adsorption" becomes more global and hence ambiguous. The reason for this ironically is that mass transfer may be convoluted with adsorption. In other words the component to be adsorbed must move from the bulk gas phase to the near vicinity of the adsorbent particle, and this is termed external mass transfer. From the near external surface region, the component must now be transported through the pore space of the particles. This is called internal mass transfer because it is within the particle. Finally, from the fluid phase within the pores, the component must be adsorbed by the surface in order to be removed from the gas. Any of these processes, external, internal, or adsorption, can, in principle, be the slowest step and therefore the process that controls the observed rate. Most often it is not the adsorption that is slow in fact, this step usually comes to equilibrium quickly (after all just think of how fast frost forms on a beer mug taken from the freezer on a humid summer afternoon). More typically it is the internal mass transport process that is rate limiting. This, however, is lumped with the true adsorption process and the overall rate is called "adsorption." We will avoid this problem and focus on adsorption alone as if it were the rate-controlling process so that we may understand this fundamentally. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

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