Equilibrium potential, liquid phase chemical

Similarly, for component 1 in a liquid mixture in equilibrium with a mixture of gases, the liquid-phase chemical potential is, from Eq. (9.3.40),... 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

There is one other three-phase equilibrium involving clathrates which is of considerable practical importance, namely that between a solution of Q, the clathrate, and gaseous A. For this equilibrium the previous formulas and many of the following conclusions also hold when replacing fiQa by fiQL, the chemical potential of Q in the liquid phase. But a complication then arises since yqL and the difference are not only... 

In Chapter 5, we considered systems in which composition becomes a variable, and defined and described the chemical potential. We showed that the chemical potential provides the condition for spontaneity or equilibrium. It is the potential that drives the flow of mass in a chemical process, A useful quantity related to the chemical potential is the fugacity. It can also be thought of as a measure of the flow of mass in a chemical process, and can be used to determine the point of equilibrium. It is often known as the escaping tendency since it can be used to describe the ease with which mass flows from one phase to another, particularly the flow from a solid or liquid phase to a gas phase. 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

There is a similar expression for polymer activity. However, if the fluid being sorbed by the polymer is a supercritical gas, it is most useful to use chemical potential for phase equilibrium calculations rather than activity. For example, at equilibrium between the fluid phase (gas) and polymer phase, the chemical potential of the gas in the fluid phase is equal to that in the liquid phase. An expression for the equality of chemical potentials is given by Cheng (12). 

The temperature at which this condition is satisfied may be referred to as the melting point Tm, which will depend, of course, on the composition of the liquid phase. If a diluent is present in the liquid phase, Tm may be regarded alternatively as the temperature at which the specified composition is that of a saturated solution. If the liquid polymer is pure, /Xn —mS where mS represents the chemical potential in the standard state, which, in accordance with custom in the treatment of solutions, we take to be the pure liquid at the same temperature and pressure. At the melting point T of the pure polymer, therefore, /x2 = /xt- To the extent that the polymer contains impurities (e.g., solvents, or copolymerized units), ixu will be less than juJ. Hence fXu after the addition of a diluent to the polymer at the temperature T will be less than and in order to re-establish the condition of equilibrium = a lower temperature Tm is required. 

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 ... 

A great many electrolytes have only limited solubility, which can be very low. If a solid electrolyte is added to a pure solvent in an amount greater than corresponds to its solubility, a heterogeneous system is formed in which equilibrium is established between the electrolyte ions in solution and in the solid phase. At constant temperature, this equilibrium can be described by the thermodynamic condition for equality of the chemical potentials of ions in the liquid and solid phases (under these conditions, cations and anions enter and leave the solid phase simultaneously, fulfilling the electroneutrality condition). In the liquid phase, the chemical potential of the ion is a function of its activity, while it is constant in the solid phase. If the formula unit of the electrolyte considered consists of v+ cations and v anions, then... 

At equilibrium the chemical potential should be equal in the gas and liquid phases. At uniform temperature and pressure, this leads to the same fugacities in the two phases. In the liquid, the fugacity may be related to the fugacity of a standard state, f°... 

The chemical potential of the solvent in the liquid phase in equilibrium with its vapor is equal to the chemical potential of the solvent vapor. Thus,... 

For most of the situations encountered in solvent extraction the gas phase above the two liquid phases is mainly air and the partial (vapor) pressures of the liquids present are low, so that the system is at atmospheric pressure. Under such conditions, the gas phase is practically ideal, and the vapor pressures represent the activities of the corresponding substances in the gas phase (also called their fugacities). Equilibrium between two or more phases means that there is no net transfer of material between them, although there still is a dynamic exchange (cf. Chapter 3). This state is achieved when the chemical potential x as... 

The chemical potential of a solute in equilibrium with its solid or liquid phase is equal to the chemical potential of the neat compound. It is therefore hardly surprising that some papers claiming to report on on water chemistry merely report reactivity comparable to reactivity of the neat reactants. In the author s opinion, the term on water chemistry would preferably be reserved for those processes for which additional rate-enhancing effects are found. 

At equilibrium, equality of chemical potentials of a component in two liquid phases Li and L2 leads to... 

Equations (A5) and (A6) describe the interspecies equilibrium in the liquid phase for some composition xlt x2. If this liquid is on the liquidus surface of Ax BuC(s), then the liquidus equations given by Eqs. (12) and (13) must also be satisfied. Using the general relation between relative chemical potential and activity coefficient given by Eq. (60), the liquidus equations are... 

T is temperature, P is pressure, and / is the fugacity of the component. In Equation 3 subscript k refers to each component of the system. In the present discussion the fugacity 42) is employed in preference to the chemical potential 21). Earlier in the history of the petroleum industry, Raoult s 55) and Dalton s laws were applied to equilibrium at pressures considerably above that of the atmosphere. These relationships, which assume perfect gas laws and additive volumes in the gas phase and zero volume for the liquid phase, prove to be of practical utility only at low pressures. Henry s law was found to be a useful approximation only for gases which were of low solubility and at reduced pressures less than unity. 

The papers in the second section deal primarily with the liquid phase itself rather than with its equilibrium vapor. They cover effects of electrolytes on mixed solvents with respect to solubilities, solvation and liquid structure, distribution coefficients, chemical potentials, activity coefficients, work functions, heat capacities, heats of solution, volumes of transfer, free energies of transfer, electrical potentials, conductances, ionization constants, electrostatic theory, osmotic coefficients, acidity functions, viscosities, and related properties and behavior. 

The original equation for salt effect in vapor-liquid equilibrium, proposed by Furter (7) and employed subsequently by Johnson and Furter (8), described the effect of salt concentration on equilibrium vapor composition under the condition of a fixed ratio of the two volatile components in the liquid phase. The equation, derived from the difference in effects of the salt on the chemical potentials of the two volatile components, with simplifying approximations reduces to the form... 

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. 

In general, equality of component fugacities, ie, chemical potentials, in the vapor and liquid phases yields the following relation for vapor—liquid equilibrium ... 

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a liquid mixture into two liquid phases (I and II) occurs when a single liquid phase is thermodynamically unstable. The equilibrium condition of equal fiigacities (and chemical potentials) for each component in the two phases allows the fugacitiesy and f in phases I and II to be equated and expressed as ... 

The thermodynamic quantity 0y is a reduced standard-state chemical potential difference and is a function only of T, P, and the choice of standard state. The principal temperature dependence of the liquidus and solidus surfaces is contained in 0 j. The term is the ratio of the deviation from ideal-solution behavior in the liquid phase to that in the solid phase. This term is consistent with the notion that only the difference between the values of the Gibbs energy for the solid and liquid phases determines which equilibrium phases are present. Expressions for the limits of the quaternary phase diagram are easily obtained (e.g., for a ternary AJB C system, y = 1 and xD = 0 for a pseudobinary section, y = 1, xD = 0, and xc = 1/2 and for a binary AC system, x = y = xAC = 1 and xB = xD = 0). 

By using the procedures just outlined, the reduced standard-state chemical potential can be estimated for all compounds. This value of Gy is valid for any solid-liquid phase equilibrium problem that contains the compound... 

What are the correct values of the potentials In the metal the potential is the same everywhere and therefore 99 has one clearly defined value. In the electrolyte, the potential close to the surface depends on the distance. Directly at the surface it is different from the potential one Debye length away from it. Only at a large distance away from the surface is the potential constant. In contrast to the electric potential, the electroc/zmz caZpotential is the same everywhere in the liquid phase assuming that the system is in equilibrium. For this reason we use the potential and chemical potential far away from the interface. 

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... 

Whenever the total pressure is increased on a condensed phase, the chemical potential of the phase is increased. As a consequence, the pressure of the vapor in equilibrium with the condensed phase must also increase. The discussion here is limited to the liquid phase, but the basic equations that are developed are applicable also to a solid phase. 

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