Ideal solution chemical equilibrium

Note that, as in the case of ideal solutions, the equilibrium constant involves a ratio of factors for the equilibrium concentration variables qj, raised to the appropriate power however, in the present case this ratio is preceded by a host of factors, enclosed in curly brackets, that attend to the nonideality of the constituents engaged in the chemical reaction. Departures from ideality of the pure components are discussed below for intermixed components one must look up in appropriate tabulations values of the various activity coefficients FjiT, P, qj) = Yj T, P, qj)a T, P). In principle, then, we have arrived at an appropriate formulation of the equilibrium constant for nonideal cases. 

Consider an ideal solution in equilibrium with its vapor at a fixed temperature T. For each component, the equilibrium condition is pi = where pi is the chemical potential... 

The theory of fluctuations in ideal solutions at equilibrium is well formulated and discussed in Chandrasekhar s classic article [23] and other sources [24, 25]. We consider a spatially homogeneous chemical reaction system in an excitable stationary state which is at constant temperature. The volume of the entire system is V, and we are concerned with fluctuations occurring in a selected small volume, v, which is in equilibrium with the larger volume surrounding it, equilibrium being taken with respect to concentration fluctuations. 

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. 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

We can find the reaction s equilibrium point from Equation 3.3 as soon as we know the form of the function representing chemical potential. The theory of ideal solutions (e.g., Pitzer and Brewer, 1961 Denbigh, 1971) holds that the chemical potential of a species can be calculated from the potential pg of the species in its pure form at the temperature and pressure of interest. According to this result, a species chemical potential is related to its standard potential by... 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

The previous summary of activities and their relation to equilibrium constants is not intended to replace lengthier discussions [1,18,25,51], Yet it is important to emphasize some points that unfortunately are often forgotten in the chemical literature. One is that the equilibrium constants, defined by equation 2.63, are dimensionless quantities. The second is that most of the reported equilibrium constants are only approximations of the true quantities because they are calculated by assuming the ideal solution model and are defined in terms of concentrations instead of molalities or mole fractions. Consider, for example, the reaction in solution ... 

C-t, which means, of course, that the ideal solution model is adopted, no matter the nature or the concentrations of the solutes and the nature of the solvent. There is no way of assessing the validity of this assumption besides chemical intuition. Even if the activity coefficients could be determined for the reactants, we would still have to estimate the activity coefficient for the activated complex, which is impossible at present. Another, less serious problem is that the appropriate quantity to be related with the activation parameters should be the equilibrium constant defined in terms of the molalities of A, B, and C. As discussed after equation 2.67, A will be affected by this correction more than A f//" (see also the following discussion). 

Seawater has high concentrations of solutes and, hence, does not exhibit ideal solution behavior. Most of this nonideal behavior is a consequence of the major and minor ions in seawater exerting forces on each other, on water, and on the reactants and products in the chemical reaction of interest. Since most of the nonideal behavior is caused by electrostatic interactions, it is largely a function of the total charge concentration, or ionic strength of the solution. Thus, the effect of nonideal behavior can be accoimted for in the equilibrium model by adding terms that reflect the ionic strength of seawater as described later. 

Consider now two practically immiscible solvents that form two phases, designated by and ". Let the solute B form a dilute ideal solution in each, so that Eq. (2.19) applies in each phase. When these two hquid phases are brought into contact, the concentrations (mole fractions) of the solute adjust by mass transfer between the phases until equilibrium is established and the chemical potential of the solute is the same in the two phases ... 

We define the standard state of a liquid as ay = 1 and for gases as an ideal gas pressure of 1 bar, Pj = I- For ideal liquid solutions (activity coefficients of unity), we write ay = Cy so at chemical equilibrium... 

Henry-Louis Le Chdtelier was a French chemist. He devised Le Chdtelier s principle, which explains the effect of a change in conditions on a chemical equilibrium. He also worked on the variation in the solubility of salts in an ideal solution. 

Charge transfer resistance, 1056 Charge transfer overpotential, 1231 Charge transfer, partial. 922. 954 Charges in solution, 882 chemical interactions, 830 Charging current. 1056 Charging time, 1120 Chemical catalysis, 1252 Chemical and electrochemical reactions, differences, 937 Chemical equilibrium, 1459 Chemical kinetics, 1122 Chemical potential, 937, 1058 definition, 830 determination, 832 of ideal gas, 936 interactions, 835 of organic adsorption. 975 and work function, 835... 

This relationship constitutes the basic definition of the activity. If the solution behaves ideally, a, =x, and Equation (18) define Raoult s law. Those four solution properties that we know as the colligative properties are all based on Equation (12) in each, solvent in solution is in equilibrium with pure solvent in another phase and has the same chemical potential in both phases. This can be solvent vapor in equilibrium with solvent in solution (as in vapor pressure lowering and boiling point elevation) or solvent in solution in equilibrium with pure, solid solvent (as in freezing point depression). Equation (12) also applies to osmotic equilibrium as shown in Figure 3.2. 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

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). 

The corresponding derivation for ideal solutions is a little simpler. The chemical potential for the isomer group and for an individual isomer at chemical equilibrium are given by... 

Martin has considered the chemical and physical factors affecting partition coefficients.20 Restricting his discussion to ideal solutions, he considered a solute, A, distributed between two phases in equilibrium with each other. The partition coefficient, a, of the solute A is related to the free energy required to transport one mole of A from one phase to... 

In order to find the vapor pressure of a component of an ideal solution, we equate its chemical potential in the solution with that in the vapor in equilibrium with the solution. Assuming that the vapor is ideal, this gives... 

Liquid-phase extraction is a procedure by which some fraction of a solute is taken out of solution by shaking the solution with a different solvent (in which the solute usually has greater solubility). The analysis of this process assumes that the shaking is sufficient so that equilibrium is established for the solute, i, between the two solutions. At equilibrium, the chemical potentials of the solute in the two solutions are equal. Assuming ideally dilute solutions, we can write... 

Historically, the state of reaction at chemical equilibrium was evaluated for fairly simple reactions, with only a few species, from the "Law of Mass Action. 1 In recent years, high-temperature reactions, including many possible species (as many as 20 or more), have become of interest and newer techniques suitable for numerical solution on high-speed digital computers have been developed.2 Initially, we will discuss chemical equilibrium from the vantage point of the "Law of Mass Action." It states that the rate at which a chemical reaction proceeds is proportional to the "active" masses of the reacting substances. The active mass for a mixture of ideal gases is the number density of each react-... 

Chemical Potentials for Ideal Dilute Solutions at Equilibrium... 

L = %a=%a- (75) The formulas refer (71) to thermal conductance (72) to electric current (73) to diffusion in the two-component mixture (74) to dissolution in the ideal solution (75) to the single-stage chemical reaction. In the formulas k is a coefficient of thermal conductance r is the electric resistance per unit of conductor length D1 is a coefficient of diffusion of substance soluble v and n are the specific volume and the number of moles, respectively Rf eq and Rr eq are the rates of the forward and reverse reactions in the equilibrium state. 

When combined with the ideal-gas and ideal-solution models of phase behavi the criterion of vapor/liquid equilibrium produces a simple and useful equati known as Raoult s law. Consider a liquid phase and a vapor phase, both compris of N chemical species, coexisting in equilibrium at temperature T and pressil P, a condition of vapor/liquid equilibrium for which Eq. (10.3) becomes... 

The properties of mixtures of ideal gases and of ideal solutions depend solely on the properties of the pure constituent species, and are calculated from them by simple equations, as illustrated in Chap. 10. Although these models approximate the behavior of certain fluid mixtures, they do not adequately represent the -behavior of most solutions of interest to chemical engineers, and Raoult s law is not in general a realistic relation for vapor/liquid equilibrium. However, these models of ideal behavior—the ideal gas, the ideal solution, and Raoult s law— provide convenient references to which the behavior of nonideal solutions may be compared. 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gihhs free energy or chemical potential of the solute in solution. This chemical potential depends on the solution conditions the temperature, pressure, and solution composition. A standard thermodynamic analysis of equilibrium concludes that the chemical potential in a local region of a system is independent of spatial position. The ideal and excess contributions to the chemical potential determine the driving forces for chemical equilibrium, solute partitioning, and conformational equilibrium. This section introduces results that will be the object of the following portions of the chapter, and gives an initial discussion of those expected results. 

Activity and Activity Coefficient. —When a pure liquid or a mixture is in equilibrium with its vapor, the chemical potential of any constituent in the liquid must be equal to that in the vapor this is a consequence of the thermodynamic requirement that for a system at equilibrium a small change at constant temperature and pressure shall not be accompanied by any change of free energy, i.e., (d( )r. p is zero. It follows, therefore, that if the vapor can be regarded as behaving ideally, the chemical potential of the constituent i of a solution can be written in the same form as equation (7), where p,- is now the partial pressure of the component in the vapor in equilibrium with the solution. If the vapor is not ideal, the partial pressure should be replaced by an ideal pressure, or fugacity, but this correction need not be considered further. According to Raoult s... 

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