Derivation of the Equilibrium Equations

The equilibrium properties of an adsorbed layer can be examined based on the chemical or electrochemical potentials of the constituents of this layer and the equilibrium equations derived in the section above. This is the simplest approach, although problems might appear in the description of the adsorbed layer properties during a surface phase transition [18]. Alternatively, the chemical potentials may be used for the determination of the grand ensemble partition function of the adsorbed layer, which in turn is used for the derivation of the equilibrium equations. This approach is mathematically more complex, but it leads to a better description of an adsorbed layer when it undergoes a phase transformation [18]. The present analysis for simplicity is restricted to the first approach. 

The above derivatives must be evaluated at the interface composition before use in computing the Jacobian elements. This additional complexity in evaluating the derivatives of the vapor-phase mass transfer rate equations arises because we have used mass fluxes and mole fractions as independent variables. If we had used mass fractions in place of mole factions the derivatives of the rate equations would be simpler, but the derivatives of the equilibrium equations would be more complicated. For simplicity, we have ignored the dependence of the mass transfer coefficients themselves on the mixture composition and on the fluxes. 

Expressions for the partial derivatives of the equilibrium equations with respect to the interface temperature and compositions are the same as those given in Example 11.5.2. The partial derivatives of the mole fraction summation equations are either unity or zero (cf. Example 11.5.2). 

The derivation of the equilibrium equations for SmC liquid crystals parallels that outlined in Section 2.4 for nematic and cholesteric liquid crystals, this approach being based on work by Ericksen [73, 74]. The energy density will be described in terms of the vectors a and c, and the equilibrium equations and static theory will be phrased in this formulation these vectors turn out to be particularly suitable for the mathematical description of statics and dynamics. We assume that the variation of the total energy at equihbrium satisfies a principle of virtual work for a given volume V of SmC liquid crystal of the form postulated by Leslie, Stewart and Nakagawa [173]... 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

B. The Henderson-Hasselbalch equation is derived from the rearrangement of the equilibrium equation for dissociation of a weak acid. 

The Young Equation. The principle of balancing forces used in the derivation of the Laplace equation can also be used to derive another important equation in surface thermodynamics, the Young equation. Consider a liquid droplet in equilibrium... 

In our derivation of the Kelvin equation, the radius is measured in the liquid. For a gas bubble in a liquid, the same equilibrium is involved, but the bubble radius is measured on the opposite side of the surface. As a consequence, a minus sign enters Equation (40) when it is applied to bubbles. Since y is on the order of millijoules while RT is on the order of joules, Equation (40) predicts that In (p/p0) is very small. It is important to realize, however, that y/R is also divided by the radius of the spherical particle and therefore becomes more important as Rs decreases. For water at 20°C, the Kelvin equation predicts values of p/p0 equal to 1.0011, 1.0184, 1.1139, and 2.94 for drops of radius 10 6, 10"7, 10 8, andl0 9m, respectively. For bubbles of the same sizes in liquid water, p/p0 equals 0.9989, 0.9893, 0.8976, and 0.339. These calculations show that the effect of surface curvature, while relatively unimportant even for particles in the micrometer range, becomes appreciable for very small particles. 

In the thermodynamic derivation of the equilibrium constant, each quantity in Equation 6-2 is expressed as the ratio of the concentration of a species to its concentration in its standard state. For solutes, the standard state is 1 M. For gases, the standard state is I bar (= 105 Pa 1 atm = 1.013 25 bar), and for solids and liquids, the standard states are the pure solid or liquid. It is understood (but rarely written) that [A] in Equation 6-2 really means [A]/( 1 M) if A is a solute. If D is a gas, [D] really means (pressure of D in bars)/( 1 bar). To emphasize that [D] means pressure of D, we usually write Pn in place of [D. The terms of Equation 6-2 are actually dimensionless therefore, all equilibrium constants are dimensionless. 

A simple derivation of the Kelvin equation is presented by Broekhoff and van Dongen [16]. Imagine a gas B in physical adsorption equilibrium above a flat, a convex, and a concave surface, respectively (see Fig. 12.8). Considering a transfer of dN moles of vapour to the adsorbed phase at constant pressure and temperature, equilibrium requires that there will be no change in the free enthalpy of the system. 

Around 1967, Broekhoff and de Boer [16], following Derjaguin [17], pointed out that the supposition introduced in the derivation of the Kelvin equation, viz. the equality of the thermodynamic potential of the adsorbed multilayer to the thermodynamic potential of the liquified gas (see Eqn. 12.28), cannot be correct. This can be seen immediately from an inspection of the common t curve (Fig. 12.5) at each t value lower than, say, 2 run, the relative equilibrium pressure is lower than 1, the equilibrium pressure of the liquefied gas. 

It is important to be aware of certain assumptions that are implicit in the derivation of the above equations. Most importantly, it is assumed that 5C and 8H are themselves independent of concentration and temperature effects. While this can often be experimentally verified for 8H, it is generally impossible to verify for 5C unless the slow-exchange limit can be attained. Another potential error is the failure to include activity coefficients in the equilibrium expressions, even though concentrations often exceed 1 M. Since iterative nonlinear curve fitting often involves locating a relatively shallow minimum, effects such as these can lead to significant error in derived K. 

Figure 1.4. Displacement of a triple line around its equilibrium position that allows derivation of the Young equation. Only a small region close to the triple line is taken into account to neglect the...

The original derivation of the BET equation was an extension and generalization of Langmuir s treatment of monolayer adsorption. This derivation is based on kinetic considerations—in particular on the fact that at equilibrium the rate of condensation of... 

The expression that that quantitatively describes the active fraction as a function of the ligand concentrationcan quite easily be derived using the equilibrium equations, together with the two equations above ... 

At equilibrium the rates of adsorption and desorption are Identical. In the kinetic derivation of the Langmuir equation the assumptions are made that the rate of adsorption is proportional to the bulk concentration of the adsorptive and to the empty fraction of the surface. 

Figure 5.10. Derivation of the Young equation for the equilibrium contact angle.

Step 3 Evaluation of the Jacobian matrix [J]. The elements of [J] are obtained quite straightforwardly using the expressions derived above. The only derivatives that need further discussion are those of the equilibrium equations, and F5. 

Derivation of the thermodynamic equations for an electrolyte system with ion pairing follows the same procedure given for a weak electrolyte. However, in the following the ion pairing equilibrium is defined in terms of an association process. For a 1-1 electrolyte, ion pairing is described as... 

Although the derivation of reaction rates using the steady state assumption is more exact, often the rapid equilibrium assumption is used because it allows a simple derivation of the rate equation from the relevant enzyme-substrate complexes (see below) and allows fitting of the kinetic data. The following explanations are based on the rapid equilibrium assumption, and therefore all following constants K are used as dissociation constants with the component dissociating from the enzyme as the subscript, e.g. Ka, Kb, and the component remaining at the enzyme as second subscript (e.g. Kf, see below). 

These two approaches for the determination of the excess chemical potential of the substance of the dispersed phase, Ap,. and Ap,., are used in the analysis of different aspects related to the equilibrium state of disperse systems. The first approach was utilized in Chapter 1,3 in the derivation of the Kelvin equation, when we examined the equilibrium between the dispersed particle and the continuous phase. The second approach accounts for the involvement of particles in thermal motion and therefore envisions both generation and disappearance of a particle as a whole, and thus allows one to describe the equilibrium between particles of different sizes. The equilibrium particle size distribution corresponds to a condition of constant chemical potential for particles of different sizes (including those of molecular dimensions), i.e. Ap/ = const. The expression for the equilibrium number of particles of a given radius2, r, can be obtained from eq. (IV. 12) as... 

In the derivation of the rate equation, it is assumed that the surface reaction is irreversible and rate determining, whereas the adsorption steps of hydrogen and aldol are rapid enough for the quasi-equilibrium hypothesis to be applied. The desorption step of triol is assumed to be irreversible and very rapid (cr —>0). 

Derivation of the Kohler equation is based on a combination of two expressions the Kelvin equation, which governs the increase in water vapor pressure over a curved surface and modified Raoult s law, which describes the water equilibrium over a flat solution ... 

It is known that the surface energy depends not only on the composition of the surface layer, but also on that of the bulk phases [130]. To formulate the Gibbs law for the non-equilibrium chemical potential, additional so-called cross-chemical potentials (the partial derivatives of the surface free energy with respect to the component concentrations in the bulk phases) have been introduced. Rusanov and Prokhorov [131] derived the Gibbs equation and the expression for the free energy of the surface layer in terms of the ordinary chemical potentials by dividing the transition layer adjacent to the surface into n thin layers. For each layer an equilibrium state was assumed. The expression for surface energy was derived by the summation of the equilibrium equations over all these layers. Further, the expression for the additional contribution to the surface tension due to the non-equilibrium diffusion layer was derived in [48, 132]... 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

Since protolytic reactions are usually very fast (especially in the presence of buffers, where proton-transfer reactions occur readily), all the protolytic steps can be assumed to be in equilibrium throughout the course of the reaction. The derivation of the rate equation is exactly as before except that the enzyme-conservation equation is given by... 

Analysis of the kinetics of CRP reactions is more complex than for conventional radical polymerizations consequently, a detailed derivation of the basic equations will not be given here. The fundamental activation-deactivation pseudoequilibria that control the living characteristics of the various CRPs have been outlined in Equation 3.31 to Equation 3.33, and if the steady state is to be achieved rapidly, then the rates of activation and deactivation must be considerably larger than the rates of the initiation and termination reactions. In successful CRP reactions, the time taken to reach the equilibrium steady state is estimated to be in the range 1 to 100 ms. 

The numerator of the equilibrium-constant expression is the product of the concentrations of all substances on the product side of the equilibrium equation, each raised to a power equal to its coefficient in the balanced equation. The denominator is similarly derived from the reactant side of the equilibrium equation. Thus, for the Haber process, N2(g) + 3 H2(g) v 2 NH3(g), the equilibrium-constant expression is... 

D24.3 The Eyring equation (eqn 24.53) results from activated complex theory, which is an attempt to account for the rate constants of bimolecular reactions of the form A + B iC -vPin terms of the formation of an activated complex. In the formulation of the theory, it is assumed that the activated complex and the reactants are in equilibrium, and the concentration of activated complex is calculated in terms of an equilibrium constant, which in turn is calculated from the partition functions of the reactants and a postulated form of the activated complex. It is further supposed that one normal mode of the activated complex, the one corresponding to displaconent along the reaction coordinate, has a very low force constant and displacement along this normal mode leads to products provided that the complex enters a certain configuration of its atoms, which is known as the transition stale. The derivation of the equilibrium constant from the partition functions leads to eqn 24.51 and in turn to eqn 24.53, the Eyring equation. See Section 24.4 for a more complete discussion of a complicated subject. 

The radiative equilibrium of this system needs to be set up. Only diffuse radiation that fractions are considered in accordance with the optical parameters for diffuse radiation that are to be used. The problem is relatively complex and confusing. The single steps for the derivation of the transport equations cannot be discussed explicitly. They are only explained in the corresponding diagrams (Figs 5.21 and 5.22). 

The liquid phase model proposed below considers the mass transfer inside the droplet and the changes in liquid phase properties due to the temperature and composition changes. In derivation of the following equations, it has been assumed that liquid circulation is absent, the droplet surface is at local thermodynamic equilibrium state, momentum, energy and mass transfer are spherically symmetric within the droplet, and the two liquids (fuel and water) are immiscible. With these assumptions, the conservation equations for the total mass, mass of water, and the energy equation are written as follows [14] ... 

If Vmax = afccatEo. and the derivation of the velocity equation is based on a rapid equilibrium assumption, we shall obtain... 

It is possible, now, to transform Eq. (14.35) into a rate equation showing the pH independent kinetic parameters and their dependence on pH. The derivation of the velocity equation is simpMed by treating the steps with protons as quasi-equihbria, because rapid equilibrium assumptions may be justified by the... 

Although the extended Langmuir equations can be derived from the kinetic approach presented in the last section, it can be derived from thermodynamics, which should be the right tool to study the equilibria (Helfferich, 1992). The two approaches, however, yield identical form of the equilibrium equation. What to follow is the analysis due to Helfferich. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

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