Chemical equilibria subscripts

In a balanced chemical equation (commonly called a chemical equation ), the same number of atoms of each element appears on both sides of the equation, chemical equilibrium A dynamic equilibrium between reactants and products in a chemical reaction, chemical formula A collection of chemical symbols and subscripts that shows the composition of a substance. See also condensed structural formula empirical formula,- molecular formula structural formula. 

The boundary conditions for these differential equations are to be applied at x = — oo (where properties will be identified by the subscript 0) and at x = -h oo (where properties will be identified by the subscript oo). At X = — 00, all properties are uniform djdx 0) (see, however, Section 2.1.2) and the values of all variables except Vq are known (compare Section 2.1.3). At x -hoo, properties again become uniform djdx = 0) and chemical equilibrium is reached. In Chapter 2 (Section 2.1) it was shown that these conditions imply that relationships exist between the properties at X = -h 00 and those at x = — oo. 

Note the absence of the subscript a on in the equation above, indicating that this is not a true equilibrium constant. Instead, K a is referred to as an apparent equilibrium constant and is a measured quantity that depends on the solution conditions (ionic strength, pH, etc.), unlike the thermodynamic equilibrium constant, which depends only on the standafSli tate and temperature. This apparent equilibrium constant is one of the concentration chemical equilibrium ratios defined in Table 13.1-3. As... 

The motion of particles in a fluid is best approached tlirough tire Boltzmaim transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In otlier words, our starting point will be the statistical themiodynamic treatment above, and we will consider the effect of botli the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,(3,.. . and let f (r, c,f)AV A be the number of molecules of type a located m... 

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

The effect of the medium (solvent) on the dissolved substance can best be expressed thermodynamically. Consider a solution of a given substance (subscript i) in solvent s and in another solvent r taken as a reference. Water (w) is usually used as a reference solvent. The two solutions are brought to equilibrium (saturated solutions are in equilibrium when each is in equilibrium with the same solid phase—the crystals of the dissolved substance solutions in completely immiscible solvents are simply brought into contact and distribution equilibrium is established). The thermodynamic equilibrium condition is expressed in terms of equality of the chemical potentials of the dissolved substance in both solutions, jU,(w) = jU/(j), whence... 

Provided that equilibrium is maintained between the aqueous and micellar pseudophases (designated by subscripts W and M) the overall reaction rate will be the sum of rates in water and the micelles and will therefore depend upon the distribution of reactants between each pseudophase and the appropriate rate constants in the two pseudophases. Early studies of reactivity in aqueous micelles showed the importance of substrate hydropho-bicity in determining the extent of substrate binding to micelles for example, reactions of a very hydrophilic substrate could be essentially unaffected by added surfactant, whereas large effects were observed with chemically similar, but hydrophobic substrates (Menger and Portnoy, 1967 Cordes and Gitler, 1973 Fendler and Fendler, 1975). 

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. 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

In this paper we have used the quantity (1 — vp0) in writing equations for sedimentation equilibrium experiments. Some workers prefer to use the density increment, 1000(dp/dc)Tfn, instead when dealing with solutions containing ionizing macromolecules. This procedure was first advocated by Vrij (44), and its advantages are discussed by Casassa and Eisenberg (39). Nichol and Ogston (13) have used the density increment in their analysis of mixed associations. The subscript p. means that all of the diffusible solutes are at constant chemical potential in the buffer... 

When a liquid solution and a solid solution are in equilibrium, both components exist in both phases. We again develop the pertinent equations for only one component, because the equations for the second component are identical except for the change of subscripts. The condition of equilibrium is the equality of the chemical potential of the component in the two phases, so... 

The equation for the second component is the same with change of subscripts. The problem is to determine values of xt and zl at specified temperatures by the use of the two equations. We define the quantity A [T, P] as equal to both sides of Equation (10.198). This quantity is the change of the chemical potential on mixing for the liquid phase that has been defined previously however, for the solid phase the standard state is now the pure liquid component. A similar definition is made for Ap%[T, P], The conditions of equilibrium then become... 

Many vapors of substances, such as those of certain metals, are mixtures of monomers and dimers and, possibly, of even higher polymers. This must be taken into account when the chemical potential of the substance in a condensed phase, either pure or a solution, is determined by a study of the vapor-liquid equilibrium. We choose to consider only the components in the condensed phase, and use the molecular mass of the monomer to determine the mole numbers. We designate the component whose chemical potential is to be determined by the subscript 1. The condition of equilibrium is then... 

The triplets p, q, r are known as the stoichiometric coefficients of the chemical components in the formation equation of the complex, and are often quoted as subscripts next to the symbol used for the formation constant, which in this case is (3, so that the equilibrium formation constant would be defined as... 

An equilibrium constant must always be accompanied by a chemical equation. This equation is often used without the subscript eq that reminds us that the concentrations are equilibrium values. Strictly speaking, this equation should be written as K = TT(ci/c0), but the standard concentration c° = 1 M will be omitted, as mentioned before equation 3.1-8. Thus the equilibrium constant will be treated as a dimensionless quantity, as, of course, it must be if we are going to take its logarithm. 

Equations (33), (35), (37), and (40) comprise N 5 equations in the N 5 unknowns, Y-, p, u, and Vq. These equations contain the functions f, P, q, V, and w., which must be related to the other dependent and independent variables if the system is to form a closed set of equations. The f- are specified by the nature of the external force field, if any, the w. are determined by the chemical kinetics, and q, p and Vj are the transport properties investigated in Appendix E. It will be seen in Appendix E that the transport properties can rigorously be related to Yj, p, u, and Vq only for near-equilibrium flows. In quoting equations (33), (35), (37), and (40) in Chapter 1, the subscript 0 is omitted from Vq, the bar is omitted from V, and the subscript x is omitted from Vj, since the molecular velocity never appears as an independent variable in the applications. 

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