Stepwise equilibria

The thermodynamic stability of a species is a measure of the extent to which this species will be formed from other species under certain conditions, provided that the system is allowed to reach equilibrium. Consider a metal ion M in solution together with a monodentate ligand L, then the system may be described by the following stepwise equilibria, in which, for convenience, coordinated water molecules are not shown ... 

Equation 33 is the familiar statistical relation between the equilibrium constants in a series of stepwise equilibria as derived by N. Bjerrum (II) for polyprotic acids and applied by J. Bjerrum (12) to complex ion equilibria. Substituting Equation 33 for K, into Equations 23 and 24 for 0f° and 0o° gives... 

Metal ions such as Cu2+ are Lewis acids that combine with Lewis bases such as H20 to form complex ions, e.g., CuCOH ". The number of ligands, the coordination number, varies from one metal ion to another the most common coordination numbers are 4 and 6, but 2, 3, 5, 7, and 8 are also known. In the presence of another Lewis base such as NH3, the H20 ligands can be replaced in stepwise equilibria ... 

Precipitation equilibria are described in part in Chapter 7. An important type of precipitation equilibrium considered here involves a coordination reaction of a metal ion with univalent anions to form an uncharged chelate that is insoluble. In general, it is necessary to consider stepwise equilibria ... 

Stepwise equilibria must be considered if successive values differ by a factor of > 10, which is normally the case. Values of A a are given in Table 10.9 in the text. 

When we add two adjacent stepwise equilibria, we multiply the two equilibrium constants to obtain the equilibrium constant for the resulting overall reaction. Thus, for the first two dissociation equilibria for H3PO4, we write... 

Scheme 23 Stepwise equilibria for the addition of a monovalent ligand A to an n-valent receptor B.

The van Deemter approach deals with the effects of rates of nonequilibrium processes (e.g. diffusion) on the widths (ct ) of the analyte bands as they move throngh the column, and thus on the effective value of H and thns of N. Obviously, the faster the mobile phase moves through the column, the greater the importance of these dispersive rate processes relative to the idealized stepwise equilibria treated by the Plate Theory, since equilibration needs time. Thus van Deemter s approach discusses variation of H with u, the linear velocity of the mobile phase (not the volume flow rate (U), although the two are simply related via the effective cross-sectional area A of the column, which in turn is not simply the value for the empty tube but must be calculated as the cross-sectional area of the empty column corrected for the fraction that is occupied by the stationary phase particles). This approach identifies the various nonequilibrium processes that contribute to the width of the peak in the Gaussian approximation and shows that these different processes make contributions to Ox (and thus H) that are essentially independent of one another and thus can be combined via simple propagation of error (Section 8.2.2) ... 

C1 CALORIMETRIC TITRATIONS FOR THE STUDY OF STEPWISE EQUILIBRIA IN SOLUTIONS, I. Grenthe and I. Leden... 

Stepwise equilibria between Lewis acids, here metal ions, and Lewis bases, here ligands, are mathematically quite like the preceding protonic equilibria. Water is the most concentrated ligand in most solutions. The competition of other ligands for the metal ions is important in the understanding of solution species distributions which play a role in many areas of chemistry, biochemistry, and geochemistry. 

A number of important acids, such as carbonic acid, sulfuric acid and phosphoric(v) acid, release more than one proton per molecule and are called polyprotic acids. Their stepwise equilibria are discussed as extension on the website that accompanies this book. 

When the chemical interactions between complexes and ligands proceed with a sufficiently high rate, it is common to treat their concentrations as close to equilibrium ones. Assumptions of such kind have been of frequent use and can be found in different problems of electrochemical kinetics. Then, the relations between concentrations might be determined by cumulative (fij) or stepwise (Kj) stability constants of complexes and protonated ligands. An expression for Pj is given in Chapter 1 stepwise equilibria, such as (3.16) and (3.17), can be characterized by... 

Stepwise equilibria must be considered if successive values dilTer by a factor of greater... 

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