Equilibrium Constants for a Series of Reactions

In complicated chemical systems, a product of one chemical reaction can he a reactant in a subsequent reaction. Because of the nature of the equilihrium expression, if we know the equilibrium constant for the two separate reactions, we can determine an overall value for the combination of the two. Consider, for example, the stepwise reaction of phosphate ions with hydrogen ions  

The third equation is the sum of the first two equations. How are their equilibrium constants related To obtain the chemical equation associated with K, we added the chemical equations for Ki and K2. To get the equilibrium constant for equation 3, however, we must multiply the expressions. 

This relationship holds true any time we add chemical equations to obtain a new equation multiply the equilibrium constants to find the equilihrium constant for the reaction of interest. 

Write the equilibrium expressions for the two reactions. Multiply them to obtain the equilibrium expression for the sum of the two reactions. Check your answer by adding the two reactions and comparing the sum to the equilibrium expression you determined. 

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The value of K has no dimensions because the concentrations are actually approximations for a dimensionless quantity called an activity. The law of mass action is good for all chemical equations, including non-elementary equations. In other words, for equilibrium constants, use the chemical equation coefficients as the exponents of the concentrations regardless of molecularity. Notice that the equilibrium constant is a capital K and the rate constant is represented by lowercase k. Also notice that the equilibrium constant for the reverse reaction is the reciprocal of the equilibrium constant of the forward reaction. This is true regardless of whether or not the reaction is elementary. Following this same line of reasoning will demonstrate that the equilibrium constant for a series of reactions is equal to the product of the equilibrium constants for each of its elementary steps. Since the rate constant depends upon temperature, the equilibrium constant must also depend upon temperature. 

In principle, if the X values for a set of like reactions are similar to one another, and W r) is small or constant, a plot of AG versus AG° will be linear and have a slope of 0.5. As noted above, it is rarely possible to measure AG values directly. An alternative option for plotting data using a linear relationships is to use Equation 1.15 and the definition of the equilibrium constant, K = A exp( AG° // 7), in which A is a constant. If the equilibrium constants for a series of reactions can be measured or calculated, one can plot In A versus In AT (Equation 1.15). A linear result with a slope of 0.5 is indicative of a common outer-sphere electron transfer mechanism. 

The Equilibrium (Mass Action) Expression Gas Phase Equilibria Kp vs. Kp Homogeneous and Heterogeneous Equilibria Numerical Importance of the Equilibrium Expression Mathematical Manipulation of Equilibrium Constants Reversing the Chemical Equation Adjusting the Stoichiometry of the Chemical Reaction Equilibrium Constants for a Series of Reactions Units and the Equilibrium Constant... 

A plot of the logarithm of a rate constant (or an equilibrium constant) for one series of reactions versus the logarithm of the rate constant (or the equilibrium constant) for a related series of reactions. (Recall that at constant temperature and pressure the logarithm of an equilibrium constant is proportional to AG°, and the logarithm of a rate constant is proportional to AG ). An example of a linear free energy relationship is provided by the Hammett crp-equation. With equilibrium constants, this relationship is given by the expression ... 

Now if we are comparing the equilibrium constants for a series of similar reactions, in which the bonds being formed or broken are the same in each case, it is easily seen that the terms in the first two pairs of parentheses in Eq. (15) will be the same for each reaction. We can then write AE in the form ... 

Mechanistic Studies.- A number of substitution reactions of alcohols, phenols, or amines with 1,3,2-dioxaphospholans, e.g. (73), oxazaphospholans, and diazaphos-pholans have been followed by n.m.r. and shown to involve H-phosphoranes, e.g. (74). The reactions are run in toluene or without solvent, and without addition of an acidic catalyst in some systems, with 4-chlorophenol as the nucleophile, an equilibrium was established between an H-phosphorane (75) and a phosphonium salt (76), but the authors still favour (75) as the true intermediate. The equilibrium constants for a series of exchange reactions between (thio)phosphites (77) and (thio)phosphoro-dichloridites (78), and some analogous bromidites and a fluoridite, have been measured.The constants increase with increased electron donor ability of R. 

Equilibrium constants for a series of redox reactions (105) have been measured by Svanholm and Parker (1973), and arc helpful in estimating how well one cation radical will serve in making another in the series. 

Guthrie JP (1978) Equilibrium constants for a series of simple aldol condensations, and linear free energy relations with other carbonyl addition reactions. Can J Chem 56 962-973... 

An important experimental rule for protolytic reactions was estabhshed by Johannes Nicolaus Brpnsted in 1918 (it was later extended to other reactions). He showed that for a series of reactions of the same type, the rate constants and the equilibrium constants are related simply as... 

The role of biocatalysis in two-phase systems has many parallels with the subject we have covered under extractive reactions. It appears that a two-phase system was originally considered for transformations of water insoluble substances like steroids. Now, a series of treatises are available which teach us that the maximum value of the apparent equilibrium constant for a second-order reaction in a two-phase system can exceed the equilibrium... 

Examining the relationship between the hydrolysis rate constants (A ,) and the equilibrium constant (Aj) for a series of reactions of the type (4.28) and (4.29) involving charged ligands X" has been very helpful in delineating the type of / mechanism. 

Another approach used in the empirical characterization of liquid polarity is the study of the outcome of a chemical reaction. Earle et al. [216] report a preliminary study of the keto-enol tautomerization of pentane-2,4-dione, and create an empirical correlation between the degree of tautomerization and the dielectric constant of molecular liquids. They then predict dielectric constants for a series of ILs based on the observed keto-enol equilibrium the values range from 40 to 50, slightly higher than those of short-chain alcohols. A more detailed study by Angelini et al. [217] considers the tautomerization of a nitroketone complex in a series of five imidazolium-based ILs. The results, parameterized as a linear free energy analysis of the behavior of the equilibrium constant, indicates an overall polarity comparable to that of acetonitrile, consistent with the partitioning and spectroscopic studies referenced above. 

This relationship shows that determining the equilibrium constant at a series of temperatures 7, and plotting In A) vs. 1/7) should yield a straight line having a slope -AH/R, thus enabling for the reaction AH to be determined. Once again it is observed that there is a linear relationship between the natural logarithm of some property and 1 IT. 

Although a statistical factor contributes to the equilibrium constants for many types of reactions, it is in reactions of the type being considered here—the replacement of one neutral ligand by another neutral ligand— that this factor may be the principal factor in causing variation of Kn with n. Other series of reactions of this type are the formation of ammonia complexes in aqueous solution. The variation of Kn with n observed for the chromium (III)/water-methyl alcohol system is slightly smaller than observed for ammonia complexes of cobalt(II) (18) and nickel(II) (19) ... 

The equilibrium constant for a proton transfer reaction from a series of proton donors (SH) to base (B) (Equation 18) has a peq value of unity for the Bronsted dependence on the pX of SH (pX, ). 

Table 12.1 gives typical values of the diffusion constant for a series of thermosetting matrices where the effect of polarity of the resin on the equilibrium on moisture concentration is illustrated. Epoxy resins are the network product of the reaction of a multifunctional epoxide monomer with a hardener. One of the important aspects of composite materials which needs to be recognised is that the material is synthesised at the same time as the component is manufactured. The precise chemistry of the final network can be uncertain but is a function of the chemical stmctures of the epoxide and hardener and/or catalyst, which determine the mechanism of cure. In this way, the cured resins can have differing polarities. 

The equilibrium law is very much an experimentally determined one. The basic structure of the expression for a particular reaction is confirmed by the value of remaining constant for a series of experiments at a given temperature. Table 7.2a shows the values for obtained in such a series of experiments when hydrogen and iodine are reacted in a sealed container at 700 K. These results show that a constant value of 54 is obtained when the equilibrium concentrations are fed into the expression ... 

The relationship between thermodynamics and kinetics in chemical reactions is usually expressed by the Br0nsted equation (Eq. 3.70 in Section 3.4) k =gK, where k is the rate constant, fCis the equilibrium constant of the elementary stage, and and a (Polanyi parameter) are constant values for a series of reactions. These constants are determined by parameters characterizing the elementary mechanism (composition and structure ofthe activated complexes, etc.), thus allowing for the existence of an optimum catalyst, on which the rate of catalytic reaction per unit of surface has a maximum value. Equations of the type (Eq. 3.70) were used for the explanation of volcano-curves, when catalytic activity as a function of thermodynamic characteristics follows a curve with a maximum. An example for a volcano curve in methanation of CO is given in Fig. 7.6. 

The most common manifestation of extrathermodynamic relationships is a linear correlation between the logarithms of rate or equilibrium constants for one reaction series and the logarithms of rate or equilibrium constants of a second reaction series, both sets being subjected to the same variation, usually of structure. For illustration, suppose the logarithm of the rate constants for a reaction series B is linearly correlated with the logarithm of the equilibrium constants for a reaction series A, with substituent changes being made in both series. The empirical correlation is... 

For the purpose of systematizing kinetic and equilibrium data, for literally hundreds of reactions, it is desirable to have a single reference series for all. Hammett adopted as the standard the acid ionization constants for substituted benzoic acids in aqueous solution at 25 °C. This choice was fortunate because the compounds are stable and for the most part readily available. Also, their pA"a s can easily and precisely be measured for nearly every substituent. Thus, one constructs a plot according to either of the following equations, in which Eq. (10-4) constitutes a further example ... 

The Hammett equation is the best-known and most widely studied of the various linear free energy relations for correlating reaction rate and equilibrium constant data. It was first proposed to correlate the rate constants and equilibrium constants for the side chain reactions of para and meta substituted benzene derivatives. Hammett (37-39) noted that for a large number of reactions of these compounds plots of log k (or log K) for one reaction versus log k (or log K) for a second reaction of the corresponding member of a series of such derivatives was reasonably linear. Figure 7.5 is a plot of this type involving the ionization constants for phenylacetic acid derivatives and for benzoic acid derivatives. The point labeled p-Cl has for its ordinate log Ka for p-chlorophenylacetic acid and for its abscissa log Ka for p-chloroben-zoic acid. The points approximate a straight line, which can be expressed as... 

Equilibrium constants Kj have been determined for a series of displacement reactions (13.3, in DCE) in which isocyanides compete for binding to the ClAu fragment [37] ... 

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