Combining equilibrium constants

The appropriate equilibrium constant, combined with eq. (XI) depends on temperature and ionic strength ... 

From equation (13) you can see that whereas free energy changes combine additively, equilibrium constants combine multiplicatively. If the equilibrium constants for... 

The measurement of cell emfs gives you yet another way to obtain equilibrium constants. Combining the previous equation, AG° = —nFEceih with the equation AG° = -i rin K from Section 19.6, you get... 

A connnon approach has been to measure the equilibrium constant, K, for these reactions as a fiinction of temperature with the use of a variable temperature high pressure ion source (see section (Bl.7.2)1. The ion concentrations are approximated by their abundance in the mass spectrum, while the neutral concentrations are known from the sample mlet pressure. A van t Hoff plot of In K versus /T should yield a straight Ime with slope equal to the reaction enthalpy (figure B1.7.11). Combining the PA with a value for basicityG at one temperature yields a value for A.S for the half-reaction involving addition of a proton to a species. While quadnipoles have been tire instruments of choice for many of these studies, other mass spectrometers can act as suitable detectors [19, 20]. 

Decades of work have led to a profusion of LEERs for a variety of reactions, for both equilibrium constants and reaction rates. LEERs were also established for other observations such as spectral data. Furthermore, various different scales of substituent constants have been proposed to model these different chemical systems. Attempts were then made to come up with a few fundamental substituent constants, such as those for the inductive, resonance, steric, or field effects. These fundamental constants have then to be combined linearly to different extents to model the various real-world systems. However, for each chemical system investigated, it had to be established which effects are operative and with which weighting factors the frmdamental constants would have to be combined. Much of this work has been summarized in two books and has also been outlined in a more recent review [9-11]. 

Equilibrium constants for complexation reactions involving solids are defined by combining appropriate Ksp and K expressions. Eor example, the solubility of AgCl increases in the presence of excess chloride as the result of the following complexation reaction... 

The equilibrium position for any reaction is defined by a fixed equilibrium constant, not by a fixed combination of concentrations for the reactants and products. This is easily appreciated by examining the equilibrium constant expression for the dissociation of acetic acid. 

Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A , must equal the weak acid s initial concentration, Cha- ... 

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an-... 

Ionic polymers may exist as undissociated, unsolvated ion pairs undissociated ion pairs solvated to some extent solvated ions dissociated to some extent or some combination of these. The propagation rate constant kp and the dissociation equilibrium constant K of the lithium salt of anionic... 

Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 ppj- Equilibrium constants can he calculated hy combining AhJ values from Table 2-221, h-r— ho is from Table 2-222, and 5 values from the above, using the formula In kp = —AG/(RT), where AG = AhJ + hj — /i29s) T . 

This can be integrated, although the result is not simple. King has shown that Scheme II is more easily described in terms of a quantity A defined as the displacement of a concentration from its equilibrium value. Let Ca = Ca + A then we also have cb = c -f A and Cz = c — A. The equilibrium constant is K = kdk-i = c /Ca c. Algebraic combination of these relationships with Eq. (3-5) gives Eq. (3-6). 

The sensitivity of the equilibrium constant to temperature, therefore, depends upon the enthalpy change AH . This is usually not a serious limitation, because most reaction enthalpies are sufficiently large and because we commonly require that the perturbation be a small one so that the linearization condition is valid. If AH is so small that the T-jump is ineffective, it may be possible to make use of an auxiliary reaction in the following way Suppose the reaction under study is an acid-base reaction with a small AH . We can add a buffer system having a large AH and apply the T-jump to the combined system. The T-jump will alter the Ka of the buffer reaction, resulting in a pH jump. The pH jump then acts as the forcing function on the reaction of interest. 

Since the domain explored will always be a very small part of the possible cases of tautomerism, it is essential to have general rules for families of compounds, substituents, and solvents. This chemical approach is maintained in this chapter, although the importance of the calculations is recognized. The following discussion begins with calculation of tautomeric equilibrium constants, followed by the combined use of theoretical calculations and experimental results (an increasingly expanding field) and ends with the calculations of the mechanisms of proton transfer between tautomers. 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

A wide range of nitroxidcs and derived alkoxyamincs has now been explored for application in NMP. Experimental work and theoretical studies have been carried out to establish structure-property correlations and provide further understanding of the kinetics and mechanism. Important parameters are the value of the activation-deactivation equilibrium constant K and the values of kaa and (Scheme 9.17), the combination disproportionation ratio for the reaction of the nilroxide with Ihe propagating radical (Section 9.3.6.3) and the intrinsic stability of the nitroxide and the alkoxyamine under the polymerization conditions (Section 9.3.6.4). The values of K, k3Cl and ktieact are influenced by several factors.11-1 "7-"9 ... 

Salts of diazonium ions with certain arenesulfonate ions also have a relatively high stability in the solid state. They are also used for inhibiting the decomposition of diazonium ions in solution. The most recent experimental data (Roller and Zollinger, 1970 Kampar et al., 1977) point to the formation of molecular complexes of the diazonium ions with the arenesulfonates rather than to diazosulfonates (ArN2 —0S02Ar ) as previously thought. For a diazonium ion in acetic acid/water (4 1) solutions of naphthalene derivatives, the complex equilibrium constants are found to increase in the order naphthalene < 1-methylnaphthalene < naphthalene-1-sulfonic acid < 1-naphthylmethanesulfonic acid. The sequence reflects the combined effects of the electron donor properties of these compounds and the Coulomb attraction between the diazonium cation and the sulfonate anions (where present). Arenediazonium salt solutions are also stabilized by crown ethers (see Sec. 11.2). 

The equilibrium equation for the first step is shown in Scheme 5-2. Introducing the equilibrium constant Kw of water (Kw = [H+][0H ]/[H20] leads to the equation shown in Scheme 5-3. ATW can be combined with the constant K (defined by Kx = K[KW) to give the equation of Scheme 5-4. In the same way, the second step can be expressed as in Scheme 5-5. 

The rate constant combinations and panel designations are those listed in Table 4-2. The exact solution is shown as A3 in A-D, and as exact in E-F, where the full solution to Eq. (4-34) is required. Each panel also displays various approximations improved steady-state, steady-state, and prior-equilibrium. Some of the solutions coincide. 

The main polymerization method is by hydrolytic polymerization or a combination of ring opening as in (3.11) and hydrolytic polymerization as in (3.12).5,7 9 11 28 The reaction of a carboxylic group with an amino group can be noncatalyzed and acid catalyzed. This is illustrated in the reaction scheme shown in Fig. 3.13. The kinetics of the hydrolytic polyamidation-type reaction has die form shown in (3.13). In aqueous solutions, die polycondensation can be described by second-order kinetics.29 Equation (3.13) can also be expressed as (3.14) in which B is die temperature-independent equilibrium constant and AHa the endialpy change of die reaction5 6 812 28 29 ... 

H,(g) HzO(g) + CO(g) at that temperature (h) Show that the manner in which equilibrium constants are calculated is consistent with the manner in which the ACr° values are calculated when combining two or more equations by determining AGr° for reactions (1) and (2) and using those values to calculate AGr° and K3 for reaction (3). 

We saw in Section 9.3 that the standard reaction Gibbs free energy, AGr°, is related to the equilibrium constant of the reaction by AGr° = —RT In K. In this chapter, we have seen that the standard reaction Gibbs free energy is related to the standard emf of a galvanic cell by AGr° = —nFE°, with n a pure number. When we combine the two equations, we get... 

The equilibrium constant of a reaction can be calculated from standard potentials by combining the equations for the balf-reactions to give the cell reaction of interest and determining the standard potential of the corresponding cell. 

We will see functions like the one occurring under the logarithm operator quite often. For efficiency, this is generally written as In(Products)/(Reactants), where (Products) and (Reactants) denote the partial pressures of the species relative to the standard state pressure raised to a power that is equal to the stoichiometric coefficients. Kp is the equilibrium constant in terms of pressures. Since all pressures are in the same units, Rp is dimensionless. Note that in some literature there may be a combination of some power of P with Kp to obtain an equilibrium constant with pressure units. In this case. 

One may combine the appropriate expressions for G for equilibria involving reactants in different phases to obtain a general expression, which relates the equilibrium constant to the... 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

Combining these two equilibria leads to cancellation of HF and F, so the sum of the two is the water equilibrium. How are the equilibrium constants for these three equilibria related ... 

Having estimated the sticking coefficient of nitrogen on the Fe(lll) surface above, we now consider the desorption of nitrogen, for which the kinetic parameters are readily derived from a TPD experiment. Combining adsorption and desorption enables us to calculate the equilibrium constant of dissociative nitrogen adsorption from... 

For gas-phase reactions, when combining Eqns. (5.4-8) and (5.4-9), the equilibrium constant is expressed as ... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

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