Equilibrium constant calorimetric measurements

Much work has been devoted to the halide complexation of these elements in non-aqueous media. Equilibrium and calorimetric measurements for the formation of the [MX ](n-2) (M = Zn or Cd X = Cl, Br, I or SCN n = 1-4) anions in dimethyl sulfoxide (DMSO) have shown that stability constants follow the same order, but are much larger than those found for aqueous solution zinc exhibits an enhanced hardness as an acceptor in DMSO as compared to cadmium. Calorimetric measurements indicate a change from octahedral to tetrahedral coordination with increasing halide concentrations.1002-1006... 

In Section 14.3 we showed how to evaluate K from calorimetric data on the pnre reactants and products. Occasionally, these thermodynamic data may not be available for a specific reaction, or a quick estimate of the value of K may suffice. In these cases we can evaluate the equilibrium constant from measurements made directly on the reaction mixture. If we can measure the equilibrium partial pressures of all the reactants and products, we can calculate the equilibrium constant by writing the eqnilibrinm expression and substituting the experimental values (in atmospheres) into it. In many cases it is not practical to measnre directly the equilibrium partial pressure of each separate reactant and prodnct. Nonetheless, the equilibrium constant can usually be derived from other available data, although the determination is less direct. We illustrate the method in the following two examples. 

As seen in previous sections, the standard entropy AS of a chemical reaction can be detemiined from the equilibrium constant K and its temperature derivative, or equivalently from the temperature derivative of the standard emf of a reversible electrochemical cell. As in the previous case, calorimetric measurements on the separate reactants and products, plus the usual extrapolation, will... 

Carell and Olin (58) were the first to derive thermodynamic functions relating to beryllium hydrolysis. They determined the enthalpy and entropy of formation of the species Be2(OH)3+ and Be3(OH)3+. Subsequently, Mesmer and Baes determined the enthalpies for these two species from the temperature variation of the respective equilibrium constants. They also determined a value for the species Be5(OH) + (66). Ishiguro and Ohtaki measured the enthalpies of formation of Be2(OH)3+ and Be3(OH)3+ calorimetrically in solution in water and water/dioxan mixtures (99). The agreement between the values is satisfactory considering the fact that they were obtained with different chemical models and ionic media. 

Table 5 lists equilibrium data for a new hypothetical gas-phase cyclisation series, for which the required thermodynamic quantities are available from either direct calorimetric measurements or statistical mechanical calculations. Compounds whose tabulated data were obtained by means of methods involving group contributions were not considered. Calculations were carried out by using S%g8 values based on a 1 M standard state. These were obtained by subtracting 6.35 e.u. from tabulated S g-values, which are based on a 1 Atm standard state. Equilibrium constants and thermodynamic parameters for these hypothetical reactions are not meaningful as such. More significant are the EM-values, and the corresponding contributions from the enthalpy and entropy terms. 

We are at a loss to explain the discrepancy in the BF3 enthalpies of interaction with the sulfur donors. Steric effects may be operative, but this is far from the whole story for the BCI3 interaction is much larger than BF3 with these donors. Furthermore, using the tentative ( 113)3 parameters to estimate those of ( 2115)3 , we calculate an enthalpy from E and of 11.1 k.cal mole- for the BF3-P( 2H6)3 adduct compared to a measured value of 9.5 k.cal mole i. The authors report much difficulty with the sulfur donor system, but their error estimates could not possibly account for the difference between our calculated and the observed result. The behavior of ( 2115)35 compared to ( 2115)3 is clearly inconsistent with the behavior of these two donors toward ( 2H5)sAl where both enthalpies are correctly predicted with our parameters. It may be that the BF3-( 2115)25 system has an even lower equilibrium constant than reported and is completely dissociated over the temperature range studied. (This would require a very different entropy if the — AH predicted by E and were correct.) A slight impurity (reported to be less than 0.1%) or decomposition product could interact appreciably with BF3 and changing pressure contributions from this adduct with temperature could be attributed incorrectly to the sulfur donor adduct. The actual BF3-sulfur donor adduct would then be a very common example of an adduct which cannot be studied by the vapor pressure technique because it is completely dissociated at the temperatures at which one of the components has appreciable vapor pressure. We have examined the reaction of BF3 ( 2Hs) 2O with large excess of ( H2) 4S in dichloroethane solution at 25 ° and have found the equilibrium constant to be too low to be measured calorimetrically. 

In the Tables, the thermodymamic functions referring to the consecutive steps are denoted by AG° , AH and ASj, respectively. The values of AGn and AH are in kj mole i, those of AS in JK-i. The ionic strength I is given in M (mole l i). Generally, the measurements refer to perchlorate media, the exceptions are indicated in the Tables. All values refer to a temperature of 25 °C, if not otherwise stated. The values of AHn have been determined calorimetrically in nearly all of the quoted investigations. This method is inherently more accurate than determinations using the temperature coefficients of the equilibrium constants Kn (for a comprehensive collection of data illustrating... 

AH values for various monomers. The AS values fall in a narrower range of values. The methods of evaluating AH and AS have been reviewed [Dainton and Ivin, 1950, 1958], These include direct calorimetric measurements of AH for the polymerization, determination by the difference between the heats of combustion of monomer and polymer, and measurements of the equilibrium constant for the polymerization. The overall thermodynamics of the polymerization of alkenes is quite favorable. The value of AG given by... 

The measurement of an equilibrium constant requires the assumption of a non-thermodynamic measuring technique. Procedures that have been employed to measure K for reaction (15.37) include calorimetric, potentiometric, conductimetric, NMR, light absorption, and polarographic methods. The calorimetric measurements are especially useful, because they give ArH° for the reaction in addition to K. Since... 

Biochemists often refer to enthalpies obtained in this manner as van t Hoff enthalpies and attach the subscript i/H to the AH to distinguish it from an enthalpy obtained directly from calorimetric measurements. In practice, one need not actually obtain the K values to extract the enthalpy change. Rather, there is a relationship between da/d T and d nK/dT. For the equilibrium of equation (16.15) (at constant pressure so that the partial derivative can be replaced by the total derivative),... 

Figure 18.10 Equilibrium constants as a function of temperature for several association reactions. The constants were obtained from flow calorimetric measurements at a pressure equal to the saturation vapor pressure.

The calorimetric method gives equilibrium constants that agree reasonably well with values obtained from other methods, such as conductance measurements or cell EMF measurements. The reliability is increased when a combination of calorimetric measurements with conductivity or cell EMF measurements is used in establishing the equilibrium conditions, especially when more than one reaction is significant. 

Titration calorimetry has been successfully employed in the determination of thermodynamic parameters for complexation (Siimer et al., 1987 Tong et al., 1991a). The technique has the advantage of employing direct calorimetric measurements and has been proposed as the most reliable method (Szejtli, 1982). It should be noted that the information derived from multistep series reactions is macroscopic in nature. In contrast to spectrophotometric methods that provide information concerning only the equilibrium constant(s), titration calorimetry also provides information about the reaction enthalpy that is important in explaining the mechanism involved in the inclusion process. 

This discussion has not included the more accurate calculations that can be made when Cp values of species are known (see equation 3.5-18). These values are not known for many species of biochemical interest. The effects of heat capacity terms are discussed in Chapter 10 because the existing information on ArCp° comes primarily from calorimetric data. In principle, ArCp° can be calculated from measurements of apparent equilibrium constants over a range of temperatures. Over short ranges of temperature, K can be represented by... 

The determination of standard transformed enthalpies of biochemical reactions at specified pH, either from temperature coefficients of apparent equilibrium constants or by calorimetric measurements, makes it possible to calculate the corresponding standard transformed entropy of reaction using... 

Calorimetric measurements yield enthalpy changes directly, and they also yield information on heat capacities, as indicated by equation 10.4-1. Heat capacity calorimeters can be used to determine Cj , directly. It is almost impossible to determine ArCp° from measurements of apparent equilibrium constants of biochemical reactions because the second derivative of In K is required. Data on heat capacities of species in dilute aqueous solutions is quite limited, although the NBS Tables give this information for most of their entries. Goldberg and Tewari (1989) have summarized some of the literature on molar heat capacities of species of biochemical interest in their survey on carbohydrates and their monophosphates. Table 10.1 give some standard molar heat capacities at 298.15 K and their uncertainties. The changes in heat capacities in some chemical reactions are given in Table 10.2. 

Equation (49) provides a means of determining the heat of a reaction without performing any calorimetric measurements. If an equilibrium can be established for the reaction with measurable amounts of reactants and products over a range of temperature, then AnnH° can be obtained from the slope of a graph of the logarithm of the equilibrium constant versus T x. Equation (49) can also be written as... 

The critical thermochemical quantities for prediction of propeUant performance are the enthalpies of formation of the product and to a degree the reactant species. The enthalpy of formation is essential to the calculation of the enthalpy of reaction and since it also appears in the expression for the equilibrium constant, as it is the basis for relating the Gibbs functions of different species, influences the calculated product equilibrium compositions. It is most desirable to measure enthalpies of formation directly from calorimetric experiments, but often the enthalpies must be... 

By these methods, solutions of highly stabilized (e.g., trityl cations) as well as of relatively unstable carbocations (e.g., sec-alkyl cations) have been produced. Although the precision of the calorimetric measurements is smaller than that of most equilibrium determinations, it is an advantage of Arnett s approach that very different types of carbocations can be studied by the same method (Scheme 3). Error propagations, which may be introduced when a series of equilibrium constants or overlapping scales are connected, are thus eliminated. 

In equation (49), which is the van t Hoff equation, —dH/de may be replaced by — AH, since these two quantities are equal for ideal-gas reactions. Relationships analogous to equation (49) may be derived for each of the equilibrium constants defined in Section A. 3, but for reactions in systems other than ideal-gas mixtures, — AH and — dH/de may not, in general, be equated in these expressions. Heats of reaction can be determined directly either by spectroscopic measurements followed by the application of statistical mechanics (for ideal-gas reactions) or by calorimetric measurements of Q (for arbitrary reactions). Since the measurement of equilibrium compositions may be simpler than either of the above procedures, in practice equation (49) is often used to obtain heats of reaction from experimental values of Kp at neighboring temperatures. 

Calorimetric measurements on a reaction like the hydrolysis of ATP yields Ar // ° at the experimental T, pH, and ionic strength. The calorimetric heat of reaction Aj/Zc must be corrected for the heat effect of the hydrogen ions produced by the enzyme-catalyzed reaction on the acid dissociation of the buffer, as described in Chapter 15. If Zf is measured at several temperatures and the acid dissociation constants of all the reactants are known at these temperature, the equilibrium constant K for the reference reaction can be calculated at each temperature. Plotting InAT versus 1/Tyields, which is given by... 

The calculations in Chapters 3 to 5 have been based on the use of Legendre transforms to introduce pH and pMg as independent intensive variables. But now we need to discuss the reverse process - that is the transformation of Af G ° values calculated from measured apparent equilibrium constants in the literature to Af G° values of species and the transformation of Af° values calculated from calorimetric measurements in the literature to AfH° of species. This is accomplished by use of the inverse Legendre transform defined by (7) ... 

Apparent equilibrium constants cannot be determined experimentally on reactions that go nearly completion. Calorimetric measurements of enthalpies of reaction do not have this problem. Proteins may be reactants in enzyme-catalyzed reactions. When apparent equilibrium constants can be measured on reactions involving proteins, the thermodynamic properties of the reaction site in the protein can be calculated. 

Direct calorimetric methods or temperature dependence of equilibrium constants can be used to measure enthalpies and entropies of acid-base reactions. The following section gives more details on use of data from these measurements. 

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