The Dependency of K on Temperature

According to Eqs 12.5.10 and 12.5.12 evaluation of the equilibrium constant K for a given reaction - at a specified temperature T - requires values for the standard free energies of the components of the reaction, also at T. The latter can be replaced by the standard free energy of formation, dGj, of each component from its elements, as we will see in Example 15.1. 

Such values are available for a large variety of compounds, but at 298.2 K and the ideal gas state at P = 1 atm (or 1 bar). Values for selected compounds are presented in Table 15.1 for other compounds, see the same reference, Reid et al (1987), or the references given by Kyle (1984). 

We proceed, therefore, to describe a method for evaluating K values at temperatures other than 298.2 K from the free energy of formation values of the components of the reaction at 298.2 K. To this purpose, we  

demonstrate that the variation of K with T d ends on the standard enthalpy of the reaction AH°, which itself is a ftmction of temperature  

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The experimentally observed pseudo-first order rate constant k is increased in the presence of DNA (18,19). This enhanced reactivity is a result of the formation of physical BaPDE-DNA complexes the dependence of k on DNA concentration coincides with the binding isotherm for the formation of site I physical intercalative complexes (20). Typically, over 90% of the BaPDE molecules are converted to tetraols, while only a minor fraction bind covalently to the DNA bases (18,21-23). The dependence of k on temperature (21,24), pH (21,23-25), salt concentration (16,20,21,25), and concentration of different buffers (23) has been investigated. In 5 mM sodium cacodylate buffer solutions the formation of tetraols and covalent adducts appear to be parallel pseudo-first order reactions characterized by the same rate constant k, but different ratios of products (21,24). Similar results are obtained with other buffers (23). The formation of carbonium ions by specific and general acid catalysis has been assumed to be the rate-determining step for both tetraol and covalent adduct formation (21,24). 

Temperature Dependence of Kinetic Constant. The dependence of k on temperature for an elementary reaction follows the Arrhenius equation... 

Although the relationship given above for the dependence of K on temperature is only intended to hold over a limited temperature range,... 

The problems of interest are finding the conditions for onset of vaporization, the bubble-point for the onset of condensation, the dewpoint and the compositions and the relative amounts of vapor and liquid phases at equilibrium under specified conditions of temperature and pressure or enthalpy and pressure. The first cases examined will take the A, to be independent of composition. These problems usually must be solved by iteration, for which the Newton-Raphson method is suitable. The dependence of K on temperature may be represented adequately by... 

The dependence of K on temperature plays an important role in establishing the relationship between the relative roles of enthalpy- AH) and entropy (AS) change in the antibody-antigen interaction. This relationship is expressed by the van t Hoff equation (Eq. (9.14)), which is obtained by differentiating Eq. (9.13) (taking into account that AG = AH - TAS) ... 

K, can be calculated from the free energy change of the reaction. Using the van t Hoff relation, we obtain the dependence of K, on temperature ... 

The value of from Example 17-1 is much different from the value given earlier for the same reaction at 25°C. For this reaction, products are favored at the lower temperature (K. = 3.6 X 10 at 25°C), whereas reactants are favored at the higher temperature K = 0.286 at 500°C). The dependence of K. on temperature will be discussed later in this chapter for now, we note that it can depend strongly on temperature. 

The dependence of K on temperature can show both a linear and a non-linear relation between logeAT and T or logeAT and l/T. Sometimes, as in the case of some carboxylic acids, a maximum in K is found as the temperature is varied. This is discussed in Worked Problems 2.5 and 8.9. 

The dependence of K on temperature and pressure can give a lot of information about details of the reaction involved. The kinetic analogues likewise give much information about the mechanism of the reaction involved. 

The most direct way, and possibly the most accurate method, for determining A// and A// for reactions in solution is a direct calorimetric determination. However, the dependence of K on temperature also enables AH to be found. 

For a group of compounds, the dependence of K on temperature would be determined by the properties of the selected sol vent-water systems, which determine A77o and AH. For example, when the polarity of the organic solvent becomes increasingly close to that of water, as in the butanol-water system (65), the value of AHq — be relatively small and the partition coefficients of... 

The enthalpy of association of one surfactant with a micelle proper, obtained from the dependence of k on temperature, is negative, indicating an exothermal process. ... 

The dependence of lifetime on temperature in the range above RT shows an activation energy in the order of 10-25 meV [Bu3, Ool]. This was proposed to be a consequence of the exchange splitting of the exciton between the singlet and the triplet state. While at RT both states are populated, only the lower triplet state is populated at temperatures below 20 K. However, it has been shown that even for crystallites of low symmetry the calculated values of the exchange splitting are too low compared with experimental observations [De3]. Calculations of the radiative lifetime of the triplet exciton that take into account spin-orbit interactions are reported to be consistent with experimental results [Nal]. 

Karty et al. [21] pointed out that the value of the reaction order r and the dependence of k on pressure and temperature in the JMAK (Johnson-Mehl-Avrami-Kolmogorov) equation (Sect. 1.4.1.2), and perhaps on other variables such as particle size, are what define the rate-limiting process. Table 2.3 shows the summary of the dependence of p on growth dimensionality, rate-limiting process, and nucleation behavior as reported by Karty et al. [21]. 

At low temperatures, the broadening factors F1/2 do not deviate too much from unity. The dependence of k°° on the temperature T is expressed by a power function (except for cases with an established potential energy barrier), viz. 

The roller coaster patterns mentioned on p. 210 are normally observed in plots of activation parameters against x2, calculated from the dependence of rate constant on either temperature or pressure, rather than in the dependence of k on x2. Thus a careful examination is usually made of the dependence of k on T at fixed p and x2, and on p at fixed T and x2. The analysis of these separate dependences has been discussed by Kohnstam (1967) who points out the various pitfalls which await the unsuspecting. Indeed, a growing... 

X parameter, a perfect correlation was not to be expected. From the correlation curve so-established the swelling of experimental vulcanizates could be predicted with reasonable accuracy from the easily determined x parameter. The GC method diould be particularly valuable for testing experimental samples, whenever only small amounts are available. For non-crosslinked materials, the magnitude of the x parameter is a direct indication of the solubility of the polymer in any given solvent. The dividing line between solvents and non-K>lvent of the polymer can be drawn at approximately x = 0.5, the smaller the values of x below this limit, the better the solvent. It is a simple matter to measure the retention characteristics of a series of probes and thus determine a suitable >lvent for any pdymer. It should be noted, however, that the temperature at which tfie GC determination is possible (T>Tg+ 50) may sometimes ermeed the temperature of interest and the dependence of x on temperature may have to be assessed. 

Fig. 1 shows the dependence of k, on d for different temperatures. It is seen that k, increases with a decrease in the particle size and that an increase in temperature leads to a decrease of the influence of the size effect on k. ... 

The solution of Eqs. (11-46) to (11-48) and (11-72) to (11-74) gives the concentration and temperature profiles within the pellet. A numerical solution is necessary because Eqs. (11-46) arid (11-72) are coupled through the nonlinear dependence of k on temperature k = A Neverthe-... 

The dependence of solubility on temperature varies. We will consider here the range of zero to about 50°C at still higher temperature unfolding may occur. For hydrophilic proteins, the solubility may increase with temperature, by up to 4% per K. For more hydrophobic proteins, solubility decreases with increasing temperature, by up to 10% per K. This is in accordance with the strong temperature dependence of hydrophobic bonds in the range considered (Fig. 3.4). Low temperature may also cause dissociation of quaternary structures. 

Practically, it is important to reduce the sharp dependence of k on temperature. In other words, it is important to broaden the permittivity versus temperature peaks as much as possible. One significant advantage of ceramic ferroelectrics is the ease with which their properties can be modified by adjusting composition and/or microstrueture. For example, the substitution of Ti by other cations results in a shift in Tq, as shown in Fig. 15.16. Replacing Ti" by Sr ions reduces Tq while the substitution of Pb" increases it. This is very beneficial because it allows for the tailoring... 

A// for the adsorption of a particular compound in a system of interest (or if we know K" for that compound at two different temperatures). Unfortunately, few heals of adsorption have been reported for chromatographic systems. Consequently, a less rigorous expression for the dependence of /l on temperature is required if we are to avoid the necessity of knowing or measuring values of AW for every situation of interest. We shall now proceed to the derivation of such a relationship. 

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