Equilibrium constant, definition temperature coefficients

For the calcite decomposition reaction, the equilibrium constant. Kg, has the same definition as the distribution coefficient, K, given in equation (5.7) giving the equilibrium partial pressure of CO2. The dependence of the equilibrium constant on temperature is given by the Clausius—Claperon equation [2] ... 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

Plan We can use standard enthalpies of formation to calculate AH for the reaction. We can then use Le ChateUer s principle to determine what effect temperature will have on the equilibrium constant. Recall that the standard enthalpy change for a reaction is given by the sum of the standard molar enthalpies of formation of the products, each multipUed by its coefficient in the balanced chemical equation, less the same quantities for the reactants. At 25 C, AHj for NH3( ) is —46.19 kj/mol. The AHJ values for H2(g) and N2(g) are zero by definition, because the enthalpies of formation of the elements in their normal states at 25 C are defined as zero (S tion 5.7). Because 2 mol of NH3 is formed, the total enthalpy change is... 

After equUibrium is attained, the concentrations of products and reactants remain constant, so a ratio of their concentrations should also remain constant. The ratio of the mathematical product [C] x [D] to the mathematical product [A]" x [B]" for this reaction has a definite value at a given temperature. It is the equilibrium constant of the reaction and is designated by the letter K. The foUowing equation describes the equUib-rium constant for the hypothetical equUibrium system. The brackets ([ ]) indicate the concentration of each substance as expressed in mol/L. The superscripts are the coefficients of each substance in the balanced chemical equation. 

The glass transition phenomenon has been the object of many molecular theories, that of Ferry [29], further developed by Bueche [30], on the free volume concept being widely accepted in the polymer field. The volume occupied by a chain of amorphous polymer consists partly of free space, i.e., the volume excluded by the movements of segments about their equilibrium position. As shown in Fig. 1.8, the temperature coefficient for the polymer volume at constant pressure dV dT)p is higher for the viscoelastic state (curve b) than for the glassy state (curve a), and it changes abruptly at Tg this is the mathematical definition of the glass transition temperature. It can also be defined as the... 

The tunnel correction is not now a fundamentally defined number rather it is defined by the equation Q = kobJk, where kobs is the observed rate constant for a chemical reaction and k is that calculated on the basis of some model which is as good as possible except that it does not allow tunnelling. In this chapter the definition used for k is that calculated by absolute reaction rate theory [3], i.e., k = KRT/Nh)K where X is the equilibrium constant for the formation of the transition state. The factor k, the transmission coefficient, is also a quantum correction on the barrier passage process, but it is in the other direction, that is k < 1. We shall here follow the customary view (though it is not solidly based) that k is temperature-independent and not markedly less than unity. The term k is used following Bell [1] the s stands for semi-classical, that is quantum mechanics is applied to vibrations and rotations, but translation along the reaction coordinate is treated classically. 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

This introduces a new unknown (and free at this stage) coefficient K. In the case of a system of molecules in a thermal bath (definitely not the one we consider), there is a relation between D and K such that, at thermal equilibrium, the equilibrium density in the potential (h is given by Boltzmann s law. This requires that K = mD/hgT, where kg is Boltzmann s constant and T the absolute temperature. In Eq. (12) the factor p in front of in is to ensure that, if... 

Generally, the distribution constant (partition coefficient) as a function of temperature is the first measurement obtained in such a study since other constants may be derived from these data. The distribution constant, KD, is the ratio of a component in a single definite form in the stationary phase per unit volume to its concentration in the mobile phase per unit volume at equilibrium. 

In the definitions of T, two variables in addition to the ion chemical potential must also be specified as constant. In an equilibrium dialysis experiment, these are temperature and the chemical potential of water. This partial derivative is known as the Donnan coefficient. (Note that the hydrostatic pressure is higher in the RNA-containing solution.) In making connections between T and the Gibbs free energy, it is more convenient if temperature... 

The preceding discussion and Eqs. (2-l)-(2-5) assume that the sample distribution coefficient K is constant throughout separation. Quite frequently K varies with sample concentration, however. Separations in which K is independent of sample concentration are referred to as linear isotherm separations, from the definition of an isotherm as a plot of X versus (20 in an equilibrium system at a given temperature. Linearity, of the plot is equivalent to constant K. A nonlinear sample isotherm... 

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