Empirical equilibrium constant

The subscript C denotes that the reaction is carried out in solution and that the empirical equilibrium constant Kq is evaluated by directly measuring the concentration of each species in the equilibrium state of the reaction. In general, Kq has dimensions (concentration) " it will be dimensionless only for those reactions for which a + b = c + d. 

The dimensionless quantity K is the thermodynamic equilibrium constant, which Section 14.3 shows can be calculated from tabulated data on the products and reactants, even if the empirical equilibrium constant defined in Equation 14.1b is not known. Therefore, K is the preferred tool for analyzing reaction equilibria in general. The informal argument by which we replaced ICp with K is made rigorous... 

In this section we use thermodynamics to demonstrate why the mass action law takes its special mathematical form and why the thermodynamic equilibrium constant fC is a dimensionless quantity. This demonstration justifies the procedures we presented in Section 14.2 for writing down the mass action law by inspection for any chemical reaction. In addition, thermodynamics gives a method for calculating the value of K from tabulated properties of the reactants and products. Consequently, the value of K can be obtained for a reaction, even if the empirical equilibrium constant Kq or has not been measured. Thermodynamics also explains how K changes when the reaction is run under different experimental conditions. With this information, we can manipulate reaction conditions to obtain maximum yield from the reaction. 

Thermodynamic equilibrium constants are dimensionless because they are expressed in terms of activities rather than partial pressure or concentration. The convention in chemical kinetics is to use concentrations rather than activities, even for gaseous species. Therefore, the equilibrium constants Ki, K2, and introduced here are the empirical equilibrium constants described briefly in Section 14.2. These constants are not dimensionless and must be multiplied by the concentration of the reference state, = l T/Pref, raised to the appropriate power to be made equal to the thermodynamic equilibrium constant. Nevertheless, to maintain consistency with the conventions of chemical kinetics, such constants as Ki, K2, and are referred to as equilibrium constants in this section and are written without the subscript c. 

Empirical equilibrium coupling constants can be compared as a benchmark with calculated equilibrium coupling constants obtained with various methods. A comparison of these empirical equilibrium constants with calculated equilibrium constants suggested that the restricted-active-space self-consistent field (RASSCF) method is the best approach for calculating the indirect nuclear spin-spin coupling constants of small molecules, and that the second-order polarization propagator approximation (SOPPA) and DFT are similar in performance. 

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... 

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. 

Historically, the identification of a linear correlation between log k and T 1 was empirical. First described by Hood [504], the relationship was given some theoretical significance by van t Hoff [505] who expressed the influence of temperature on equilibrium constants (Ke) by an equation of similar form, viz. 

We use a different measure of concentration when writing expressions for the equilibrium constants of reactions that involve species other than gases. Thus, for a species J that forms an ideal solution in a liquid solvent, the partial pressure in the expression for K is replaced by the molarity fjl relative to the standard molarity c° = 1 mol-L 1. Although K should be written in terms of the dimensionless ratio UJ/c°, it is common practice to write K in terms of [J] alone and to interpret each [JJ as the molarity with the units struck out. It has been found empirically, and is justified by thermodynamics, that pure liquids or solids should not appear in K. So, even though CaC03(s) and CaO(s) occur in the equilibrium... 

Each reaction has its own characteristic equilibrium constant, with a value that can be changed only by varying the temperature (Table 9.2). The extraordinary empirical result, which we justify in the next section, is that, regardless of the initial composition of a reaction mixture, the composition tends to adjust until the... 

Solubility equilibria are described quantitatively by the equilibrium constant for solid dissolution, Ksp (the solubility product). Formally, this equilibrium constant should be written as the activity of the products divided by that of the reactants, including the solid. However, since the activity of any pure solid is defined as 1.0, the solid is commonly left out of the equilibrium constant expression. The activity of the solid is important in natural systems where the solids are frequently not pure, but are mixtures. In such a case, the activity of a solid component that forms part of an "ideal" solid solution is defined as its mole fraction in the solid phase. Empirically, it appears that most solid solutions are far from ideal, with the dilute component having an activity considerably greater than its mole fraction. Nevertheless, the point remains that not all solid components found in an aquatic system have unit activity, and thus their solubility will be less than that defined by the solubility constant in its conventional form. 

The equilibrium constant of hexaphenylethane dissociation, in striking contrast to the rate constant for dissociation, varies considerably with solvent. The radical with its unpaired electron and nearly planar structure probably complexes with solvents to a considerable extent while the ethane does not. Since the transition state is like the ethane and its solvation is hindered, the dissociation rate constants change very little with solvent.12 13 From an empirical relationship that happens to exist in this case between the rate and equilibrium constants in a series of solvents, it has been calculated that the transition state resembles the ethane at least four times as much as it resembles the radical. These are the proportions that must be used if the free energy of the transition state in a given solvent is to be expressed as a linear combination of the free energies of the ethane and radical states.14... 

Heat effects accompanying chemical reaction influence equilibrium constants and compositions as well as rates of reaction. The enthalpy change of reaction, AHr, is the difference between the enthalpies of formation of the participants. It is positive for endothermic reactions and negative for exothermic ones. This convention is the opposite of that for heats of reaction, so care should be exercised in applications of this quantity. Enthalpies of formation are empirical data, most often known at a standard temperature, frequently at 298 K. The Gibbs energies of formation, AGfl likewise are empirical data. 

An important advance in the understanding of the chemical behaviour of glasses in aqueous solution was made in 1977, when Paul (1977) published a theoretical model for the various processes based on the calculation of the standard free energy (AG ) and equilibrium constants for the reactions of the components with water. This model successfully predicted many of the empirically derived phenomena described above, such as the increased durability resulting from the addition of small amounts of CaO to the glass, and forms the basis for our current understanding of the kinetic and thermodynamic behaviour of glass in aqueous media. 

Therefore, one must accept that the description of the solvent effect is rather complex and cannot be simplistically made on the basis of single physical parameters. A large number of parameters (including empirical parameters) must be considered which derive from thermodynamic calculations (equilibrium constant) and kinetic calculations (rate constants) performed on a large number of chemical reactions. 

Fractionation factors are calculated for a large variety of trigonal-planar (XY3) and tetrahedral XY4) molecules and molecule-like complexes, with a particular focus on metal halides. Empirical force-held models (MUBFF) are used to estimate vibrational frequencies for the rarer isotopic forms of the substances studied, and aqueous and crystalline moleculelike species are modeled as gas-phase molecules. In the tabulation below the original equilibrium constants have been converted to fractionation factors (a R-x)-... 

S. Adair, H. S. Sinuns, K. Linderstrom-Lang, and, especially, J. Wyman. These treatments, however, were empirical or thermodynamic in content, that is, expressed from the outset in terms of thermodynamic equilibrium constants. The advantage of the explicit use of the actual grand partition function is that it is more general it includes everything in the empirical or thermodynamic approach, plus providing, when needed, the background molecular theory (as statistical mechanics always does). 

Equilibrium constants are also dependent on temperature and pressure. The temperature functionality can be predicted from a reaction s enthalpy and entropy changes. The effect of pressure can be significant when comparing speciation at the sea surface to that in the deep sea. Empirical equations are used to adapt equilibrium constants measured at 1 atm for high-pressure conditions. Equilibrium constants can be formulated from solute concentrations in units of molarity, molality, or even moles per kilogram of seawater. 

Comparison of various methods For the first three methods, it is necessary to know how the equilibrium constant of the reaction depends on temperature (and often on the composition of the phase), the reaction rate law, and how the rate coefficients depend on temperature (and the composition). The empirical method directly relates cooling rate with cooled species concentrations. The first three methods have better extrapolation capabilities, whereas the empirical method does not have much extrapolation ability. The empirical method, hence, only works on a cooling timescale of several years or less. 

The relative importance of these surface species varies greatly as a function of pH. Hence, accurate predictions of the sorption of such organic ligands on mineral oxides requires applying more than one empirical surface reaction equilibrium constant to calculate the contributions of each bound species (see Evanko and Dzombak, 1999 for examples). 

TABLE 13.3 T-dependent Equilibrium Constant (KT), Gibbs Free Energy of Reaction (AGT), and Overall Entropic Shift (AAG = AG12oo — AG90o) for the Water Gas Shift Reaction (cf. Tables 13.1, 13.2, and Text), as Determined from Theoretically ( B3LYP ) or Empirically ( Hill ) Evaluated Statistical Thermodynamic Formulas Versus Experiment ( Exp. )... 

Difficulties associated with measuring the activities of ions on a solid phase led many workers to suggest empirical relationships similar to eq 16 in an attempt to define the equilibrium constant, K. A relationship developed by Vanselow (70) assumes that surface activity is proportional to the mole fraction of an ion. For exchange between cations M and N, the surface activity of M is defined by... 

A useful empirical approach to the design of heterogeneous chemical reactors often consists of selecting a suitable equation, such as one in Table 3.3 which, with numerical values substituted for the kinetic and equilibrium constants, represents the chemical reaction in the absence of mass transfer effects. Graphical methods are often employed to aid the selection of an appropriate equation140 and the constants determined by a least squares approach<40). It is important to stress, however, that while the equation selected may well represent the experimental data, it does not... 

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