Equilibrium constant approach

There are two basic approaches to the solution of a species distribution problem (1) The equilibrium constant approach, and (2) the Gibbs free energy approach. Most... 

The equilibrium state is generated by minimizing the Gibbs free energy of the system at a given temperature and pressure. In [57], the method is described as the modified equilibrium constant approach. The reaction products are obtained from a data base that contains information on the enthalpy of formation, the heat capacity, the specific enthalpy, the specific entropy, and the specific volume of substances. The desired gaseous equation of state can be chosen. The conditions of the decomposition reaction are chosen by defining the value of a pair of variables (e.g., p and T, V and T). The requirements for input are ... 

The data can be correlated using an equilibrium constant approach with most deviations being less than + 0.1 pH unit. 

Prototropic tautomerism of 2-alkylated tetrahydro species 21 is solvent dependent, tautomers 21a and 21b being identified as the major species present in DMSO-dimethyl sulfoxide) and deuterochloroform, respectively <2000HCA1693> by NOE experiments. Acetylated analogue 22 appears to behave similarly, although in this case the spectrum is complicated by the presence of amide rotamers. As judged from the N-H signals observed in H NMR spectrum, the major species in deuterochloroform is 22a, whereas, in DMSO-i4> the tautomeric equilibrium constant approaches unity, and both tautomers are present. 

As [02] approaches infinity, this apparent equilibrium constant approaches a limiting value because the reaction becomes... 

From this expression it can be seen that the modulus rcv transforms to the standard Thiele modulus (eq 27) when the equilibrium constant approaches infinity. Additionally, it is obvious that the effectiveness factor decreases when, at a given value of the forward rate constant k+, the reverse reaction becomes increasingly important (Fig. 18). This holds for all types of reversible reactions [31, 91]. Therefore, the effectiveness factor of a truly reversible reaction might be considerably overestimated if the reaction is treated as irreversible. 

The standard method of solving the problem by the equilibrium constant approach is to use linearized matrix inversion. Convergence assumes, of course, that the solution not only exists but that it is unique. If a system can have several thermodynamically metastable states (local minima in the Gibbs function) then several nonunique solutions are possible. 

The basic mathematics of the equilibrium constant approach was derived in general form by Brinkley (11,12) and Kandiner and Brinkley ( n). The development of the equations was well suited for adaptation to digital computers and Feldman, et al. 

Solving the equilibrium problem means finding the minimum in the free energy curve for the defined system. Two general approaches have been used the equilibrium constant approach and the free-energy minimization approach. As the names suggest they primarily use equilibrium... 

Two approaches to the treatment of large systems have been developed the Gibbs energy minimization method and the equilibrium constant approach. Both are based on a knowledge of the chemical potentials of species under standard conditions and under actual conditions of composition of conditions in the chemical system. In the Gibbs energy minimization approach (Clason,... 

In subsequent chapters we will make frequent use of the algebraic langu ige of the equilibrium constant approach components, species, stoichiometric matrices, and equilibrium constant vectors in describing multispecies systems. This language will be found helpful for system visualization and essential for numerical solution of multispecies problems. 

Chemical equilibrium dictates the extent of reaction possible for a given temperature and pressure. For single simple reactions, an equilibrium constant approach can be used to determine the equilibrium concentration of gases for a given reaction. At equilibrium, the forward and the reverse reaction rates are equal. The equilibrium constant is calculated from the Gibbs free energy, as follows ... 

The equilibrium constant approach works well when single simple reactions occur, but not when there are competing reactions. The formal definition of chemical equilibrium is that the total Gibbs free energy is at a minimum ... 

As the electrolyte concentration becomes very small, concentration-based equilibrium constants approach their thermodynamic values A p, K.. ... 

The effect of rf coordination on the arenes was studied in the context of the phe-nol-ketodiene equilibrium . It was shown that this equilibrium for the free ligands favors heavily the phenol tautomer vide supra) whereas for the complexes [Os(NH3)5-2,3- -arene)] + (arene = phenol 2-, 3-, 4-methylphenol 3,4-dimethylphenol) the corresponding equilibrium constants approach unity (20 °C). The conversion of phenol 67 into the 2,4-cyclohexadien-l-one 68 was kinetically favored over the formation of the 2,5-isomer 69, although the latter is the thermodynamically favored product (equation 26). It was assumed that osmium rehybridizes the C(5) and C(6) atoms to form a metallocy-clopropane. This removes much of the resonance energy and therefore destabilizes the enohc form of the free hgand. The free energies of ketonization (25 °C) for the /j -phenol complex in comparison with free phenol are shown in equations 27 and 28 . 

The equilibrium constant approach is based on Equation (4.6), and it is useful if the chemical reaction is relatively simple, involving a small number of gaseous and solid species. This method requires a knowledge of all chemical reactions and information of species involved, including all the species, chemical reaction paths and their corresponding equilibrium constants. 

The reason for this goes back to the underlying theoretical foundations of the two approaches —. It is not surprising that an equilibrium constant approach is a rather limited way to sum over all possible binary and ternary interactions, particularly when one recalls that there are interactions between ions of like sign as well as those of opposite sign. 

Equilibrium constant approach. ponent system can be expressed as... 

For more complex reactions, the Michaelis constants for all components of the system must be included. When the substrates and products have similar affinities for the enzyme, the equilibrium constant approaches the ratio of the maximum rates, which can often be measured as initial rates. Examples will be shown later of reactions in which the correction of the ratio of rates by the ratio of Michaelis constants is more than an order of magnitude. 

As these equations indicate, hydrations of aldehydes and ketones are reversible. The equilibrium lies to the left for ketones and to the right for formaldehyde and aldehydes bearing inductively electron-withdrawing substituents. For ordinary aldehydes, the equilibrium constant approaches unity. 

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