Pressure dependence of the equilibrium constant

The pressure-jump (P-jump) method is based on the pressure dependence of the equilibrium constant, Eq. (4-28), where AV is the molar volume change of the reaction. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

Note that the equations for estimating the pressure dependencies of 7 and aw (Eqs. 2.87 and 2.90) depend on the Pitzer equations (Eqs. 2.76, 2.80, and 2.81) but this is not the case for the pressure dependence of the equilibrium constants (Eq. 2.29) the latter equation is based entirely on partial molar volumes at infinite dilution, which are independent of concentration. Also, compared to the pressure-dependent equation for the equilibrium constant (Eq. 2.29), the pressure equations for activity coefficients (Eq. 2.87) and the activity of water (Eq. 2.90) do not contain compressibilities (K) because the database for these terms and the associated Pitzer parameters are lacking at present (Krumgalz et al. 1999). The consequences of truncating Eqs. 2.80 and 2.81 for ternary terms and Eqs. 2.87 and 2.90 for compressibilities will be discussed in Sect. 3.6 under limitations. 

Because of the great depth of the ocean, the most important physical property determining the solubility of carbonate minerals in the sea is pressure. The pressure dependence of the equilibrium constants is related to the difference in volume, AV, occupied by the ions of Ca and in... 

The principle of the pressure-jump method is based on the pressure dependence of the equilibrium constant that is. 

It is clear that from the integrated form of Equation (4) the volume of reaction can be obtained if the equilibrium constant can be determined over a range of pressure. If the volume of activation is not experimentally accessible for one of the directions of the reaction, A V can be used to calculate its value. Under certain conditions and with suitable properties of reactants and/or products it may be possible to determine their partial molar volumes, hence allowing development of a volume profile on an absolute volume basis, as noted above. Even if A V can be determined either from the pressure dependence of the equilibrium constant and/or from use of Equation (5), it may be possible to confirm its value by determination of the partial molar volumes from density measurements. The conditions for conducting successful determinations of partial molar volumes are rather stringent and will be described in Section 2. The method depends on measuring the density, d, of several solutions of different concentrations of the reactant or product. The following equation is used to obtain... 

The reaction volume could be obtained from the pressure dependence of the equilibrium constant as -47.7 cm3 mol-1, in excellent agreement with the value obtained from the kinetically determined value of-48.3 cm3 mol-1 (reaction... 

Table 2. Volume changes (mL/mol) for the formation of molecular complexes obtained from the pressure dependence of the equilibrium constants [75].

High pressure stopped-flow experiments upon the reaction of hexaaquairon(III) and promazine (R = CH3CH2CH2N(CH3)2, see reaction given below) yielded AV values of -6.3 and -12.5 cm mof for the forward and reverse reactions, respectively. [139] The derived reaction volume of 6.2 cm mof has been confirmed by calculation from the pressure dependence of the equilibrium constant obtained from the variation of the UV/Visible spectra as a function of pressure. The volume profile (see Figure 9) displays the fact of a compact transition state, and the positive reaction volume was suggested to arise from charge dilution in proceeding from the reactants to Fe(aq) and the promazine cation radical. 

The isothermal pressure dependence of the equilibrium constant for (12) is given by n Pp.t... 

A series of related reports have appeared on the equilibria between five- and six-coordinate species and possible adduct formation in Co(iii) corrinoids, on the thermodynamic and kinetic properties for what is termed the base-on/ base-off equilibration of alkyl cobalamins, " and on the kinetics and thermodynamics of parallel equilibria of alkyl-13-epicobalamins. In the first report, the pressure dependence of the UV/visible spectra of the five-coordinate (yellow)/six-coordinate (red) equilibrium for both methylcobalamin and vinylcobinamide was obtained. Water is the ligand that converts the five- to a six-coordinate species. The reaction volumes were obtained from the pressure dependence of the equilibrium constant. The values of AF of —12.5 1.2 and —12.5 l.Ocm moF for the methyl and vinyl complexes are close to the values 13cm moF advocated and accepted for the displacement or... 

Let us now discuss the pressure dependence of the equilibrium constant in cases where the standard state selected is the pure constituent at the temperature and pressure of the system. From relation (3.22) follows... 

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