Constants, thermodynamic activity

The following relationship may be derived using the general expression for the distribution coefficient D in conjunction with the conditional fy values and assuming constant thermodynamic activities for the species ... 

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

While the dilated van Laar model gives a reliable representation of constant-pressure activity coefficients for nonpolar systems, the good agreement between calculated and experimental high-pressure phase behavior shown in Fig. 14 is primarily a result of good representation of the partial molar volumes, as discussed in Section IV. The essential part of any thermodynamic description of high-pressure vapor-liquid equilibria must depend,... 

Rate constants and thermodynamic activation parameters. The rate constant for the reaction between C2H4 and HCN catalyzed by a nickel(0) complex was studied over a range of -50 to -10 °C in toluene.31 These authors give the activation parameters A//1 = 36.7kJmor andAS = -145 J mol-1 K I when the reaction rate was expressed using concentrations in the units molL-1 and time in the unit seconds. 

The thermodynamic activity equilibrium constant (Ka) is expressed in terms of mole fraction (X) and activity coefficient (y) by the following equation ... 

The changeover to thermodynamic activities is equivalent to a change of variables in mathematical equations. The relation between parameters and a. is unambiguous only when a definite value has been selected for the constant p. For solutes this constant is selected so that in highly dilute solutions where the system p approaches an ideal state, the activity will coincide with the concenttation (lim... 

Considering only the aqueous phase of the biocatalytic system, the equilibrium constant for the reaction is given as a function of thermodynamic activities of the components shown ... 

The understanding that the computers controlling the equipment might possess could be contained within a kinetic and thermodynamic model that encapsulates the detailed chemistry of the process. This would include a description of all the reactions that might be expected to occur under the conditions achievable in the plant, together with a list of the relevant rate constants and activation energies for each reaction. In addition, process variables, such as maximum flow rates or pump pressures that are needed for a full description of the behavior of the system under all feasible conditions, would be provided. 

Equilibrium constants do not have units because in the strict thermodynamic definition of the equilibrium constant, the activity of a component is used, not its concentration. The activity of a species in an ideal mixture is the ratio of its concentration or partial pressure to a standard concentration (1 M) or pressure (1 atm). Because activity is a ratio, it is unitless and the equilibrium constant involving activities is also unitless. 

We omit concentrations of pure solids and pure liquids from equilibrium constant expressions because their activity is taken to be 1 and the thermodynamic equilibrium constant involves activities, rather than concentrations. 

This permits provisional calculation of the compositional dependence of the equilibrium constant and determination of provisional values of the solid phase activity coefficients (discussed below). The equilibrium constant and activity coefficients are termed provisional because it is not possible to determine if stoichiometric saturation has been established without independent knowledge of the compositional dependence of the equilibrium constant, such as would be provided from independent thermodynamic measurements. Using the provisional activity coefficient data we may compare the observed solid solution-aqueous solution compositions with those calculated at equilibrium. Agreement between the calculated and observed values confirms, within the experimental data uncertainties, the establishment of equilibrium. The true solid solution thermodynamic properties are then defined to be equal to the provisional values. 

The p/<, of a base is actually that of its conjugate acid. As the numeric value of the dissociation constant increases (i.e., pKa decreases), the acid strength increases. Conversely, as the acid dissociation constant of a base (that of its conjugate acid) increases, the strength of the base decreases. For a more accurate definition of dissociation constants, each concentration term must be replaced by thermodynamic activity. In dilute solutions, concentration of each species is taken to be equal to activity. Activity-based dissociation constants are true equilibrium constants and depend only on temperature. Dissociation constants measured by spectroscopy are concentration dissociation constants." Most piCa values in the pharmaceutical literature are measured by ignoring activity effects and therefore are actually concentration dissociation constants or apparent dissociation constants. It is customary to report dissociation constant values at 25°C. 

Such a chemical approach which links ionic conductivity with thermodynamic characteristics of the dissociating species was initially proposed by Ravaine and Souquet (1977). Since it simply extends to glasses the theory of electrolytic dissociation proposed a century ago by Arrhenius for liquid ionic solutions, this approach is currently called the weak electrolyte theory. The weak electrolyte approach allows, for a glass in which the ionic conductivity is mainly dominated by an MY salt, a simple relationship between the cationic conductivity a+, the electrical mobility u+ of the charge carrier, the dissociation constant and the thermodynamic activity of the salt with a partial molar free energy AG y with respect to an arbitrary reference state ... 

As with all speciation calculations, the results for the trace metals depend on the values used for the thermodynamic equilibriiun constants and activity coefficients. Determination of these constants is very difficult because direct detection of trace metal species is complicated by their low concentrations and analytical interferences... 

Buccal dosage forms can be of the reservoir or the matrix type. Formulations of the reservoir type are surrounded by a polymeric membrane, which controls the release rate. Reservoir systems present a constant release profile provided (1) that the polymeric membrane is rate limiting, and (2) that an excess amoimt of drug is present in the reservoir. Condition (1) may be achieved with a thicker membrane (i.e., rate controlling) and lower diffusivity in which case the rate of drug release is directly proportional to the polymer solubility and membrane diffusivity, and inversely proportional to membrane thickness. Condition (2) may be achieved, if the intrinsic thermodynamic activity of the drug is very low and the device has a thick hydrodynamic diffusion layer. In this case the release rate of the drug is directly proportional to solution solubility and solution diffusivity, and inversely proportional to the thickness of the hydrodynamic diffusion layer. 

The equilibrium thus established is a Frenkel defect. In both the Schottky and Frenkel equilibria, the stoichiometry of the crystal is unaltered (figure 4.2). Assuming that the thermodynamic activity of the various species obeys Raoult s law, thus corresponding to their molar concentrations (denoted hereafter by square brackets), the constant of the Schottky process is reduced to... 

When the partial derivative reaches zero, the two potentials coincide and component i is present unmixed as a pure term. If unmixing takes place in solvus conditions, the thermodynamic activity of component i remains constant for the entire solvus field. However, in the case of spinodal decomposition, the activity of i within the spinodal field plots within a maximum and minimum (cf sections 3.11 and 3.12). 

This methodology has been applied for determining the relative composition of alloys [225], amalgams [226], and mixed crystals [227], among others [74-78], based on peak current measurements. The essential requisite is that both electroactive components behave independently—i.e., that the components of a mechanical mixture do not influence each other with respect to their thermodynamic activities in electrochemical reactions [77]. In Fig. 4.4, theoretical calibration plots for the absolute peak current, when the amount of mixture is constant for each measurement (left) and for the percentage peak current (right), are shown [77, 228]. 

However, if the valences of the exchanging cations are equal, the selectivity coefficient or pseudo-equilibrium constant is not affected by concentration. As already mentioned, one isotherm corresponds to a specific temperature in the case of adsorption or ion exchange of equal valence ions, whereas additionally, the same normality is required for the existence of only one isotherm in the case of ion exchange of different valence ions, due to the concentration-valence effect (Helfferich, 1962). The determination of the true equilibrium constant should be based on the thermodynamic activities (activity coefficients) of the species rather than concentrations. It is clear that the difficulties in the determination of activity coefficients also complicate the determination of the true equilibrium constant (Culfaz and Yagiz, 2004). 

The competition between transferase and hydrolysis reactions can be described in terms of nucleophile (acceptor) selectivities of the enzymes, and selectivity constants can be defined. These constants are meant to quantify the intrinsic selectivity of the enzymes. Selectivity constants in combination with the concentrations (or thermodynamic activities) of the competing nucleophiles give the transferase/hydrolysis ratio. The selectivity constants are defined as follows [38, 39] ... 

The influence of the level of availability of the different organic components was also studied in a solid/gas reactor by varying their thermodynamic activity, and the result was that E remained constant. 

In principle, the nature of biopolymer interactions in solution can also be determined from sedimentation equilibrium of a single biopolymer component at constant temperature T and angular velocity co. The thermodynamic activity on a weight-concentration basis, z, = yiC/M at radial distance r is related to that at a selected reference radial position (r ) by the following formal expression (Deszczynski et al., 2006) ... 

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