Equation, thermodynamic activity

A.queous Solubility. SolubiHty of a chemical in water can be calculated rigorously from equiHbrium thermodynamic equations. Because activity coefficient data are often not available from the Hterature or direct experiments, models such as UNIFAC can be used for stmcture—activity estimations (24). Phase-equiHbrium relationships can then be appHed to predict miscibility. Simplified calculations are possible for low miscibiHty however, when there is a high degree of miscibility, the phase-equiHbrium relationships must be solved rigorously. 

Trustworthy thermodynamic data for metal solutions have been very scarce until recently,25 and even now they are accumulating only slowly because of the severe experimental difficulties associated with their measurement. Thermodynamic activities of the component of a metallic solution may be measured by high-temperature galvanic cells,44 by the measurement of the vapor pressure of the individual components, or by equilibration of the metal system with a mixture of gases able to interact with one of the components in the metal.26 Usually, the activity of only one of the components in a binary metallic solution can be directly measured the activity of the other is calculated via the Gibbs-Duhem equation if the activity of the first has been measured over a sufficiently extensive range of composition. 

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... 

A possible reason for the departures from Pick s first law is the fact that the diffusion process tends to level chemical potentials (thermodynamic activities) rather than concentrations of the substances involved. Hence, the equation sometimes is written as... 

The higher thermodynamic activity, a, of the amorphous form compared to that of the crystalline form explains the higher initial dissolution rate per unit surface area (intrinsic dissolution rate, J) and the higher solubility, s, of the amorphous form compared to that of the crystalline form, according to a simple form of the Noyes-Whitney equation [15],... 

Within the two metastable ranges one of the binary components can have an apparent thermodynamic activity larger than 1. The maximum will be reached at x = xgp. In this study x p was derived as a function of W/2.303 RT by iterative procedures using the relevant equations given by Meyering (6 1 ). Subsequently, the thermodynamic activities of the two components were calculated at the extremes which can be reached for variable xsp (Figure 9). Apparently, such high values as log a= 2 are reached only for x > 0.93. 

The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as ... 

Ideally, thermodynamic activities of the reactants should be used in the equation, but since concentrations are normally easier to measure these are often used instead. The use of the activity of water (which can be measured fairly easily) and the concentrations of the other reactants has been recommended for studies of enzyme catalyzed reactions in organic media (Hailing, 1984). In order to increase the synthesis of the ester, the water concentration (or activity) should be reduced. This can be achieved by replacing part of the water with a water miscible solvent. 

Following the definition (Equation 1), / -values can be obtained by measurement of the fesp-values for the enzymatic reaction of the enantiomers. Considering practical limitations, i.e. the availability of chirally pure enantiomers, E-values are more commonly determined by evaluating the ee-values as a function of the extent of conversion in batch reactions (see [28, 29] and [30] for overviews). It must be emphasized that the relationship between E, the intrinsic property, and , the realization of this property, shows a (complex) dependence on the concentrations (more precisely, the mass-action equivalents or thermodynamic activities) of the... 

For continuous systems, molar flow rates Q can be used instead of n. The thermodynamic activity (ax) can be calculated according to Equation 2, but requires knowledge of the saturation pressure of the pure compound (Ppsatx). This data can be obtained from the saturation curves (vapor-liquid equilibrium curves) and is taken at the working temperature of the gas stream. The thermodynamic activity is then calculated using the following equation ... 

Thermodynamic activities are then calculated according to Equation 6. 

In equation (5.37), v/ is the partial specific volume of the biopolymer and ps is the solvent density. Making use of the relationship between the thermodynamic activity of the biopolymer in solution and its osmotic second virial coefficient (see chapter 3 for more details), one can get the following relationship (Deszczynski et al., 2006) ... 

Nonlinear regression analysis of the dependence of c2(r) upon (r), a transform of the radial distance i leads to evaluation of the reference thermodynamic activity, Mzzz(rf), and the osmotic second virial coefficient, Au/Mi, expressed on a weight basis (litre/g) rather than a molar basis (litre/mol). Furthermore, the values of Miy v/ and ps can be obtained by curve-fitting the sedimentation equilibrium distribution for low biopolymer concentrations (M fr) cf for all r) to the equations (5.36) and (5.37) in order to deduce the quantity [Mj( 1 - v/ps)] from the coefficient of the exponent (Winzor et al., 2001 Deszczynski et al., 2006). 

In Equations 2, 4, and 6, ax represents thermodynamic activities based on molar concentrations Cj of the species indicated, y represents mean ionic activity coefficients, i/ha is the activity coefficient of HA(S) molecules, and the activity of water is chosen to be one in all solvents. Consequently values of K, AG°, and AS° are based on these choices regarding standard states. 

The basis of the ideal solution model is that the thermodynamic activities of the components are the same as their mole fractions. Implicit in this assumption is the idea that the activity coefficients are equal to unity. This is at best an approximation and has been found to be invalid in most cases. Solutions in which activity coefficients are taken into account are referred to as "real" solutions and are described by equation 3.4. 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

Where K is the equilibrium constant and a-, is the thermodynamical activity of each entity (macro and microcomponent) involved in the equilibrium. Equation 3 can also be expressed in terms of concentrations (corrected with activity coefficients). 

---