Excess functions definition

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

For solutions (or rather mixtures) of s at higher concentrations beyond the realms of Henry s law, we depart from the traditional notion of solvation and must use the definition as presented in section 7.2. There exists a variety of data which measures the extent of deviation from dilute ideal solutions. These include tables of activity coefficients, osmotic coefficients, and excess functions. All of these may be used to compute solvation thermodynamic quantities. 

Partial molar properties play a central role in phase-equilibrium thermodynamics, and ii is convenient to broaden their definition to include partial molar residua) junctions and partial molar excess functions. Hence, we define, analogous to Eq- (1-2-5),... 

By choosing, following the procedure of Gibbs, and to make the excess quantities F and f equal to zero, only independent variables are retained in this equation. Then the definitions of the excess functions may be considered to imply that the actual properties of the system are compared to the properties of a hypothetical system with the volume V — -t- F " in which... 

Here Affi and dSi are the relative partial molar heats of dilution such that AGi=Affi-TASi=(fii—iJ.i. The use of a squared term in these definitions is simply recognition that the excess functions must involve two-body (and higher) interactions. Equations (3.31) and (3.32) when coupled with equation (3.30) yield... 

Section II. Because this evolution equation (5.112) also can be derived by requiring the excess function to be positive definite [19], we call this form of the theory the positive (or partition) function renormalization group (PFRG). 

Again an important aspect in this definition is the choice of the independent variables. From eq 2.76 two different definitions of excess functions can be obtained ... 

The excess functions used here are in excess of those which apply to a particular kind of ideal solution viz. that (essentially) given by Raoult s law. This choice of Ideality is arbitrary and for some situations a different definition of ideal solution may be more suitable. Further, choosing (essentially) Raoult s law as our definition of an ideal solution, we are naturally led to Che use of mole fraction x as our choice of composition variable. That is not necessarily the best choice and there are several cases (notably, polymer solutions and solutions of electrolytes) where other measures of composition are much more convenient. 

The definition of the terms can be found in Refs. 91, 92. The excess chemical potential, computed from the direct correlation function of the... 

Having established the definitions and conventions for the activity function and for the excess Gibbs function in Chapter 16, we are in a position to understand the experimental methods that have been used to determine numeric values of these quantities. 

The regular solution approximation is introduced by assuming definition) that the excess entropy of mixing is zero. This requires that the excess free energy equal the excess enthalpy of mixing. For binary mixtures the excess enthalpy of mixing is ordinarily represented by a function of the form... 

The definition of surface excess, on the other hand, starts with the concentration profile, but involves an integration between limits Kq. (6.65)]. Once the integration is done and limits are inserted, one obtains a number (so many moles per square centimeter) and loses all knowledge of the function c((x). In other words, after the integration is carried out and the surface excess evaluated, the concentration profile... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

HMG CoA reductase inhibitors can be associated with small rises in alanine transaminase activity, but have not been definitely associated with severe morbidity involving altered hepatic function. The results of randomized trials do not suggest that statins in standard doses are hepato-toxic. In none of the large randomized studies in which standard doses were assessed (atorvastatin 10 mg/day, fluvastatin 40-80 mg/day, pravastatin 40 mg/day, simvastatin 20-40 mg/day) was there any clear excess risk of hepatitis or any other serious liver-related adverse events. Long-term large randomized trials have confirmed an excess of persistent rises in transaminases with atorvastatin 80 mg/day compared with lower doses or placebo, and similarly some excess with simvastatin 80 mg/day, but hepatitis and liver failure were not reported (4). 

In the previous sections concerning reference and standard states we have developed expressions for the thermodynamic functions in terms of the components of the solution. The equations derived and the definitions of the reference and standard states for components are the same in terms of species when reactions take place in the system so that other species, in addition to the components, are present. Experimental studies of such systems and the thermodynamic treatment of the data in terms of the components yield the values of the excess thermodynamic quantities as functions of the temperature, pressure, and composition variables. However, no information is obtained concerning the equilibrium constants for the chemical reactions, and no correlations of the observed quantities with theoretical concepts are possible. Such information can be obtained and correlations made when the thermodynamic functions are expressed in terms of the species actually present or assumed to be present. The methods that are used are discussed in Chapter 11. Here, general relations concerning the expressions for the thermodynamic functions in terms of species and certain problems concerning the reference states are discussed. 

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