THERMODYNAMIC FUNCTIONS DERIVED FROM ACTIVITY

THERMODYNAMIC FUNCTIONS DERIVED FROM ACTIVITY COEFFICIENTS... 

In the literature, thermodynamic excess functions derived from activity coefficients at infinite dilution may be encountered [14, 46]. 

Activity coefficient is a function of the state of a mixture. An activity-coefficient equation is required to calculate the fugacities of real solutions. The interrelationship of the activity coefficients through the Gibbs-Duhem equation implies that the activity-coefficient equations of aU components are derivatives of a common thermodynamic function. Since the activity coefficient is an expression of the nonideal behavior of a component, a thermodynamic function is needed to express the nonideality of the total solution and then to obtain from it the activity-coefficient equation. 

Nitta et. al. ( 7) extended the group interaction model to thermodynamic properties of pure polar and non-polar liquids and their solutions, including energy of vaporization, pvT relations, excess properties and activity coefficients. The model is based on the cell theory with a cell partition function derived from the Carnahan-Starling equation of state for hard spheres. The lattice energy is made up of group interaction contributions. 

With the development of Equation 5.12 relating the partition function and the macroscopic properties, all of the macroscopic thermodynamic properties may be derived from Equation 5.7. For example, differentiating In E with respect to the absolute activity (A.) of./, provides the total number of guest molecules J over all the cavities i... 

The most popular lands of the diols for asymmetric synthesis are bis-secondary diols that have a C2 axis of symmetry [212]. The presence of the symmetry axis avoids the formation of diastereoisomeric esters or acetals [213], (1R, 27 )-Cyclohexanediol 1.34 (n = 1) has been used as an auxiliary in asymmetric cyclopropanation [214] and (IS, 2S)-cycloheptanediol 1.34 (n = 2) in 1,4-addition of cuprates[157], Dioxolane derivatives of 1.34 have been used for asymmetric P-ketoester alkylations [215] and cuprate 1,4-additions [216]. Linear 1,2-diols 1.35 (R = Me, i-Pr, c-CgH j, Ph) and functionalized 1,2-diols 1.36 (Y = COOalkyl, CONR 2, CH2OR ) are readily available from optically active tartaric acids 1.36 (Y = COOH). Acetals derived from these diols are valuable reagents m asymmetric synthesis [173, 213, 217], as the related 1,3-diols 1.37. Acetals of 1,3-butanediol 137 (R = Me, R = H) have also been used. When these acetals are formed from aldehydes under thermodynamic conditions, one 1,3-di-oxane stereoisomer often predominates. In this favored isomer, the substituent from the aldehyde and the methyl group from 1.37 are both in equatorial orientar... 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

Equations (24)-(26) are explicit equations for proton adsorption at the S—MO interface derived from rigorous Gibbs-Lewis thermodynamics. From a knowledge of and K 2 mole fraction of amphoteric surface sites can be calculated as a function of solution pH with the exception of the activity coefficient quotient, Z As an approximation, Z can be set to unity so that = Z =i2 ss l, which eliminates all nonideal... 

The terms Ca and Ct are defined by equations (3 9a) and (3-9) respectively. The hydrogen and hydroxyl ion activity terms have been placed in the equation to emphasize that these are assumed to be the measured quantities derived from the pH measurements. Whilst it is correct to use the hydrogen ion activity in equation (3-22), the hydrogen ion concentration is required for equation (3-22a). It is apparent, therefore that mixed constants would result from the solution of equation (3 -22). A further complication is the ionic strength which, in this case, cannot be calculated directly from the stoicheiometric concentration used, for example, in equation (3-19). To obtain the thermodynamic values, K and KI, the activity functions must be calculated from estimates of the ionic strength. How this can be achieved will now be described for dibasic acids followed by a similar method for ampholytes and diacidic bases. 

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