Partial molar properties among

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

RELATIONSHIPS AMONG PARTIAL MOLAR PROPERTIES OF A SINGLE COMPONENT... 

For characterization of the interrelation between the composition and extensive properties of the solution the outstanding American physico-chemist Gilbert Newton Lewis (1875-1946) introduced additional intensive parameters under the common mme partial molar quantity. Among them are partial molar volume, partial molar heat capacity, chemical potential, etc. 

The partial derivative is a linear operator therefore, the partial molar derivative (3.4.5) may be applied to all those expressions given in 3.2, producing partial molar versions of the fundamental equations. In particular, when we apply the partial molar derivative to the integrated forms (3.2.29)-(3.2.31) of the fundamental equations, we obtain the following important relations among partial molar properties ... 

KP and v can, in contrast to kp, not be determined via the concentration gradient for binary and ternary mixed micelles, because for the calculation of the Nemstian distribution a constant CMC and an almost constant partial molar volume must be assumed. The calculation of aggregation constants of simple bile salt systems based on Eq. (4) yields similar results (Fig. 8b). Assuming the formation of several concurrent complexes, a brutto stability constant can be calculated. For each application of any tenside, suitable markers have to be found. The completeness of dissolution in the micellar phase is, among other parameters, dependent on the pH value and the ionic strength of the counterions. Therefore, the displacement method should be used, which is not dependent on the chemical solubilization properties of markers. For electrophoretic MACE studies, it is advantageous for the micellar constitution (structure of micelle, type of phase micellar or lamellar) to be known for the relevant range of concentrations (surfactant, lipids). 

In a similar fashion a large collection of relations among the partial molar quantities can be developed. For example, since dCj cT)p M — —S- for a pure fluid, one can easily show that (8G /dT)pj j. — —5j for a mixture. In fact, by extending this argument to other mixture properties, one finds that for each relationship among the thermodynamic variables in a pure fluid, there exists an identical relationship for the partial molar thermodynamic properties in a mixture ... 

These mathematical properties have a simple practical result they ensure that the same relationships can be written among all forms of a thermodynamic function, including molar, total, residual, partial molar, or residual partial molar. 

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