Partial molar property free energy

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

Although the partial molar Gibbs free energy (G<) or the chemical potential (pi) defined by the equation given below pertains to the individual components of the system, it is a property of the system as a whole. The value of the partial molar free energy depends not only the nature of the particular substance in question but also on the nature and relative amounts of the other components present as well. 

Partial molar volumes are of interest in part through their thermodynamic connection with other partial molar quantities such as partial molar Gibbs free energy, known also as chemical potential. An important property of chemical potential is that for any given component it is equal for all phases that are in equilibrium with each other. Gonsider a system... 

Those properties that depend only on the concentration of solute molecules and not on the nature of the solute are called colligative. A colligative property is also a measure of the chemical potential (partial molar Gibbs free energy) of the solvent in the solution. We consider ideal solutions first and then show how allowances are made for real solutions. 

The most important partial molar property is the chemical potential. It is the partial molar Gibbs free energy and is given by... 

The gas-liquid chromatography is a convenient technique for studying the thermodynamic properties of liquid crystals and liquid crystalline solutions. The basis for such applications is the following relation between the activity coefficient 7 and the partial molar excess free energy Gf of the solute at infinite dilution... 

Derivation of a partial molar Helmholtz free energy equation for an ideal solution will provide a tool by which ideal and real solution behavior can be differentiated. Specifically, we will make use of the fact that the partial molar enthalpy of a real solution will depend on the type and eoncentration of solutes in a solution while for an ideal solution, the partial molar enthalpy for a solute is independent of the solution composition [18]. As a brief proof of this ideal solution property, consider the defining Eq. (12) for the chemical potential of a solute, Y y, in an ideal solution ... 

From earlier discussions, it was found that this free energy difference represents a difference in screened and unscreened partial molar Helmholtz free energies. Thus, the activity coefficient can be seen to result from the partial molar free energy, which is due to the screening of the counter ion atmosphere surrounding each ion in any real solution. This definition is consistent with the view that the activity coefficient predicts electrostatic screening in real solutions or the difference between real and ideal solution properties. 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Hence, for a pure substance, the chemical potential is a measure of its molar Gibbs free energy. We next want to describe the chemical potential for a component in a mixture, but to do so, we first need to define and describe a quantity known as a partial molar property. 

We will follow the IUPAC recommendation that surface properties per unit surface area be represented by the lower case (g = Gibbs free energy, u = energy, h = enthalpy, etc.) with a superscript.s designating that the property is for the surface. The quantities gs,us,hs... for the surface are in many ways comparable to molar properties (or partial molar properties for mixtures) in the bulk phase. 

When the partial molar property Gi is the partial molar free energy fu, equation (26.8) becomes... 

Since any partial molar property of a pure substance is simply the corresponding molar property, the chemical potential of a component i in pure form, denoted by p°, is evidently equal to the molar Gibbs free energy G° of pure component i at the same temperature and pressure. 

The chemical potential /z, is a partial molar property of the Gibbs free energy because, as illustrated by the mole number coefficient of G in equation (29-24r/), temperature, pressure, and all other mole numbers are held constant during differentiation with respect to Ni. 

One of the most important consequences of Euler s integral theorem, as applied to stability criteria and phase separation, is the expansion of the extensive Gibbs free energy of mixing for a multicomponent mixture in terms of partial molar properties. This result is employed to analyze chemical stability of a binary mixture. 

Extension to a nonideal system Minimization of free energy Thermodynamics of reactions in solution Partial molar properties Medium and substituent effects on standard free energy change, equilibrium constant, and activity coefficient General considerations Solvent and solute operators Comments... 

We have seen in Chapter 11 that all partial molar properties must satisfy the Gibbs-Duhem equation, Eq. 11.6.3, which when applied to the molar excess Gibbs free energy yields (Problem 13.52) ... 

---