Partial molar quantities Gibbs free energy

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

We divide by Avogadro s number to convert the partial molar Gibbs free energy to a molecular quantity, and the minus sign enters because the force and the gradient are in opposing directions. Recalling the definition of chemical potential [Eq. (8.13)], we write jUj + RT In aj = ii2 + RT In 7jC, where aj... 

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. 

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. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gibbs free energy or chemical potential of the solute in solution. This chemical potential reflects the conformational degrees of freedom of the solute and the solution conditions (temperature, pressure, and solvent composition) and provides the driving force for solute conformational transitions in solution. For a simple solute with no internal structure (i.e., no intramolecular degrees of freedom), this chemical potential can be expressed as... 

The last member of Equation (2) shows that n, is the partial molar quantity associated with the Gibbs free energy, G. Euler s theorem gives then... 

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. 

This shows that the chemical potential of a component is just its partial molar Gibbs free energy. Note that the definitions of the chemical potential in terms of other thermodynamic variables, given in Chapter 6, Eq. (8), are not partial molar quantities because pressure and temperature are not the variables held constant in these derivatives. 

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... 

The chemical potential is the partial molar Gibbs free energy. Partial molar quantities figure importantly in the theory of solutions and are defined at constant temperature and pressure thus, the Gibbs free energy is a natural state function for their derivation. As an example, the partial molar volume is found from the Maxwell relation... 

Consider a physical property (such as the total Gibbs free energy G) of a continuous mixture, the value of which depends on the composition of the mixture. Because the latter is a function of, say, the mole distribution n(x), one has a mapping from a function to (in this case) a scalar quantity G, which is expressed by saying that G is given by afunctional of n(x). [One could equally well consider the mass distribution function m(x), and consequently one would have partial mass properties rather than partial molar ones.] We use z for the label x when in-... 

In an earlier section the free energy of a phase and the free energy of a total system were discussed generally in terms of the potentials (e.g., equation 48). With the definition of the chemical potential as a function of activity in hand, we will now consider the Gibbs energy of a system. In a similar fashion, the enthalpy and entropy of a system can be computed using the partial molar quantities and the mole numbers of each phase. 

The partial molar quantities most commonly encountered in the thermodynamics of polymer lutions are partial molar volume Vi and partial molar Gibbs free energy Gi- The latter quantity is of special significance since it is identical to the quantity called chemical potential, pi, defined by... 

Thermodynamically, if there is an equilibrium between a solution and a solid state, the Gibbs free energies AG of a polymer species in solution and in the solid state are equal. The equilibrium concentration of the polymer species in solution is then called the saturated concentration, Cs. In the case of a polydisperse polymer, there will be polymer species with different MWs, both in solution and in the solid state, all of which will feature unique energies and saturated concentrations. To describe their contributions, chemical potentials of species /i, are used, which are partial molar quantities and represent the change in the overall Gibbs energy of the system upon addition of one mole of the species in question. 

The partial molar quantities of importance for polymer solutions are the partial molar Gibbs free energy (G,), the chemical potential (/i,), and the partial molar volume (F,). 

Solubility of water in chlorine. Expressing the free energy of a system in terms of partial molar quantities (chemical potentials) and comparing the result of differentiation of the whole function with an expression for the rate of change of free energy with composition leads to what is known as the Gibbs-Duhem relation [55,56]. The Gibbs-Duhem equation can be written as... 

For example, you could get p, by measuring either how U depends on Nj with V and S held constant, or how G depends on N, with T and p constant. However, partial molar quantities are defined specifically to be quantities measured at constant T and p. So only the derivative of the Gibbs free energy dGIdrij is called a partial molar quantity. 

Let us note here some of the basic principles underlying models, theories, and calculations of hydrophobic effects. The quantity of first interest is the free energy associated with the interaction of the solute with the aqueous environment. This is the chemical potential or partial molar Gibbs free energy of the solute. This quantity provides a driving force for rearranging molecules in thermodynamic systems and many quantities of interest are fundamentally connected to this free energy. Consider first an atomic solute of type A. For example, perhaps A = Ar. The chemical potential at extreme dilution may be expressed as... 

Such quantities Z, are called partial molar quantities and with polymer solutions we are normally concerned with partial molar Gibbs free energies Gi and partial molar volumes V",. In single component systems, such as pure solvent, the partial molar quantities are identical to the corresponding molar quantity which we will denote by a lower case symbol, z,. 

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