Gibbs energy partial molar functions

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... 

Corresponding partial molar Gibbs energies for 500 °C were calculated using Eq. (3.33). From the temperature dependence of the potentials partial molar entropies were calculated using Eq. (3.36). Finally, partial molar enthalpies were obtained using Eq. (3.37). Values of the partial molar functions of Ag as a function of composition are summarized in Table 3.5. 

An equation for the activity coefficient can be obtained by fitting an appropriate equation to the experimental data. Rather than fitting each individual activity coefficient to its own function, the preferred procedure is to fit the excess Gibbs energy as a function Xi. The activity coefficients are obtained from this fit by noting from eg. that In is in fact a partial molar property specifically, it is the partial molar... 

Figure (5-V. Schematic represen la lion of the molar Gibbs energy as a function of the volume fraction of the solute for a partially miscible system.

First, observe that no real mixtures have this simple an equation for Gibbs energy as a function of composition however, this equation leads to simple mathematics, so please bear with it. Here we know that we have two relations to satisfy, namely that the partial molar Gibbs energy of a is the same in both phases and that the partial molar Gibbs energy of /3 is the same in both phases. 

In the case of reciprocal systems, the modelling of the solution can be simplified to some degree. The partial molar Gibbs energy of mixing of a neutral component, for example AC, is obtained by differentiation with respect to the number of AC neutral entities. In general, the partial derivative of any thermodynamic function Y for a component AaCc is given by... 

Before discussing all these biopolymer applications, we first take this opportunity to remind the reader that, in general, any thermodynamic variable can be expressed as the sum of two functions, one of which depends only on the temperature and pressure, and another which depends on the system composition (expressed as the mole fraction xt of the /-component). Therefore, for example, the chemical potential fM of the /-component of the system at constant temperature T and pressure p (the general experimental conditions), /. e., partial molar Gibbs free energy (dG/dn TtP may be expressed as (Prigogine and Defay, 1954) ... 

The standard partial molar Gibbs free energy of solution is related to the enthalpy and entropy functions at the column temperature T by the expression... 

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-... 

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

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. 

Refer to Eq. (2) where the chemical potential is given as a function G at constant T, P, and composition rij. Here, the subscript j indicates that all compositions but rii are to be held constant along with T and P. Thus, Eq. (2) defines the partial molar Gibbs free energy G. 

It is sometimes convenient to reformulate the generalized diffusional driving forces dg either in terms of mass or molar functions, using the partial mass Gibbs free energy definition, Gg = hg —Tsg, and the chain rule of partial differentiation assuming that the chemical potential (i.e., /Xg = Gg) is a function of temperature, pressure and concentration (Slattery [89], sect. 8.4). dg can then be expressed in several useful forms as listed below. Expressing the thermodynamic functions on a mass basis we may write ... 

To characterize the thermodynamic behavior of the components in a solution, it is necessary to use the concept of partial molar or partial specific functions. The partial molar quantities most commonly encountered in the thermodynamics of polymer solutions 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... 

Here the first two derivatives follow from Eqs. 6.2-12 for the pure fluid, and the last from the definition of the partial molar Gibbs energy. Historically, the partial molar Gibbs energy has been called the chemical potential and designated by the symbol Since the enthalpy can be written as a function of entropy and pressure (see Eq. 

The fugacity function has been.introduced because its relation to the Gibbs energy makes it useful in phase equilibrium calculations. The present criterion for equilibrium between two phases is that G- = Gf for all species i, with the restriction that the temperature and pressure be constant and equal in both phases. Using Eqs. 9.2-10 and the equality of partial molar Gibbs free energies yields... 

The equations developed for the partial molar volume and enthalpy can be generalized to all state variables, but given the importance of the Gibbs free energy function, it will be convenient to have some of these equations in their free energy form here. 

Figure 6-12. Schematic representation of the molar Gibbs energy (above) and the demixing temperature (below) as a function of the volume fraction of the solute for a partially miscible system, st. Stable region m, metastable region u, instable region b, binodals, sp, spinodals Ti, temperature for which the upper diagram is applicable.

For an alloy one has to describe the mixture of two or more metals in one phase. This can be done by the introduction of partial molar thermodynamic functions for each component. This will be described for a binary alloy A B, . The partial molar Gibbs energy for component A is defined for the gas phase. 

---