Gibbs function partial molar

In a similar way, we now look at the molar Gibbs function of each component i within a mixture. Component i could be a contaminant. But because i is only one part of a system, we call the value of Gm for material i the partial molar Gibbs function. The partial molar Gibbs function is also called the chemical potential, and is symbolized with the Greek letter mu, p. 

Figure 5.18 depicts graphically the relationship in Equation (5.12), and shows the partial molar Gibbs function of the host material as a function of temperature. We first consider the heavy bold lines, which relate to a pure host material, i.e. before contamination. The figure clearly shows two bold lines, one each for the material when solid and another at higher temperatures for the... 

Figure 5.19 The chemical potential /j, (the partial molar Gibbs function) of a species in a mixture is obtained as the slope of a graph of Gibbs function G as a function of composition...

Therefore the state in which the solute has a partial molar Gibbs function of p.2 can be found from the following limit, because 72 approaches unity as m2 approaches zero ... 

It should be pointed out that Gi" and Si" Sj, where G and S- are the partial molar Gibbs function and partial molar entropy of the solute in an ideal solution and G and 5 are the molar Gibbs function and molar entropy of the pure liquid solute, respectively. However, when usin the above mentioned choice of standard states it also holds that AG, (LLC) = AG -AGll, and 4G, (GLC) = 4G s + G d, so that even here... 

Go is the partial molar Gibbs function associated with cavity formation and Gi is the partial molar Gibbs function for the solute-solvent interaction. Pierotti used an expression for Go derived from scaled-particle theory assuming a hard-sphere potential. Go is thus a function of pi, the solvent number density, and of the diameters of the solvent and of the cavity. Ui and hence Gi were estimated from the expression ... 

This condition of chemical equilibrium can also be expressed in terms of chemical potentials /x (the partial molar Gibbs functions), and, for example, for an A —> B reaction we would obtain ... 

In this equation mi, si and v are the partial molar energy, entropy and volume of the species identified by subscript 1. Since the total Gibbs function, G, is the sum of the type of functions shown in (19), it is as if each species has its own private Gibbs function. Equation 19 shows that chemical potential has the same form as the Gibbs function, hence it is also called the partial molar Gibbs function. 

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... 

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... 

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. 

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. 

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... 

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. 

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. 

Values of partial molar thermodynamic functions of silver and gold in the Ag Au alloy as a function of composition for 500 °C. The values for gold were obtained by integration of the Gibbs-Duhem equation (Section 3.2.4, data for 0.169 were omitted for the integration of the... 

From first the excess function and then the partial molar Gibbs energy of gold can... 

Figure 3.14 Excess function of the partial molar Gibbs energy of silver in the silver-gold alloy plotted as function of y in Agj,Au. The solid line is a logarithmic fit of the experimental data.

Figure 3.16 Partial molar Gibbs energy of Au in the alloy Ag u, as function of the mole fraction of Ag. Two sets of values are shown The first set (squares) was calculated with the Gibbs-Duhem equation, as described in Section 3.2.3, and the second set of data (circles) was calculated with the Duhem-Margules equation, temperature 500 °C.

The formation reaction (Eq. (4.62)) can be used to calculate thermodynamic functions of the UPD phase. From the cell voltage AB = E — Eq one can calculate the partial molar Gibbs energy AG pp of the UPD modification of the metal B... 

In order to find this function, we only need to know the chemical potentials, because they are the partial molar Gibbs energies, and we have ... 

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