Excess thermodynamic functions free energy

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

When non-ideal liquid solutions are considered, we use excess thermodynamic functions, which are defined as the differences between the actual thermodynamic mixing parameters and the corresponding values for an ideal mixture. For constant temperature, pressure and molar fractions, excess Gibbs free energy is given as... 

Excess thermodynamic functions show the deviations from ideal solution behavior and there is of course a relation between GE and the activity coefficients. Similar to Equation (369), if we write the actual Gibbs free energy of mixing (AGmi[)actuai in terms of activities,... 

The basic problem of the conversion from the LR to the MM system is in the relation of A to G , the excess Gibbs free energy of a solution per kilogram of solvent, which is used in Section 3. The principal features of the two systems of excess thermodynamic functions are summarized in Table 1. 

Now let us consider the non-electrolytes. Here we have two very distinct types of behaviour. There are the so-called hydro-phobic, and the hydrophilic effects. The hydrophobic effect can be shown schematically from a consideration of the thermodynamics of hydrocarbon solutions. Usually a non-ideal solution arises because the two components either strongly attract each other or strongly repel each other the effects are shown in the enthalpy. Figure 8 shows various types of behaviour, as reflected in the excess thermodynamic functions (Rowlinson, 1969). The drawn out lines are free energies, the broken lines are enthalpies and the dotted lines are the entropy curves. A positive free energy means a positive deviation from ideal behaviour. In normal systems AG follows the AH curve fairly benzene-MeOH. In... 

This relation can be inverted to yield fip(n) as function of c.p(nr) (eh Appendix A 4.1). Substituting the result into 77[/xp], we find the osmotic pressure as function of the concentration, which is the standard form of the osmotic equation of state. Also all the other thermodynamic quantities can be calculated from n[fip]. The excess free energy due to the solute, for instance, takes the form... 

The difference in thermodynamic functions between a non-ideal solution and a comparative perfect solution is called in general the thermodynamic excess function. In addition to the excess free enthalpy gE, other excess functions may also be defined such as excess entropy sE, excess enthalpy hE, excess volume vE, and excess free energy fE per mole of a non-ideal binary solution. These excess functions can be derived as partial derivatives of the excess free enthalpy gE in the following. 

Burton (39) has calculated properties of Ar clusters containing up to 87 atoms. He finds that the vibrational entropy per atom becomes constant for about 25 atoms. The entropy per atom for spherical face-centered cubic structures exceeds that of an infinite crystal and reaches a maximum between 19 and 43 atoms. An expression for the free energy of the cluster as a function of size was derived and shows that the excess free energy per atom increases with cluster size up to the largest clusters calculated. Although this approach yields valuable thermodynamic information on small clusters, it does not give electronic information. 

The thermodynamic characteristics of solutions are often expressed by means of excess functions. These are the amounts by which the free energy, entropy, enthalpy, etc. exceed those of a hypothetical ideal solution of the same composition (Denbigh, 1981). The excess free energy is closely related to the activity coefficients. The total free enthalpy (Gibbs free energy) of a system is ... 

The first two terms here constitute the van der Waals approximation as discussed above. The succeeding term is a correction that lowers this free energy. The thermodynamic excess chemical potential is then obtained by averaging the Boltzmann factor of this conditional result using the isolated solute distribution function sj 0l ). 

Equations 8 and 16 provide the basic thermodynamic equations which can predict the dependence on salinity, of the equilibrium radius r and the volume fraction at the transition between the region in which a microenulsion phase forms alone and that in which it coexists with an excess dispersed phase. Any addition to the system of excess dispersed phase having the same composition as the globules will change neither nor r in the microenulsion, as soon as the transition point is reached. To carry out such calculations, explicit expressions are needed for the interfacial tensionY as a function of the concentrations of surfactant and cosurfactant in the continuous phase, of salinity and radius r, as well as expressions for C and for the free energy Af. The interfacial tension depends on the radius for the following two reasons If the radius were increased at con-... 

The effect of substitutional impurities on the stability and aqueous solubility of a variety of solids is investigated. Stoichiometric saturation, primary saturation and thermodynamic equilibrium solubilities are compared to pure phase solubilities. Contour plots of pure phase saturation indices (SI) are drawn at minimum stoichiometric saturation, as a function of the amount of substitution and of the excess-free-energy of the substitution. SI plots drawn for the major component of a binary solid-solution generally show little deviation from pure phase solubility except at trace component fractions greater than 1%. In contrast, trace component SI plots reveal that aqueous solutions at minimum stoichiometric saturation can achieve considerable supersaturation with respect to the pure trace-component end-member solid, in cases where the major component is more soluble than the trace. 

Fig. 17.19. (a) The relative surface excess for two flat plates as a function of the distance of separation (b) characteristic free energy of interaction arising from depletion effects according to the thermodynamic approach. 

In defining surface thermodynamic functions, the difficulty over the absence of a unique surface plane is circumvented by defining these functions in terms of surface excess— total minus bulk value of the property concerned [46,47]. Thus the Gibbs surface free energy is defined as... 

Here A - Ajg is the excess Helmholtz free energy with respect to an ideal gas at the same temperature, volume, and number density of each species. Thus, because of the minus sign, the factor kT, and the factor V in the first equality, si can be regarded as a negative dimensionless excess free energy density for the system. Since both A and Aig are extensive thermodynamic properties of the system, A/V and A JV are functions only of the intensive independent variables. Thus si has been expressed as a function of only the temperature and the number density of each species. (Moreover, we have chosen to use j8 = l/Ztr, rather than T, as the independent temperature variable.) It is this quantity si which has a simple representation in terms of graphs, which will be given below. If si can be calculated (exactly or approximately), this leads to (exact or approximate) results for A and hence for all the thermodynamic properties. 

For single electrolytes in water at 25 C the experimental excess enthalpies have been tabulated by Parker and the various free energy functions have been given in terms of parameterized equations by Pitzer. Other thermodynamic excess function data have not been so systematically collected. 

H. L. Friedman, Lewis-Randall to McMillan-Mayer conversion for the thermodynamic excess functions of solutions. Part I. Partial free energy coefficients, /. Solution Chem. 1,387 (1972). 

---