Excess thermodynamic functions 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,... 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

We have now seen that for real non-ideal solutions all the thermodynamic properties such as G, S, H, V and the internal energy U can differ significantly from the ideal values. This deviation from ideality can be conveniently expressed as a difference from the ideal quantities. The differences are called excess thermodynamic functions ... 

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... 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

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. 

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. 

The data for Hultgren, Orr, Anderson, and Kelley s compilation were prepared between 1955 and 1963. Information on 65 elements and 167 alloy systems is presented, and selected values for heat capacity, entropy, enthalpy, free energy functions , and vapour pressures of phases are given in tabular form. For alloys, the preferred values of integral, partial, and excess thermodynamic properties are listed or are presented as analytical functions. Phase diagrams and graphs are also included. 

For calculation of the equilibrium compositions of the liquid phase either the equilibrium constants of the dissociation and polycondensation reactions have to be known or they can be computed by methods which use the approach of minimizing Gibbs free energy [200-202]. In addition, ab initio modeling techniques such as density functional theory (DFT) in combination with reactive molecular dynamic (MD) simulations could be used. Once the liquid phase system is modeled, there are in principle two options to describe the vapor-liquid equilibrium. Either equations of state (EOS) or excess Gibbs free energy models (g -models) may be used to describe the thermodynamics of the liquid... 

Values of the thermodynamic functions based on experiments for the two binary systems H20-NaCl and H2O-CO2 in the fluid one-phase region at high temperatures and pressures have not yet been sufficiently determined. New work in this field is being done at present. As one example. Fig, 7 shows partial mola volumes of NaCl in H2O which were calculated recently (17) from a critical compilation of existing data. Fig. 8 gives excess Gibbs free energy values of H2O-CO2 mixtures for two supercritical temperatures. ... 

One can now introduce the notion of excess functions which are defined as the difference between the thermodynamic function of the real mixture and the corresponding function of the ideal mixture. In particular, the excess chemical potential juf and the molar excess Gibbs energy are given by ... 

The influence of the chemical composition in (5) can be derived using a simplified version of Barker s lattice theory [55]. The most important consequence of (5) is the fact that the segment-molar excess Gibbs free energy of mixing and, hence, the activity coefficients depend only on the average value (yvv) of the distribution function, but not on the distribution function itself. In continuous thermodynamics, the phase equilibrium conditions read ... 

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