Thermodynamic excess functions

The excess molar thermodynamic function Z is defined as the difference in the property Zm for a real mixture and that for an ideal solution. That is, 

For the enthalpy, internal energy, and volume changes, Am,xZJjJ = 0. Hence, 

Since these mixing processes occur at constant pressure, // is the heat evolved or absorbed upon mixing. It is usually measured in a mixing calorimeter. The excess Gibbs free energy, is usually obtained from phase equilibria measurements that yield the activity of each component in the mixtureb and S is then obtained from equation (7.17). The excess volumes are usually obtained 

The extent of deviation from ideal solution behavior and hence, the magnitude and arithmetic sign of the excess function, depend upon the nature of the interactions in the mixture. We will now give some representative examples. 

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, 

Since the activity coefficient, (pf, is given in terms of molar fraction as (pf= aJX, in Equation (168), then by inserting this expression in Equation (375), the total excess Gibbs free energy for the constant temperature, pressure and mole number of other constituents can be written as 

In practice, determination of the activity coefficients of a solvent in a solution is easy, if the solute is nonvolatile. The vapor pressure of the solution and the pure solvent are measured and aA = PJP (Equation (166) applies). However, if the solute is volatile, then the partial pressure of both the solute and the solvent should be determined. 

Determination of the activity coefficients of the non-volatile solute in a solution is difficult. If electrolytes (ions) are present, the activities can be obtained from experimental electromotive force (EMF) measurements. However, for non-electrolyte and non-volatile solutes an indirect method is applied to find initially the activity of the solvent over a range of solute concentrations, and then the Gibbs-Duhem equation is integrated to find the solute activity. If the solution is saturated, then it is easy to calculate the activity coefficient 

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  

All other thermodynamic relations can be written with excess quantities in this manner, simply substituting the excess property for the usual variable. As with all thermodynamic variables, this permits us to completely determine all excess properties simply by measuring a necessary minimum number of properties. Therefore if excess volumes, free energies, and enthalpies are known for a solution at different temperatures and compositions, then all other properties can be calculated for a range of T, P, and composition. Similarly, if we have measured excess free energies over a range of temperatures, it is not necessary to measure the excess entropy it is already known from 

Relationship Between Excess Properties and the Activity Coejficient 

Excess properties are just anotlier way of representing the activity coefficient and are used because they tend to simplify notation. We now derive the relationships between the activity coefficient and excess free energy, enthalpy, entropy, and volume. 

Taking the free energy first, consider an ion i in any non-ideal solution. Starting with 

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Figure 7.2 Excess thermodynamic functions at 7= 298.15 K for. Y1C10H22 +. Y2C6H14, an example of a system where nonpolar chain-like molecules are mixed.

Figure 7.3 Excess thermodynamic functions at 7"= 318.15 K for x +. X2CH3CN, an example of a system in which polar molecules are mixed...

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. 

As the laws of dilute solution are limiting laws, they may not provide an adequate approximation at finite concentrations. For a more satisfactory treatment of solutions of finite concentrations, for which deviations from the limiting laws become appreciable, the use of new functions, the activity function and excess thermodynamic functions, is described in the following chapters. 

An alternative approach that is particularly applicable to binary solutions of nonelectrolytes is that of excess thermodynamic functions for the solution instead of activities for the components. That approach is most useful in treatments of phase equihbria and separation processes [1], and it will be discussed in Section 16.7. 

DEVIATIONS FROM IDEALITY IN TERMS OF EXCESS THERMODYNAMIC FUNCTIONS 373... 

Excess thermodynamic functions can be evaluated most readily when the vapor pressures of both solute and solvent in a solution can be measured. 

From the experimental temperature dependence of A2 (and the corresponding inferred temperature dependence of juE the other basic excess thermodynamic functions can be determined using general thermodynamic relationships. This then provides a complete thermodynamic characterization of the system as a whole. Thus, for the determination of the excess molar enthalpy of the system at constant pressure, the following equation can be used (Prigogine and Defay, 1954) ... 

The above listed excess thermodynamic functions all relate to 1 mole of solution. For the system as a whole, containing a total number of moles... 

Dhillon, M.S. (1979). Second virial coefficient and excess thermodynamic functions in benz-ene of a polymer 2-butene-l,4-diol with adipic acid. Thermochimica Acta, 31, 375-379. 

Excess thermodynamic functions for four mixtures with increasingly negative HB, (References in the text.)... 

A very interesting development of the free-volume concept has been made in some papers by Miller29,36 His main idea is that all excess thermodynamic functions... 

Figure 17.16 (a) (Liquid + liquid) phase equilibia and (b) excess thermodynamic functions at 283.15 K for jciH20 + (CjHs N). 

K , Fluid Phase Equilib., 8, 75-86 (1982) I. Nagata and Y. Kawamura, Excess Thermodynamic Functions and Complex Formation in Binary Liquid Mixtures Containing Acetonitrile , Fluid Phase Equilib., 3, 1-11 (1979). 

An alternative way of expressing the deviation from ideal solution behavior is by means of excess thermodynamic functions. These are defined as the difference between a thermodynamic property of a solution and the thermodynamic property it would have if it were an ideal solution ... 

Figure 15. Excess thermodynamic functions for various 1 1 salts in water at 298 Kj mj = molality of salt and the dotted lines indicate the behaviour predicted by the Debye-Huckel limiting law (Fortier et al., 1974).

Figure 31. Excess thermodynamic functions of mixing for ethyl alcohol + water mixtures at 298-15 K.

Figure 36. Excess thermodynamic functions for methyldiethylamine + water mixtures at 320 K (Copp, 1955).

Figure 37. Excess thermodynamic functions for acetone + water mixtures at 298 K (Wells, 1974).

Here S is the entropy and N j is the composition of the system. It is convenient to subtract the corresponding bulk phase equations pertaining to the volumes va and vp from these relations and to introduce intensive excess thermodynamic functions. The familiar result is given by... 

Equation (4.23.1) can be derived from excess thermodynamic functions (Problem 4.23.1). Equation (4.23.1) can be modified, by analogy to the van der Waals equation for gases, to (II + nzaA z) (A rib) nRT, where a represents the intermolecular attractions within the monolayer, and b represents the excluded area. 

Scatchard s procedure is the basis for calculating excess thermodynamic functions (AH , A5 , AF )—the difference between the function for the ideal solution and actual solution of the same compo-... 

J. L. Gopp and D. H. Everett. Trans. Faraday Soc. 53, 9-18 (1957). Calculations excess thermodynamic functions, water-ternary amines. 

G. Scatchard and C. L. Raymond. J. Am. Chem. Soc. 60, 1278-87 (1938). Density, vapor pressure ethanol-CHCU, 35-55 C, excess thermodynamic functions. 

Chaudhry, M. M. Van Ness, H. C. Abbott, M. M. Excess thermodynamic functions for ternary systems. 6. Total-pressure data and G for acetone—ethanol—water at 50 °C. J. Chem. Eng. Data 1980, 25, 254-257. 

Excess Thermodynamic Functions in the Region of a Critical Solution Temperature. 

It is of interest to examine the behaviour of the excess thermodynamic functions near a critical solution temperature. We start from the equation... 

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