Solution excess properties

Molar or specific value, extensive tlrenrrodyiramic property Partial property, species i in solution Excess property == M — M "... 

Ideal solution Excess properties Activity coefficient... 

Heat copacities, Ideal gas. Residual properties. Ideal solution. Excess properties. Activity coefficients. Chemical Potential and Fugacities... 

Liquid solutions are often most easily dealt with through properties that measure their deviations, not from ideal gas behavior, but from ideal solution behavior. Thus the mathematical formaUsm of excess properties is analogous to that of the residual properties. 

If M represents the molar value of any extensive thermodynamic property, an excess property is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus,... 

Excess properties have no meaning for pure species, but for species in solution. 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

Figure 4-2 displays plots of AH, AS, and AG as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3 the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressnre, and for practical pnrposes represent sohition properties at 50°C (122°F) and low pressnre (P 1 bar [14.5 psi]).

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

In the derivation of the regular solution model the vibrational contribution to the excess properties has been neglected. However, as a first approximation the vibrational contribution can be taken as independent of the interaction between the different atoms, and this contribution can be factored out of the exponential and taken into account explicitly. The partition function of the solution is then given as... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

This latter expression allows us to compute all the excess properties of dilute electrolytic solutions for instance, the excess osmotic pressure is determined by Eq. (138). The most remarkable result is of course that all these thermodynamic properties are non-anaiytic functions of the concentration ... 

In the equations developed by Reilly and Wood (15) from the cluster Integral model (1 6), y+ is calculated in complex solutions from excess properties of single salt solutions. Note that the cluster Integral approach 1s based upon terms which represent the contributions of pair-wise ion interactions 1n various types of clusters to the potential interaction energy. Then, the partition function and the excess properties of the solution can be evaluated. The procedure is akin to the vlrial expansion 1n terms of clusters. 

Jancso, G., Rebelo, L. P. N. and Van Hook, W. A. Isotope effects in solution thermodynamics excess properties in solutions of isotopomers. Chem. Rev. 93, 2645 (1993). 

Bertrand G. L., Acree W. E. Jr., and Burchfield T. (1983). Thermodynamical excess properties of multicomponent systems Representation and estimation from binary mixing data. J. Solution. Chem., 12 327-340. 

We now extend the discussion of excess properties to examples that help us to better understand the nature of interactions in a variety of nonelectrolyte mixtures. We will give examples showing temperature and pressure effects, including an example of solutions near the critical locus of the mixture and into the supercritical fluid region. 

Q is called die interaction parameter and independent of composition and temperature. Let s examine the properties of a regular solution using die concept of excess properties. 

The fundamental excess-property relation is derived in exactly the same as the fundamental residual-property relation and leads to analogous rest Equation (13.12), written for the special case of an ideal solution, is subtra from Eq. (13.12) itself, yielding ... 

An excess property is the difference between the actual property value of a solution and the ideal solution value at the same composition, temperature, and pressure. Therefore, excess properties represent the nonideal behavior of liquid mixtures. The major thermodynamic properties for ideal mixtures are... 

The activity coefficient is a measure of the deviation of liquid solutions from ideal behavior, and unity in ideal solutions. We have the definitions of excess properties of Gibbs energy, volume, and enthalpy, which are experimentally measurable... 

The ideal gas is a useful model of the behavior of gases and serves as a standard to which real gas behavior can be compared. This is formalized by the introduction of residual properties. Another useful model is the i(kal solution, which serves as a standard to which real solution behavior can be compared. This is formalized by introduction of excess properties. 

Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding... 

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. 

Equations (3.2) and (3.3) relate intermolecular interactions to measurable solution thermodynamic properties. The excess chemical potential is obtained from... 

---