Thermodynamic properties, solute excess

Figure 9. Isotope effects on the solute excess thermodynamic properties of calcium chloride solutions (59).

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase... 

This should certainly be a good approximation as the isotopomers are quite similar, one to the other. Even so we will improve on it later when we consider (the quite small) nonideality of isotopomer solutions and their excess thermodynamic properties. For Raoult s law solutions... 

In real solutions, we describe the excess thermodynamic property Z. It is the excess in Z over that for the ideal solution. That is,... 

Chapters 17 and 18 use thermodynamics to describe solutions, with nonelectrolyte solutions described in Chapter 17 and electrolyte solutions described in Chapter 18. Chapter 17 focuses on the excess thermodynamic properties, with the properties of the ideal and regular solution compared with the real solution. Deviations from ideal solution behavior are correlated with the type of interactions in the liquid mixture, and extensions are made to systems with (liquid + liquid) phase equilibrium, and (fluid -I- fluid) phase equilibrium when the mixture involves supercritical fluids. 

Thus, an excess thermodynamic property is also the difference between the thermodynamic property for mixing the real and ideal solutions. For the Gibbs free energy, this becomes, using Eq. (3) and Eq. (35) of Chapter 8,... 

FIGURE 7-7 Schematic diagram of excess thermodynamic properties as a function of solution composition (various properties can be negative, depending on the system). 

Carpenter MA, McCoimell JDC, Navrotsky A (1985) Enthalpies of ordering in the plagioclase feldspar solid solution. Geochim Cosmochim Acta 49 947-966 Carpenter MA, Salje EKH, Graeme-Barber A, Wmck B, Dove MT, Kiught KS (1998) Calibration of excess thermodynamic properties and elastic constant variations associated with the a< p phase transition in quartz. Am Mineral 83 2-22... 

An excess thermodynamic property ME is equal to the difference between the actual property M and the property for an ideal solution at the same T, P and x. Here M represents extensive properties V, U, H, Cp,... 

If valid, such a crude theory of metals should in a certain sense be analogous to the cell theory of ordinary liquids, which also involves two molecular parameters e and cr. It would then be possible to develop along the same lines a theory of metallic solutions. The excess thermodynamic properties of a mixture of two metals A and B would then depend on the differences ( a b) 3-nd (ZoA — > ob)- That this theory is very rough for pure metals is not a limitation for its applicability to mixtures it is indeed well known from the theory of non-electrolyte solutions that the properties of mixtures may be reasonably analyzed from rather crude statistical models. 

The difference between the thermodynamic properties of solution, based on K, and the analogous properties of concentration from the 101-325 kPa ideal gas state may be evaluated directly from the solute activity coefficient and can be identified with the excess thermodynamic properties of solution ... 

Many of the theories and models described in this section were developed for the excess thermodynamic properties of solutions, including not only the excess partial molar volume, but also other excess properties. In the following subsections we have restricted the discussion to the volumetric properties of aqueous systems. 

In the last decades the progress of statistical mechanics has opened the possibility of treating quantitatively the effect of ionic interactions at the Mc-Millan Mayer level for clusters [8] [9] [10]. It is possible to include the non ideal contribution in the statistical formulation of the thermodynamic properties of ionic solutions [11] [12] [13]. This can be done combining the concept of ionic association to the evaluation of excess thermodynamic properties. 

Gladden, J. K. Ghaffari, F. Excess thermodynamic properties of ethylene diamine - ethylene glycol solutions at 25.deg.C J. Chem. Eng. Data 1972,17, 468-471... 

The brownian dynamics is a solvent averaged dynamics, which allows for a dynamical representation of the static Mac-Millan Mayer hamiltonian, which has been proved to be a useful tool for the structural and excess thermodynamical properties of solutions. ... 

The simple Flory-Huggins approach and the solubility parameter concept are inadequate when tested against experimental data for polymer solutions. Even for mixtures of n-alkanes, the excess thermodynamic properties cannot be described satisfactorily - Flory et In particular, changes of volume upon mixing are excluded and observed... 

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,... 

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. 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

Fig. 2.37. Phase diagram for Ca0-Na20 Si02-(Al203)-H20 system in equilibrium with quartz at 400°C and 400 bars. Plagioclase solid solution can be represented by the albite and anorthite fields, whereas epidote is represented by clinozoisite. Note that the clinozoisite field is adjacent to the anorthite field, suggesting that fluids with high Ca/(H+) might equilibrate with excess anorthite by replacing it with epidote. The location of the albite-anorthite-epidote equilibrium point is a function of epidote and plagioclase composition and depends on the model used for calculation of the thermodynamic properties of aqueous cations (Berndt et al., 1989).

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... 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

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 order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

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