Thermodynamic Properties of Ideal Solutions

Equation (17.5) is the starting point for deriving equations for AmixZJ J, the change in Zm to form an ideal mixture. For the ideal solution, 7r,/ = 1 and equation (17.5) becomes 

The change in other thermodynamic properties to form the ideal mixture is easily obtained from equation (17.6). Differentiating equation (17.6) with respect to T and with respect to p gives 

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

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Polymer solutions always exhibit large deviations from Raoult s law, though at extreme dilutions they do approach ideality. Generally however, deviation from ideal behaviour is too great to make Raoult s law of any use for describing the thermodynamic properties of polymer solutions. 

As in the nonelectrolyte case, the problem of representing the thermodynamic properties of electrolyte solutions is best regarded as that of finding a suitable expression for the non-ideal part of the chemical potential, or the excess Gibbs energy, as a function of composition, temperature, dielectric constant and any other relevant variables. 

To understand why a solute lowers the vapor pressure, we need to look at the thermodynamic properties of the solution. We saw in Section 8.2, specifically Eq. 1, that at equilibrium, and in the absence of any solute, the molar free energy of the vapor is equal to that of the pure solvent. We now need to consider the molar free energies of the solvent and the vapor when a solute is present. We shall consider only nonvolatile solutes, which do not appear in the vapor phase, and limit our considerations to ideal solutions. 

Statistical mechanics provides a bridge between the properties of atoms and molecules (microscopic view) and the thermodynmamic properties of bulk matter (macroscopic view). For example, the thermodynamic properties of ideal gases can be calculated from the atomic masses and vibrational frequencies, bond distances, and the like, of molecules. This is, in general, not possible for biochemical species in aqueous solution because these systems are very complicated from a molecular point of view. Nevertheless, statistical mechanmics does consider thermodynamic systems from a very broad point of view, that is, from the point of view of partition functions. A partition function contains all the thermodynamic information on a system. There is a different partition function... 

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 ... 

The ionization of electrolytes is clearly manifest in the thermodynamic properties of their solutions. For example, in the ideally dilute solution limit, a solution of a strong electrolyte behaves as ions, rather than molecules, interacting with solvent molecules. A NaCl solution of molality m behaves, in the limit of infinite dilution, as an ideally dilute solution of concentration 2m, as 2 mol of ions are produced from each mole of NaCl dissolved in solution. A general strong electrolyte, dissociating by the equation... 

Activities of Electrolytes.—When the solute is an electrolyte, the standard states for the ions are chosen, in the manner previously indicated, as a hypothetical ideal solution of unit activity in this solution the thermodynamic properties of the solute, e.g., the partial molal heat content, heat capacity, volume, etc., will be those of a real solution at infinite dilution, i.e., when it behaves ideally. With this definition of the standard state the activity of an ion becomes equal to its concentration at infinite dilution. 

The difference between the thermodynamic function of mixing (denoted by superscript M) for an actual system, and the value corresponding to an ideal solution at the same T and jp, is called the thermodynamic excess function (denoted by superscript E). This quantity represents the excess (positive or negative) of a given thermodynamic property of the solution, over that in the ideal reference solution. nnhn< ... 

Debye-Huckel theory. A theory advanced in 1923 for quantitatively predicting the deviations from ideality of dilute electrolytic solutions. It involves the assumption that every ion in a solution is surrounded by an ion atmosphere of opposite charge. Results deduced from this theory have been verified for dilute solutions of strong electrolytes, and it provides a means of extrapolating the thermodynamic properties of electrolytic solutions to infinite dilution. 

Using Flory-Huggins theory it is possible to account for the equilibrium thermodynamic properties of polymer solutions, particularly the fact that polymer solutions show major deviations from ideal solution behavior, as for example, the vapor pressure of solvent above a polymer solution invariably is very much lower than predicted from Raoult s law. The theory also accounts for the phase separation and fractionation behavior of polymer solutions, melting point depressions in crystalline polymers, and swelling of polymer networks. However, the theory is only able to predict general trends and fails to achieve precise agreement with experimental data. 

When analyzing thermodynamic properties of polymer solutions it is sufficient to consider only one of the components, which for reasons of simplicity normally is the solvent. It is convenient to separate Eq. (3.88) for solvent (i = 1) into ideal and nonideal (or excess) contributions by defining... 

From these results, the thermodynamic properties of the solutions may be obtained within the McMillan-Mayer approximation i.e. treating the dilute solution as a quasi-ideal gas, and looking at deviations from this model solely in terms of ion-ion interactions, we have... 

In the following sections we will quantify some of the thermodynamic properties of mechanical mixtures and ideal and non-ideal solutions. As we detail the properties of ideal solutions, it will become clear that they are strictly hypothetical another thermodynamic concept, like true equilibrium , which is a limiting state for real systems. Ideal solutions, in other words, are another part of the thermodynamic model, not of reality. It is a useful concept, because real solutions can be compared to the hypothetical ideal solution and any differences described by using correction factors (activity coefficients) in the equations describing ideal behavior. These correction factors can either be estimated theoretically or determined by actually measuring the difference between the predicted (ideal) and actual behavior of real solutions. 

In this chapter, we apply some of the general principles developed heretofore to a study of the bulk thermodynamic properties of nonelectrolyte solutions. In Sec. 11-1 we discuss conventions for the description of chemical potentials in nonelectrolyte solutions and introduce the concept of an ideal component. In Sec. 11-2, we demonstrate how the concept of solution molecular weight can be introduced into thermodynamics in a natural fashion. Section 11-3 is devoted to a study of the properties of ideal solutions. In Sec. 11-4, we discuss the properties of solutions that can be considered to be ideal when they are dilute but are not necessarily ideal when they are more concentrated. In Sec. 11-5, regular solutions are defined and some of their properties are derived. Section 11-6 is devoted to a study of some of the approximations that prove useful in the derivation of the properties of real solutions. Finally, in Sec. 11-7, some of the experimental techniques utilized for the measurement of chemical potentials and activity coefficients of components in solution are described. 

To determine the thermodynamic properties of polymer solutions, must be correlated with the statistical thermodynamic theories. According to the Hory-Huggius theory, for non-ideal polymer solutions the solute activity is given by eqn (3.91) which becomes ... 

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. 

We ha c developed a model for the thermodynamic properties of ideal and regular solutions. Two components A and B will tend to mix because of the favorable entropy resulting from the many different ways of interspersing A and B particles. The degree of mixing also depends on w hether the. 46 attractions are stronger or weaker than the AA and BB attractions. In the next chapters we will apply this model to the properties of solutions. 

The thermodynamic properties of the solution—such as the equilibrium constants of reactions involving ions—can then be derived in the same way as for ideal solutions but with activities in place of concentrations. However, when we want to relate the results we derive, we need to know how to relate activities to concentrations. We ignored that problem when discussing acids and bases and simply assumed that all activity coefficients were 1. The cytoplasm and other fluids in organisms have ion concentrations that are far too high to behave ideally, so y = 1 is a poor approximation in this chapter, we see how to improve that approximation. 

Another approach to non-ideality of aqueous solutions does not take into account the nature of interactions between constituents the simple solution concept allows one to determine thermodynamic properties of concentrated solutions of salts and also the density of various mixtures the latter parameter being required for the conversion of concentrations from the molar to molal scale [41]. 

The techniques that have been described so far to measure the molar masses of polymers in solution depend upon the equilibrium properties of the polymer solution. It is possible to relate the molar mass of the polymer to the solution properties through theoretical (e.g. thermodynamic) equations and the measurements are normally extrapolated to zero concentration where the solutions exhibit ideal behaviour. It is also possible to determine molar masses by studying the transport properties of polymer solutions which are usually analysed in terms of hydrodynamic models. These properties can be divided into two categories one of which involves the motion of the molecules through a solvent which is itself stationary (e.g. ultracentifuge) and the other deals with the effect of polymer molecules upon the motion of the whole solution (e.g. solution viscosity). The theoretical models which have been devised to explain the transport properties are by no means as well developed as those used to explain, for example, the thermodynamic properties of polymer solutions and so transport properties are normally analysed using semi-empirical... 

In the thermodynamic properties of polyelectrolyte solutions, it is always found that polyelectrolyte solutions are highly non-ideal. That is, the activity coefficient and osmotic pressure coefficient are much lower than unity. [1] Reasonable explanations on these phenomena are usually obtained by assuming that a part of counter-ions are bound on fixed charges. 

This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), apphed to an ideal solution with replaced by Gj, can be written... 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

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