Excess functions solutions

Thermodynamic Excess Functions. Solutions of a Single Electrolyte... 

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. 

Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

A great many of the difficulties (and sometimes the misunderstandings) arise from point (c). It is however important to notice that the APM describes the properties of solutions as finite differences between suitable composition-dependent averages and the properties of the pure components. Series expansions in powers of 6, p, 6, and a were introduced afterwards for the purpose of qualitative discussion and comparison with other treatments, e.g., the theory of conformal solutions.34>85>36 They introduce artificial difficulties due to their slow convergencef which have nothing to do with the physical ideas of the APM. Therefore expansions of this type should be proscribed for all quantitative applications one should instead use the compact expressions of the excess functions. 

The earlier rather complicated evidence for clathrate structures enforced by hydrophobic pairs (from EPR lineshape phenomena for paramagnetic hydrophobic solutes, (23)) and for two states of the hydrophobic bond (from thermodynamic excess functions (2k,25)) is provided with a detailed background by these important theoretical developments. 

Various functions have been used to express the deviation of observed behavior of solutions from that expected for ideal systems. Some functions, such as the activity coefficient, are most convenient for measuring deviations from ideality for a particular component of a solution. However, the most convenient measure for the solution as a whole, especially for mixtures of nonelectrolytes, is the series of excess functions (1) (3), which are defined in the foUowing way. 

Grover J. (1977). Chemical mixing in multicomponent solutions An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behaviour. In Thermodynamics in Geology, D. G. Fraser, ed. D. Reidel, Dordrecht-Holland. 

Several resins have been used frequently in reactant sequestration. Ami-nomethylpolystyrene 1 and the more highly functionalized polyamine resins 2 and 3 have been reported to sequester excesses of solution-phase electrophiles, including isocyanates, isothiocyanates, sulfonyl chlorides, acid chlorides, anhydrides, aldehydes, and imines. Cross-site reactivity is not an issue with the more densely functionalized sequestering resins so their use in an automated laboratory environment offers a significant resin and volume economy compared to less densely functionalized resins. 

In the ternary solution KCI-H2O-PEG-2OO, the quantity A, which can be considered a type of excess function, is positive except in the range of higher nonelectrolyte concentration, where it becomes negative. Its contribution to the binary data is 5-10%. 

Specific solution models therefore involve specific mathematical assumptions concerning the excess functions H%s(x), Sfs(x). (If phase is not the standard-state form of component A or B, an additional contribution is needed for the free energy phase change of each pure component, but this involves only pure-component properties and can be ignored for the present purposes.)... 

The (liquid 4- liquid) equilibria diagram for (cyclohexane + methanol) was taken from D. C. Jones and S. Amstell, The Critical Solution Temperature of the System Methyl Alcohol-Cyclohexane as a Means of Detecting and Estimating Water in Methyl Alcohol , J. Chem. Soc., 1930, 1316-1323 (1930). The G results were calculated from the (vapor 4- liquid) results of K. Strubl, V. Svoboda, R. Holub, and J. Pick, Liquid-Vapour Equilibrium. XIV. Isothermal Equilibrium and Calculation of Excess Functions in the Systems Methanol -Cyclohexane and Cyclohexane-Propanol , Collect. Czech. Chem. Commun., 35, 3004-3019 (1970). The results are from M. Dai and J.-P.Chao, Studies on Thermodynamic Properties of Binary Systems Containing Alcohols. II. Excess Enthalpies of C to C5 Normal Alcohols + 1,4-Dioxane , Fluid Phase Equilib., 23, 321-326 (1985). 

Solutions are thermodynamically classified into perfect, ideal, and non-ideal solutions. This chapter discusses the characteristics of these solutions and define the excess functions of non-ideal solutions. Also examined are electrolytic solutions which contain dissociated ions. 

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 the foregoing the excess function has been defined for one mole of the non-ideal solution. For the whole system of n moles of substances present, we then obtain Eq. 8.23 ... 

J. Grover, Chemical mixing in multicomponent solutions. An introduction to the use of Margules and other thermodynamics excess functions to represent nonideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. It follows from Eqs. 5.17 and 5.19 that, in general, In fA = Xjat and In fBA = Xjbj. Equation 5.20a is a special case of this relation for a third-order Margules expansion. 

That is, the excess function r gives the total number of ions of valence v found in the vicinity of the spherical colloid compared with the number of such ions that would occur in the bathing solution in the absence of the colloid. 

The thermodynamic characteristics of solutions are often expressed by means of excess functions. These are the amounts by which the free energy, entropy, enthalpy, etc. exceed those of a hypothetical ideal solution of the same composition (Denbigh, 1981). The excess free energy is closely related to the activity coefficients. The total free enthalpy (Gibbs free energy) of a system is ... 

Figure 16. Excess functions for three 1 1 salts in water and deuterium oxide as a function of the aquamolality of the salt solution, amj, at 298 K (Wuand Friedman, 1966).

Considerable information concerning structural effects on aqueous salt solutions has been provided by studies of the properties of mixed solutions (Anderson and Wood, 1973). In a mixed salt solution prepared by mixing YAm moles of a salt MX (molality m) with Yhm moles of a salt NX (molality m) to yield m moles of mixture in 1 kg of solvent, if W is the weight of solvent, the excess Gibbs function of mixing Am GE is given by (19) where GE is the excess function for... 

The importance of the excess entropy of mixing in aqueous mixtures explains why many of these systems show phase separation with a lower critical solution temperature (LCST). This phenomenon is rarer—though not unknown—in non-aqueous mixtures (for an example, see Wheeler, 1975). The conditions for phase separation at a critical temperature can be expressed in terms of the excess functions of mixing (Rowlinson, 1969 Copp and Everett, 1953). 

This hydrostatic approach also yields a formal closed formula for y in terms of the components of the stress tensor. When the stress tensor is expressed in terms of molecular variables, the resulting statistical mechanical formula for y provides a direct means for the calculation of surface tension. For example, it may be used directly to compute the surface tension of dilute ionic solutions (6). It also illustrates in molecular detail the iterative subtractive procedures that lead to the excess functions of the familiar phenomenological approach. 

The thermodynamics of non-ideal bulk mixtures has been considered in sec. 1.2.18. Non-idealities can be expressed in terms of activity coefficients, excess functions, pair interaction energies (as In Regular Solution theory) or through vlrlal expansions. For all these methods surface equivalents can be formulated. 

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< ... 

The above method can be apphed to a calculation of other excess functions in terms of intermolecular forces. We may note that the excess entropy of mixing is closely related to the excess volume and to the change of the free volume of the solution with composition. We shall not, however, go into any further details here. 

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