Free energy ideal solution

Since ideal conditions simplify calculations, an ideal gas at 1 atm pressure in the gas phase which is infinitely dilute in solution will be utilized. Then the total standard partial molar Gibbs free energy of solution (chemical potential), AG, can be directly related to KD, the distribution constant, by the expression... 

The Perturiiatioii Treatment.—Based on a statistical solution theory of Longuet-Higgins, a first-order perturbation approach to the free energy of solution has been developed by Luckhurst and Martire and by Tewari et al. This theory appears useful for interpreting the interaction parameters within a series of structurally related solute molecules in the same solvent. It is thus ideally suited for testing data obtained from g.l.c. experiments. The theory involves a comparison of two solutes - a reference one and a perturbed one in a common solvent. The interaction parameter x is shown to be linearly related to Ty( V y where Tf and Ff are the solute critical temperature and the solute molar volume at temperature 0.675 respecively. The term 75/( Ff ) is a measure of the depth of the interaction potential well. ... 

The calculation of an electrolyte s solubility product can therefore be based on the saturation molality of a pure solution of that electrolyte or on tabulated values of Gibbs free energies. Ideally, the solubility product should be the same no matter what the method of calculation. In reality, this is rarely the case. 

Subsequent theories of non-ideality have been mainly concerned with explaining the concentration and temperature dependences of Y and 0 (3,16). For a comparison with various other theories for the non-ideal part of free energy of solutions, see (14). The interionic attraction theory (3,5,16-18) formulated on the assumption of complete dissociation of strong electrolytes, predicted the InV vs /m linear dependence and explained the Jc dependence of A found empirically by Kohlrausch (3,14) for dilute solutions. Since the square-root laws were found to hold for dilute solutions of many electrolytes in different solvents, the interionic attraction theory gained a wide acceptance. However, as the square root laws were found to be unsatisfactory for concentrations higher than about 0.01m, the equations were extended or modified by the successive additions of more terms, parameters and theories to fit the data for higher concentrations. See e.g., (3,16) for more details. 

The absence of sereening in Eq. (55) indicates that counter ions play no role in determining the free energy of solute ions in ideal solutions. This is consistent with the derivation requirement that the free energy of ions in ideal solutions be independent of solute eoneentration. 

A fairly simple treatment, due to Guggenheim [80], is useful for the case of ideal or nearly ideal solutions. An abbreviated derivation begins with the free energy of a species... 

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

Fig. 6. Free energies of hydration calculated, for a series of polar and non-polar solute molecules by extrapolating using (3) from a 1.6 ns trajectory of a softcore cavity in water plotted against values obtained using Thermodynamic Integration. The solid line indicates an ideal one-to-one correspondence. The broken line is a line of best fit through the calculated points.

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

In the present case, each endpoint involves—in addition to the fully interacting solute—an intact side chain fragment without any interactions with its environment. This fragment is equivalent to a molecule in the gas phase (acetamide or acetate) and contributes an additional term to the overall free energy that is easily calculated from ideal gas statistical mechanics [18]. This contribution is similar but not identical at the two endpoints. However, the corresponding contributions are the same for the transfonnation in solution and in complex with the protein therefore, they cancel exactly when the upper and lower legs of the thermodynamic cycle are subtracted (Fig. 3a). 

Thus the formation of an ideal solution from its components is always a spontaneous process. Real solutions are described in terms of the difference in the molar Gibbs free energy of their formation and that of the corresponding ideal solution, thus ... 

The contribution that (46) makes to the free energy of mixing is — kT In Wc and it will be noticed that, if the right-hand side of (47) is multiplied by kT, it becomes identical with (45), which is the total change in the free energy, when an ideal solution is formed from its components. 

Here d is the disparity between the free energy per ion pair added to the non-ideal solution and the free energy per ion pair added to the corresponding ideal solution. It is the disparity between the communal term in the free energy and the cratic term in the free energy. In the solution... 

The Disparity of a Solution. We may begin to use the word disparity in a technical sense, for the quantity defined above, and to speak of d as the disparity of the solution when the mole fraction of the solute is x. In dilute ionic solutions the sign of d is always negative. The effect of the interionic forces is that ions added to a dilute solution always lose more free energy than they would when added to the corresponding ideal solution hence the total communal term is less than the cratic term. 

To find how AG changes with composition, we need to know how the molar Gibbs free energy of each substance varies with its partial pressure, if it is a gas, or with its concentration, if it is a solute. We have already seen (in Section 8.3) that the molar Gibbs free energy of an ideal gas J is related to its partial pressure, P(, by... 

Gibbs free energy, 311 molar volume, 151 ideal gas law, 147 ideal solution, 331 ilmenite, 662 iminodisuccinate ion. 675 implant, 344 incandescence, 8, 647 inch, A4... 

A very simple treatment can be carried out by assuming that the liquid phase is a series of ideal solutions of lead and thallium, and that in the solid phase isomorphous replacement of thallium atoms in the PbTl3 structure by lead atoms occurs in the way corresponding to the formation of an ideal solution. For the liquid phase the free energy would then be represented by the expression... 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

The first term in Eq. (68) represents the ideal free energy of mixing term in dilute solutions, as will be apparent below. If the entire volume were available to atl of the molecules, which would be an acceptable assumption if either 7 were very large or u were very small, it would be the only term, for then Q = Const. The second term in... 

Here G is the Gibbs free energy of the system without external electrostatic potential, and qis refers to the energy contribution coming from the interaction of an apphed constant electrostatic potential s (which will be specified later) with the charge qt of the species. The first term on the right-hand side of (5.1) is the usual chemical potential /r,(T, Ci), which, for an ideal solution, is given by... 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

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. 

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