Ideal solutions enthalpy

For the case of large mixing enthalpies, the steady state approximation for the main components (not only for vacancies) appears to be valid. For small and zero mixing enthalpies (ideal solution), the steady state approximation for the main components works worse. 

A hypothetical solution that obeys Raoult s law exactly at all concentrations is called an ideal solution. In an ideal solution, the interactions between solute and solvent molecules are the same as the interactions between solvent molecules in the pure state and between solute molecules in the pure state. Consequently, the solute molecules mingle freely with the solvent molecules. That is, in an ideal solution, the enthalpy of solution is zero. Solutes that form nearly ideal solutions are often similar in composition and structure to the solvent molecules. For instance, methylbenzene (toluene), C6H5CH, forms nearly ideal solutions with benzene, C6H6. Real solutions do not obey Raoult s law at all concentrations but the lower the solute concentration, the more closely they resemble ideal solutions. Raoult s law is another example of a limiting law (Section 4.4), which in this case becomes increasingly valid as the concentration of the solute approaches zero. A solution that does not obey Raoult s law at a particular solute concentration is called a nonideal solution. Real solutions are approximately ideal at solute concentrations below about 0.1 M for nonelectrolyte solutions and 0.01 M for electrolyte solutions. The greater departure from ideality in electrolyte solutions arises from the interactions between ions, which occur over a long distance and hence have a pronounced effect. Unless stated otherwise, we shall assume that all the solutions that we meet are ideal. 

Figure 3.3 Thermodynamic properties of an arbitrary ideal solution A-B at 1000 K. (a) The Gibbs energy, enthalpy and entropy, (b) The entropy of mixing and the partial entropy of mixing of component A. (c) The Gibbs energy of mixing and the partial Gibbs energy of mixing of component A.

In this particular case of ideal solutions the phase diagram is defined solely by the temperature and enthalpy of fusion of the two components. 

The ideal solution approximation is well suited for systems where the A and B atoms are of similar size and in general have similar properties. In such systems a given atom has nearly the same interaction with its neighbours, whether in a mixture or in the pure state. If the size and/or chemical nature of the atoms or molecules deviate sufficiently from each other, the deviation from the ideal model may be considerable and other models are needed which allow excess enthalpies and possibly excess entropies of mixing. 

One of the approaches to calculating the solubility of compounds was developed by Hildebrand. In his approach, a regular solution involves no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition when the total volume remains the same. In other words, a regular solution can have a non-ideal enthalpy of formation but must have an ideal entropy of formation. In this theory, a quantity called the Hildebrand parameter is defined as ... 

The equation for the molar enthalpy of an ideal solution derived,... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

The enthalpy change associated with formation of a thermodynamically ideal solution is equal to zero. Therefore any heat change measured in a mixing calorimetry experiment is a direct indicator of the interactions in the system (Prigogine and Defay, 1954). For a simple biopolymer solution, calorimetric measurements can be conveniently made using titra-tion/flow calorimeter equipment. For example, from isothermal titration calorimetry of solutions of bovine P-casein, Portnaya et al. (2006) have determined the association behaviour, the critical micelle concentration (CMC), and the enthalpy of (de)micellization. 

Previously we found that the enthalpies of solution of n-Bu4NBr in mixtures of DMF and n-methylformamide (NMF) are almost proportional to the solvent composition and may be regarded as close to ideal (6). In this context, ideal behavior means that AHid = XaAHa° + XbAHb0 in which AHA° and AHb° are the enthalpies of solution in two pure solvents A and B. As a consequence, the... 

A hypothetical solution that obeys Raoult s law exactly at all concentrations is called an ideal solution. In an ideal solution, the interactions between solute and solvent molecules are the same as the interactions between solvent molecules, so the solute molecules mingle freely with the solvent molecules. That is, in an ideal solution, the enthalpy of solution is 0. Solutes that form nearly ideal solutions are often similar in composition and structure to the solvent molecules. For instance, methylbenzene (toluene) forms nearly ideal solutions with benzene. 

Even in an ideal solution, a solute increases the entropy of the liquid phase we can no longer be sure that a molecule picked at random from the solution will be a solvent molecule. Because the entropy of the liquid phase is raised by a solute but the enthalpy is left unchanged, overall there is a decrease in the molar free energy of the solvent. 

FIGURE 8.32 (a) The molar enthalpy, entropy, and free energy of a pure solvent and its vapor. At equilibrium, the two molar free energies are equal, (b) When a solute is added to form an ideal solution, the molar entropy of the solvent rises, so the free energy of the solvent falls. For the vapor to remain in equilibrium, its molar free energy must fall that is achieved by a decrease in pressure. 

The physical state of each substance is indicated in the column headed State as crystalline solid (c), liquid (liq), gaseous (g), or amorphous (amorp). Solutions in water are listed as aqueous (aq). Solutions in water are designated as aqueous, and the concentration of the solution is expressed in terms of the number of moles of solvent associated with 1 mol of the solute. If no concentration is indicated, the solution is assumed to be dilute. The standard state for a solute in aqueous solution is taken as the hypothetical ideal solution of unit molality (indicated as std state, m = 1). In this state the partial molal enthalpy and the heat capacity of the solute are the same as in the infinitely dilute real solution (aq. m). 

Consider the case of an ideal solution involving a solute, which is a crystalline material at the solution temperature, dissolved in a liquid solvent. Even though the partial molal enthalpy of mixing is still zero because it is an ideal solution, there is an enthalpy requirement to overcome the crystal structure interactions. The van t Hoff equation (Adamson, 1979) gives... 

Therefore, the mole fraction ideal solubility of a crystalline solute in a saturated ideal solution is a function of three experimental parameters the melting point, the molar enthalpy of fusion, and the solution temperature. Equation 2.15 can be expressed as a linear relationship with respect to the inverse of the solution temperature ... 

The change of enthalpy on mixing, AHM[T, P, x], at constant temperature and pressure is seen to be zero for an ideal solution. The change of the heat capacity on mixing at constant temperature and pressure is also zero for an ideal solution, as are all higher derivatives of AHM with respect to both the temperature and pressure at constant composition. Differentiation of Equations (8.57), (8.59), and (8.60) with respect to the pressure yields... 

Thus in the ideal solution model of a two phase field, a knowledge of the enthalpies of fusion of the pure components at their respective melting points allows simultaneous solution of these two equations for the two unknowns, NA(l> and NA(aj at the temperature of interest. 

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