Ideal solutions mixing functions

When two chemicals are mixed, an ideal solution is defined when the properties of the mixture are a linear function of the properties of the two chemicals. The simplest relation is in the form of linear relations ... 

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... 

Excess Thermodynamic Functions The excess molar thermodynamic function Z is defined as the difference between AmjXZm, the change in Zm for mixing components to form a real solution, and AmjxZ , the change in Zm to form the ideal solution. Thus,... 

Solutions and glasses do not follow the third law. If a solid solution continues to persist on cooling a sample to the lowest possible temperature, then at this temperature the molecules of the several components are distributed in some fashion in the same crystal lattice. Under such conditions all of the molecules of the substance could not attain the same quantum state on further cooling to 0 K in the sense of the required extrapolation. Only if the solid solution was separated into the pure components would the value of zero be obtained for the entropy function at 0 K. If the molecules of the components were randomly distributed in the crystal lattice, as in an ideal solution, then the entropy of the substance at absolute zero would be equal to the ideal entropy of mixing, so... 

The volume fractions of fully functional pores as a function of template size for the different preparation conditions are plotted in Fig. 2a. In contrast to the total number of imprinted pores, it appears as if ideal solution conditions hardly favor solvation of the functional sites on the templates, and any deviation from such conditions increases the imprinting effect. However, not only do optimum conditions exist for a given size of the templates (typified by a peak at a particular r), optimum conditions that determine the excluded volume of the monomers as well as the phase segregation tendency appear to exist. Comparison of curves in Fig. 2a show that for r >2, xv=5 proves to be more efficient than both Xv=2 and ifv=8. In addition, mixing interactions (/f<0) are favorable for small templates, while segregating interactions are favorable for larger templates. 

We have already introduced in 3 of chap. XX the thermodynamic functions of mixing in the case of perfect solutions. These definitions are easily extended to non-ideal solutions. The results obtained provide a useful basis for the classification of non-ideal solutions which will be made in paragraph 5 of this chapter. 

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< ... 

Next the equations that we can write are for calculating system properties. Because equilibrium is assumed, the rate equations and, therefore, the transport and transfer properties are of no concern. In general, the thermodynamic properties of mixtures will depend on temperature, pressure, and composition, we will assume that the mixture is an ideal solution to simplify the computation of thermodynamic properties. Thus, we can write the enthalpies of the mixtures as mole fraction averages of the pure component enthalpies, without an enthalpy of mixing term. We can also write the phase equilibrium relations as functions of temperature and pressure only and not composition. The pure component enthalpies of liquids generally do not depend strongly on pressure, but there may be some effect of pressure on the vapor-phase enthalpy. We will neglect this effect for simplicity. 

In the case of non-ideal solutions the mixing functions are often referred to the value they would have in an ideal solution, thereby defining the excess... 

Excess thermodynamic functions show the deviations from ideal solution behavior and there is of course a relation between GE and the activity coefficients. Similar to Equation (369), if we write the actual Gibbs free energy of mixing (AGmi[)actuai in terms of activities,... 

The act or process by which a compound such as oxygen is molecularly mixed with a liquid (such as water) or a solid (such as a polymer) is called dissolution, and the result of the mixing is a solution. If a solution is very dilute, as is commonly found in packaging, it behaves as an ideal solution, and again the activity coefficient is approximately 1, so concentration can be substituted for activity in thermodynamic relationships. In order to describe the solubility of a compound present in a gas phase that is in contact with a solid phase, as may be the case of oxygen in air contacting a polymer, we need a relationship between the concentration in the liquid (or solid) phase and the concentration (or partial pressure) in the gas phase. In other words, we need an expression for the solubility of the substance at equilibrium, as a function of the partial pressure of the gas or vapor in the contacting gas phase. 

The value of/i can be determined from the position of the DA concentration maximum as a function of the initial concentration ratios of D and A. A simple method proceeds from solutions of equal molar concentrations of A and D, which are then mixed for various volume fractions, d>D = 1 - . For ideal solutions, with [Af]o = ([A] + [D])o ... 

In these equations /q and are functions of pressure and temperature, but do not depend on x. Obviously /q is identical with the molecular potential g of the pure solvent (x = 0). Solutions in which go Si given by the simple expressions (43) and (44) are called ideal solutions. In these solutions the vapour pressure, osmotic pressure, etc. are determined solely by the entropy of mixing, i.e., by the tendency to attain the state of maximum probability. 

Both laws hold over regions of mixing and pressure respectively in which the conditions requiring the absence of any interfering forces are fulfilled they have different mathematical form, as is evident from comparison of equations (48) and (50) the osmotic pressure of ideal solutions obeys a logarithmic law according to the molar concentration n/V, In both cases the entropy gain by the volume increase is decisive for the pressure on the wall this increase in entropy is, however, expressed each time by a different mathematical function of the variables involved. 

How should we define the simplest possible deviation from ideal solution behavior It should evidently have about the same parabolic shape as the ideal mixing curve, which means we need some kind of y a function. Then because excess properties are zero for pure components, this function should approach zero as the mole fraction, or Xg of either of the two components, approaches 1.0. Finally, in a solution of two similar components such as benzene and toluene, we might expect the solution to be most nonideal when the components are mixed in equal proportions (because that s when nonuniform interactions between the two species are maximized). That means the equation should have a maximum or minimum at the 1 1 composition. The simplest equation which satisfies all these conditions is... 

The temperature of the transition between two different phase structures is thermodynamically defined by the equality of their partial molar free enthalpies Ml and M2- Figure 13a illustrates the so-called ideal solution in a plot of the vapor pressure Pi as a function of concentration in terms of the mole fraction xi (Raoult s law, Xi Po= Pi, where Po is the vapor pressure of the pure solvent 1). In this ideal case, only the entropy of mixing, ASi = —RT In xi, changes the chemical potential of the solvent, mi > from its pure state, mi°, as expressed by... 

The well-known mean-field incompressible Flory-Huggins theory of polymer mixtures assumes random mixing of polymer repeat units. However, it has been demonstrated that the radial distribution functions gay(r) of polymer melts are sensitive to the details of the polymer architecture on short length scales. Hence, one expects that in polymer mixtures the radial distribution functions will likewise depend on the intramolecular structure of the components, and that the packing will not be random. Since by definition the heat of mixing is zero for an athermal blend, Flory-Huggins theory predicts athermal mixtures are ideal solutions that exhibit complete miscibility. 

For an ideal solution, activity is equal to mole fraction, and hence it is independent of temperature. The followinq mixing functions can be obtained for an ideal solution A-B ... 

A regular solution may be defined as that solution of which the enthalpy of mixing is different from zero, but the entropy of mixing is the same as for an ideal solution. The various mixing functions of a binary regular solution A-B may be written as... 

Activity-coefficient models, however, can only be used to calculate liquid-state fugacities and enthalpies of mixing. These models provide algebraic equations for the activity coefficient (y,) as a function of composition and temperature. Because the activity coefficient is merely a correction factor for the ideal-solution model (essentially Raoult s Law), it cannot be used for supercritical or noncondensable components. (Modifications of these models for these types of systems have been developed, but they are not recommended for the process simulator user without consultation with a thermodynamics expert.)... 

It follows that even in ideal solution of the simplest stoichiometry, a is not a linear function of the mixed solvent composition Xg. Dependence of isotherm a on the solvent composition at = 1 is presented in Figure 8.1.9a. 

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