Thermodynamically ideal solutions

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

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. 

Derive the governing equation for composition as a function of position and lime for the case of the interdiffusion of two fluids in which the molar density of the mixture is not constant, but the partial molar volumes of each species are constant and equal to the pore component values, that is, a thermodynamically ideal solution. 

The experiment is actually one of free diffusion as described in Example 2.3-1. Uuder the assumptions that the solutions form thermodynamically ideal solutions, die combination of die total mass (mole) and species balances yields Fick s Second Law as die governing equation. 

For the prediction of viscosity of well-defined mixtures of liquids, the method of Cao et al. is very good in general [27,32]. However, since hydrocarbon mixtures form almost thermodynamically ideal solutions and their viscosities are simple functions of composition, simpler methods such as those of Allan and Teja [12] or of Orbey and Sandler [13] result in almost equal accuracy without the need for binary interaction parameters. These last two, simpler models can only be used for hydrocarbons, while the model of Cao et al. is of more general applicability. 

The ideal solutions are characterized by zero excess thermodynamic functions. In thermodynamics, ideal solutions are often characterized by their excess entropy and enthalpy. Note that to obtain these, we need to assume differentiability of (4.91) with respect to temperature. This assumption is still quite weaker than the requirement that (4.91) be valid at all T and P. 

For a thermodynamically ideal solution (e.g., a very dilute solution), the osmotic pressure obeys Van t Hoff s equation ... 

Osmotic pressure is one of four closely related properties of solutions that are collectively known as colligative properties. In all four, a difference in the behavior of the solution and the pure solvent is related to the thermodynamic activity of the solvent in the solution. In ideal solutions the activity equals the mole fraction, and the mole fractions of the solvent (subscript 1) and the solute (subscript 2) add up to unity in two-component systems. Therefore the colligative properties can easily be related to the mole fraction of the solute in an ideal solution. The following review of the other three colligative properties indicates the similarity which underlies the analysis of all the colligative properties ... 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

All other thermodynamic properties for an ideal solution foUow from this equation. In particular, differentiation with respect to temperature and pressure, followed by appHcation of equations for partial properties analogous to equations 62 and 63, leads to equations 191 and 192 ... 

If M represents the molar value of any extensive thermodynamic property, an excess property is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus,... 

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. 

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase... 

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... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

In many process design applications like polymerization and plasticization, specific knowledge of the thermodynamics of polymer systems can be very useful. For example, non-ideal solution behavior strongly governs the diffusion phenomena observed for polymer melts and concentrated solutions. Hence, accurate modeling of... 

There are several different scales 011 which the activity of a solute may be defined.1 In thermodynamic expressions for a solute in a non-ideal solution the activity on the molality scale plays the same part that is played by the molality of a solute in an ideal solution. Since the activity is expressed in the same units as the molality, the ratio of the activity to the molality—the activity coefficient—is a pure number whose value is independent of these units it is also indopendont of the particular b.q.s. that has been adopted. Thus the numerical values of all activities and molalities would change in the same ratio, if at any time a new choice were made for the b.q.s. 

The excess molar thermodynamic function Z is defined as the difference in the property Zm for a real mixture and that for an ideal solution. That is,... 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

It is found empirically and can be justified thermodynamically that the freezing-point depression for an ideal solution is proportional to the molality of the solute. For a nonelectrolyte solution. 

We use a different measure of concentration when writing expressions for the equilibrium constants of reactions that involve species other than gases. Thus, for a species J that forms an ideal solution in a liquid solvent, the partial pressure in the expression for K is replaced by the molarity fjl relative to the standard molarity c° = 1 mol-L 1. Although K should be written in terms of the dimensionless ratio UJ/c°, it is common practice to write K in terms of [J] alone and to interpret each [JJ as the molarity with the units struck out. It has been found empirically, and is justified by thermodynamics, that pure liquids or solids should not appear in K. So, even though CaC03(s) and CaO(s) occur in the equilibrium... 

The term 6 is important it has the same units as temperature and at critical value (0 = T) causes the excess chemical potential to disappear. This point is known as the 6 temperature and at it the polymer solution behaves in a thermodynamically ideal way. 

As previously noted, the equilibrium constant is independent of pressure as is AG. Equation (7.33) applies to ideal solutions of incompressible materials and has no pressure dependence. Equation (7.31) applies to ideal gas mixtures and has the explicit pressure dependence of the F/Fq term when there is a change in the number of moles upon reaction, v / 0. The temperature dependence of the thermodynamic equilibrium constant is given by... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

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