Thermodynamics ideal solutions

That surfactant molecules form aggregates designed to remove unfavourable hydrocarbon-water contact is not surprising but the question that should be asked is why the aggregates form sharply at a concentration characteristic of the surfactant (the cmc). From the basic equation of ideal solution thermodynamics... 

All the theories described above are based on the ideal solution thermodynamics, on the one hand, and on a rather heuristic molecular treatment of micelles as a phase particle, on the other hand. Despite of their obvious successes in predicting micellar solution properties, these theories have some essential drawbacks. The number of adjusting parameters at the evaluation of different contributions to the free energy is too high, as well as the number of oversimplifications, which have been used in order to estimate these parameters. For example, the micellar core is considered as a very small fluid phase droplet surrounded by a second fluid phase and the free energy of micelle surface is estimated as the interfacial tension between these two fluid phases. In order to elucidate this problem Eriksson et al, [24] attempted to... 

Ebias accounts for the externally applied potentials, Lapp, and E is the energy of the initial state with respect to the energy barrier and related to the Nernst potential given by Equation (4.9) for the electrochemical reaction R + ze - P, where R stands for the reacting species and P for the product species. Eq is the standard potential in Vnhe. z is the number of electrons exchanged, F is Faraday s constant, and Up and are the activities of the products and reactants—using ideal solution thermodynamics so that their activities are based on their solution concentrations— respectively. 

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 ... 

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. 

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 ... 

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... 

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. 

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.

H (MPa) (Eq. (13)) and HA (MPa m3 mor1) (Eq. (14)) are often referred to as Henry s constant , but they are in fact definitions which can be used for any composition of the phases. They reduce to Henry s law for an ideal gas phase (low pressure) and for infinitely dilute solution, and are Henry s constant as they are the limit when C qL (or xA) goes to zero. When both phases behave ideally, H depends on temperature only for a dilute dissolving gas, H depends also on pressure when the gas phase deviates from a perfect gas finally, for a non-ideal solution (gas or liquid), H depends on the composition. This clearly shows that H is not a classical thermodynamic constant and it should be called Henry s coefficient . 

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