Ideal mixture defined

Hence, ideal -values can be determined as the ratio of fugacity of the pure components in liquid and vapour phase, and these depend only on T and P. The result is similar with ideal mixtures defined by the Raoult-Dalton law, where Kj = PjlP, P, being the vapour pressure and P the total pressure. A notable difference is that equation (6.4) may be used at high pressures, where the fugacity concept is more suitable. 

The chemical potential of a component y in an ideal mixture defined on... 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

In a general case of a mixture, no component takes preference and the standard state is that of the pure component. In solutions, however, one component, termed the solvent, is treated differently from the others, called solutes. Dilute solutions occupy a special position, as the solvent is present in a large excess. The quantities pertaining to the solvent are denoted by the subscript 0 and those of the solute by the subscript 1. For >0 and x0-+ 1, Po = Po and P — kxxx. Equation (1.1.5) is again valid for the chemical potentials of both components. The standard chemical potential of the solvent is defined in the same way as the standard chemical potential of the component of an ideal mixture, the standard state being that of the pure solvent. The standard chemical potential of the dissolved component jU is the chemical potential of that pure component in the physically unattainable state corresponding to linear extrapolation of the behaviour of this component according to Henry s law up to point xx = 1 at the temperature of the mixture T and at pressure p = kx, which is the proportionality constant of Henry s law. 

Volume Changes. The density changes of several amide-water systems have been measured using 10-cc. specific gravity bottles. These experimental values were compared with an ideal density defined in terms of additive molar volumes and are shown as a dotted line in Figures 1 through 8 for the respective amide-water mixtures. 

Such behavior occurs when the two components either form an ideal mixture or are immiscible. Before drawing conclusions concerning molecular interactions (12, 13), it is clearly important to establish that homogeneous mixed films have been formed. Any deviation from line LM is, of course, indicative of both mixing and nonideality. In discussing such effects, we define any negative deviation from LM as a "condensation and any positive deviation as an "expansion. Our results fall into three distinct categories. 

Here spr is the projected entropy of an ideal mixture. The first term appearing in it, p0 = J dop a), is the zeroth moment, which is identical to the overall particle density p defined previously. If this is among the moment densities used for the projection (or more generally, if it is a linear combination of them), then the term — Tp0 is simply a linear contribution to the projected free energy/pr(p,) and can be dropped because it does not affect phase equilibrium calculations. Otherwise, p0 needs to be expressed—via the A —as a function of the pit and its contribution cannot be ignored. We will see an example of this in Section V. 

Equation (36.30) therefore defines an ideal mixture of gases the ideality this time stemming from the fact that Dalton s Law of Partial Pressures (Frame 31 and 34) is obeyed by the gas mixture. 

In Frame 37 mole fractions were used in the discussion of the thermodynamics of ideal mixtures. In expressing the equilibrium constant, Kx, in terms of mole fractions, no division of the individual terms by standard states is required (Appendix Frame 6, section 6.3) since mole fractions are already, themselves, dimensionless. Thus equation (40.10) defines Kx ... 

Activity as a function was introduced by Lewis in 1908, and a full description was given by Lewis and Randall [74] in 1923. The activity a of a substance i can be defined [75.76] as a value corresponding to the mole fraction of the substance i in the given phase. This value is in agreement with the thermodynamic potential of the ideal mixture and gives the real value of this potential. 

Activity — The absolute activity of a substance, A, is defined as A = exp(p/RT), where p is the molar free energy. The relative activity a, is defined as a = cxp[ (p -p" )/RT, where p" is the molar free energy of the material in some defined standard state for which the activity is taken as unity. Historically, the concept of activity arose out of an attempt, initially formulated by -> Lewis, to understand the behavior of mixtures. Ideal mixtures or solutions are those for which the -> chemical potential or molar free energy of any of the component species i can be written in the form p, = p + RT nx, where xt is the mole fraction of the ith component, defined as X =, n, is the number of moles of species i... 

Matteoli and Lepori [10,11] noted that the cjGji are non-zero for ideal mixtures, which should be considered non-aggregated, and suggested that the aggregation is better reflected in the excesses (the ML excesses) defined as... 

It is of interest to note that since the mole fraction n of any gas in a mixture must be less than unity, its logarithm is negative hence ASm as defined by equation (19.32) is always positive. In other words, the mixing of two or more gases, e.g., by diffusion, is accompanied by an increase of entropy. Although equation (19.32) has been derived here for a mixture of ideal gases, it can be shown that it applies equally to an ideal mixture of liquids or an ideal solid solution. 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

Activity (of a component of an ideal mixture) A dimensionless quantity whose magnitude is equal to molar concentration in an ideal solution, equal to partial pressure (in atmospheres) in an ideal gas mixture, and defined as 1 for pure solids or liquids. 

When non-ideal liquid solutions are considered, we use excess thermodynamic functions, which are defined as the differences between the actual thermodynamic mixing parameters and the corresponding values for an ideal mixture. For constant temperature, pressure and molar fractions, excess Gibbs free energy is given as... 

The third approach, used primarily for liquid mixtures, requires reliable pure-component density values (see Section 1.2). One starts with an ideal mixture volume, defined by... 

Suppose the initial partial pressures to be 1 bar for each reactant. (The partial pressure of one component of a mixture of gases is defined as the pressure it would exert if it were alone in the available space. For an ideal mixture of perfect gases, the total pressure is the sum of the various partial pressures.) At 25°C, AG° for this reaction is -100.4 kJ/mol, and so the driving force is considerable, and reaction takes place over the catalyst with great vigour. The temperature of the reaction vessel is kept constant. As the reaction occurs, the partial pressures of ethylene and hydrogen drop, and their reduced active mass , or... 

For the special case of ideal mixtures, a simple method is available for calculating the value of froiii the terminal concentrations and Xjy This is based on the relative volatility of the two components oc g, which is defined in terms of the equilibrium concentrations... 

An ideal screen would sharply separate the feed mixture in such a way that the smallest particle in the overflow would be just larger than the largest particle in the underflow. Such an ideal separation defines a cut diameter that marks... 

An ideal mixture, which may be either a gaseous or liquid mixture, is defined to be a mixture in which... 

For convenience, we will use the K factor, defined by the relation y, = K X, in the calculations for the ideal mixtures considered here K PP [T)/P. Thus we obtain the following three equations ... 

We refer to a mixture for which Equation 58 or 59 is satisfied as an ideal mixture. Clearly this concept of an ideal mixture is an idealization that is not realized in practice. Real mixtures will show deviations from the ideal results. However, the properties of an ideal mixture are convenient reference states for thermodynamic properties. For example, it is conventional to use excess thermodynamic properties of mixing that are defined as the difference between the thermodynamic property of the mixture and those of an ideal mixture of the components at the same temperature and pressure. 

For mixtures that display significant deviations from ideality, correcting factors from the ideal mixture assumption are used. For gaseous mixtures, one defines the fugacity coefficient, as follows... 

The form of Eq. (11.14) suggests a generalization. Suppose we define an ideal mixture, or ideal solution, in any state of aggregation (solid, liquid, or gaseous) as one in which... 

Motivatedby (4.432) we use in classical thermodynamics (e.g. [129, 138, 152, 155]) the ideal mixture or the ideal solution defined by the following expression for molar chemical potential of gas or liquid... 

The behaviour of real fluid mixture may be described through deviations from ideal mixture [129, 138, 152, 154, 156] expressed by the activity coefficient / defined by... 

That basic strategy is illustrated in Table 6.1. First we define an ideal mixture whose properties we can readily determine. Then for real mixtures we compute deviations from the ideality as either difference measures or ratio measures. In one route the ideality is the ideal-gas mixture, the difference measures are residual properties, and the ratio measure is the fugacity coefficient. In the other route the ideality is the ideal solution, the difference measures are excess properties, and the ratio measure is the activity coefficient. 

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