Mole fraction description

The system of primary interest, then, is that of a condensable vapor moving between a Hquid phase, usually pure, and a vapor phase in which other components are present. Some of the gas-phase components may be noncondensable. A simple example would be water vapor moving through air to condense on a cold surface. Here the condensed phase, characterized by T and P, exists pure. The vapor-phase description requiresjy, the mole fraction, as weU as T and P. The nomenclature used in the description of vapor-inert gas systems is given in Table 1. 

Stream Description Upper bound on flowrate kg mol/s Supply composition of benzene (mole fraction) x [ Target composition of benzene (mole fraction) x j... 

Stream Description L j kg mole/s (mole fraction) r J (mole fraction) r ... 

There are several ways to describe the chemical composition of a mixture of gases. The simplest method is merely to list each component with its partial pressure or number of moles. Two other descriptions, mole fractions and parts per million, also are used frequently. 

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. 

Thermodynamically the quantitative treatment of both active and passive processes requires them to be downhill or exoergic. The description of chemical potential as a function of mole fraction follows the same form as before for a neutral species (Section 8.2) ... 

Figure 7 shows the predicted vapor-phase mole fractions of HC1 at 25°C as a function of the liquid-phase molality of HC1 for a constant NaCl molality of 3. Also included are predicted vapor-phase mole fractions of HC1 when the interaction parameter A23 is taken as zero. There are unfortunately no experimental vapor-liquid equilibrium data available for the HC1-NaCl-FLO system however, considering the excellent description of the liquid-phase activity coefficients and the low total pressures, it is expected that predicted mole fractions would be within 2-3% of the experimental values. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The empirical description of dilute solutions that we take as the starting point of our discussion is Henry s law. Recognizing that when the vapor phase is in equUibrium with the solution, p,2 in the condensed phase is equal to p,2 g, we can state this law as follows For dilute solutions of a nondissociating solute at constant temperature, the fugacity of the solute in the gas phase is proportional to its mole fraction in the condensed phase That is. 

The description of the partial pressure exerted by a sorbate, or a mixture of sorbates, when they reside on the sorbent surface, at some given temperature is what we speak of as adsorption equihbrium. For a single adsorbate (adsorbing molecular species) we require three state variables to completely describe the equilibrium the temperature, the sorbed phase concentration or loading and the partial pressure exerted by the sorbed phase are very convenient variables to use. As more adsorbable compounds are added to the problem we require additional information to adequately describe the problem. That information is the specification of the mole fractions of the adsorbable compounds in both the gas and sorbed states. 

Concentration is the most common means for describing the composition of a solution in biochemistry. Enzyme kinetic expressions are typically expressed in these concentration units. Unless otherwise noted, this is the method used throughout this text. Nevertheless, other methods for describing compositions are utilized. For example, mole fractions are often used in Job plots. Gases in solution are commonly measured in terms of partial pressures. Below is a brief description of a few of these other conventions or methods. 

Mole fraction, often symbolized by x or X followed by a subscript denoting the entity, represents the amount of a component divided by the total amount of all components. Thus, the mole fraction of component B of a solution, xb, is equal to hb/Xhi where Hb is the amount of substance B and Sni is the total amount of all substances in solution. In biochemical systems, usually the solvent is disregarded in determining mole fractions. The mole fraction, a dimensionless number expressed in decimal fractions or percentages, is temperature-independent and is a useful description for solutions in theoretical studies and in physical biochemistry. 

Cantor and SchimmeP provide a lucid description of the thermodynamics of the hydrophobic effect, and they stress the importance of considering both the unitary and cratic contributions to the partial molal entropy of solute-solvent interactions. Briefly, the partial molal entropy (5a) is the sum of the unitary contribution (5a ) which takes into account the characteristics of solute A and its interactions with water) and the cratic term (-R In Ca, where R is the universal gas constant and ( a is the mole fraction of component A) which is a statistical term resulting from the mixing of component A with solvent molecules. The unitary change in entropy 5a ... 

By contrast, the macroscopic atom model of Miedema (Miedema et al. 1975) starts with a descriptions of the solid state which is then modified to describe the liquid state (Boom et al. 1976a, 1976b). In their model the enthalpy of formation at 0.5 mole fraction, Hc=o.s is given to a first approximation by ... 

We really should use mole fraction, and not concentration, in our description of y and x, but for our work, we will just say that the term concentration refers to the percent of a component that the operator would see in the gas-chromatographic (GC) results, as reported by the lab. The equilibrium constant, assuming the ideal-gas law applies, is defined as... 

Figure 17.3a compares ff for (cyclohexane + hexane) at three temperatures. The curves are not symmetrical with mole fraction (as required for regular solution behavior), with maximum values of approximately 200 J-mol-, skewed toward the mole fraction of cyclohexane. As with previous examples, on the molecular level, we can think of the mixing process as one in which we replace A-A interactions (in cyclohexane) and B-B interactions (in hexane) with A-B interactions (between hexane and cyclohexane). The energy difference for this process is the major contributor to H . We will find this simplified qualitative description useful as we compare systems that contain different types of interactions. ... 

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. 

The dependence of HE and SE on mole fraction can be complex although the overall effect on the GE -curve is not. The importance of the entropy of mixing warrants the description of such mixtures as typically aqueous (Franks, 1968b). In the following account of the kinetics of reactions in these mixtures, we examine these systems under three headings (i) typically aqueous , TA (ii) typically... 

Wilsons equation and the modification proposed by Renon and Prausnitz (8) use the local mole fraction concept, produced because molecules in solution aggregate as a result of the variation in intermo-lecular forces. The local mole fraction concept results in a more useful description of the behavior of molecules in a non-ideal mixture. 

There is clearly a broad gap between this impossible informational requirement and the handful of variables (P, V, T, mole fraction) needed to adequately describe the thermodynamic state of the system and so determine the macroscopic behavior of the system at equilibrium (see Secs. I.l and 1.2). Even the requirements for an empirical description of a kinetic system are nowhere so formidable. 

Several fundamental models have recently been proposed (//, I7a,b, 28) in order to quantitatively accommodate the influence of solution equilibria on the distribution process. Although application of these theoretical approaches has yet to provide a comprehensive general description of retention behavior for peptides, they have provided useful insight into those mobile-phase effects arising from changes in the nature and mole fraction of the organic solvent modifier. 

Semiempirical expression was derived for the description of the retention of chaotropic counteranions in reversed-phase conditions [165]. Overall expression for the description of the retention dependencies of analyte ions versus eluent composition will have only four unknowns and allow numerical approximation of experimental retention data (shown as a function of the mole fraction of organic eluent component). 

However, going back to the discrete description, consider the special case where all the mole fractions are equal to each other, X/ = N this composition maximizes the absolute value of G IRT. One obtains (j IRT = ln(l/N) and one... 

The conceptual point is as follows. The discrete case X/ = l/N requires the mole fractions of all components to be equal to each other. That does not, however, correspond to a constant X (x) in the continuous description, because ) (x)dx is the mole fraction of species between x and x + dx, and one would need to require (x)dx to be constant. But this can only be done if one has chosen a specific scaling for the label x Any label x that is given by a monotonous function x (x) = X would be legitimate, and of course x)dx could be taken as constant for only one such scale. In other words, in a continuous description one has chosen some label x. The form of the mole fraction distribution must then satisfy certain constraints such as the one discussed earlier for the gamma distribution. The problem is related to the more general problem of the correct generalization to a continuous description of nonlinear formulas. 

In the general description of LLE, any number of species may be considered, and pressure may be a significant variable. We treat here a simpler (but important) special case, that of binary LLE either at constant pressure or at reduced temperatures low enough that the effect of pressure on the activity coefficients may be ignored. With but one independent mole fraction per phase. 

Tj with oxygen mole fraction x, cannot be satisfactorily described using the simple extension of the IBI formulation outlined here. This difBculty seems more likely to be related to the adequacy of the description of the fluid mixture than to a breakdown of the basic IBI hypotheses which appear to be well justified in the above case. 

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