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Thermodynamic mole fractions

There are two ways to arrive at the relationship between aj and the concentration expressed as, say, a mole fraction. One is purely thermodynamic and involves experimental observations the other involves a model and is based on a statistical approach. We shall examine both. [Pg.510]

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 ... [Pg.542]

In thermodynamics the formal way of dealing with nonideality is to introduce an activity coefficient 7 into the relationship between activity and mole fraction ... [Pg.546]

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... [Pg.256]

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). [Pg.502]

Thermodynamic Relationships. A closed container with vapor and liquid phases at thermodynamic equiUbrium may be depicted as in Figure 2, where at least two mixture components ate present in each phase. The components distribute themselves between the phases according to their relative volatiUties. A distribution ratio for mixture component i may be defined using mole fractions ... [Pg.156]

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. [Pg.236]

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. [Pg.236]

Postiilate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. Tnese molar or unit-mass properties, represented by the plain symbols U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V U S ) do depend on system size, and are extensive. For a system containing n moles of fluid, M = nM, where M is a molar property. [Pg.514]

Usually called simply a i< -value, it adds nothing to thermodynamic knowledge of X T.E. However, its use may make for computational convenience, allowing formal ehmination of one set of mole fractions [yi] or (x,) in favor of the other. Moreover, it characterizes lightness of a constituent species. For a light species, tending to concentrate in the vapor phase, K> 1 for a heavy species, tending to concentrate in the liquid phase, K<1. [Pg.538]

Thermodynamically the partition ratio K° is derived in mole fractions... [Pg.1450]

The protonation equilibria for nine hydroxamic acids in solutions have been studied pH-potentiometrically via a modified Irving and Rossotti technique. The dissociation constants (p/fa values) of hydroxamic acids and the thermodynamic functions (AG°, AH°, AS°, and 5) for the successive and overall protonation processes of hydroxamic acids have been derived at different temperatures in water and in three different mixtures of water and dioxane (the mole fractions of dioxane were 0.083, 0.174, and 0.33). Titrations were also carried out in water ionic strengths of (0.15, 0.20, and 0.25) mol dm NaNOg, and the resulting dissociation constants are reported. A detailed thermodynamic analysis of the effects of organic solvent (dioxane), temperature, and ionic strength on the protonation processes of hydroxamic acids is presented and discussed to determine the factors which control these processes. [Pg.40]

Thermodynamic data show that the stabilities of the caesium chloride-metal chloride complexes are greater than the conesponding sodium and potassium compounds, and tire fluorides form complexes more readily tlrair the chlorides, in the solid state. It would seem that tire stabilities of these compounds would transfer into tire liquid state. In fact, it has been possible to account for the heats of formation of molten salt mixtures by the assumption that molten complex salts contain complex as well as simple anions, so tlrat tire heat of formation of the liquid mixtures is tire mole fraction weighted product of the pure components and the complex. For example, in the CsCl-ZrCU system the heat of formation is given on each side of tire complex compound composition, the mole fraction of the compound... [Pg.349]

To extract a desired component A from a homogeneous liquid solution, one can introduce another liquid phase which is insoluble with the one containing A. In theory, component A is present in low concentrations, and hence, we have a system consisting of two mutually insoluble carrier solutions between which the solute A is distributed. The solution rich in A is referred to as the extract phase, E (usually the solvent layer) the treated solution, lean in A, is called the raffinate, R. In practice, there will be some mutual solubility between the two solvents. Following the definitions provided by Henley and Staffin (1963) (see reference Section C), designating two solvents as B and S, the thermodynamic variables for the system are T, P, x g, x r, Xrr (where P is system pressure, T is temperature, and the a s denote mole fractions).. The concentration of solvent S is not considered to be a variable at any given temperature, T, and pressure, P. As such, we note the following ... [Pg.320]

Interpreting results of the pinch diagram As can be seen from Fig. 3.12, the pinch is located at the corresponding mole fractions (y,Xi.jc ) - (0.(K)10, 0.0030, 0.0010). The excess capacity of the process MSAs is 1.4 x lO" kg mol benzene/s and cannot be used because of thermodynamic and practical-feasibility limitations. This excess can be eliminated by reducing the outlet compositions and/or flowrates of the process MSAs. Since the inlet composition of S2 corresponds to a mole fraction of 0.0015 on the y scale, the waste load immediately... [Pg.56]

A gaseous emission has a flowrate of 0.02 kmole/s and contains 0.014 mole fraction of vinyl chloride. The supply temperature of the stream is 338 K. It is desired to recover 80% of the vinyl chloride using a combination of pressurization and cooling. Available for service are two refrigerants NH3 and Nj. Thermodynamic and economic data are provided by problem 10.1 and by Dunn etal. (1995), Design a cost-effective energy-induced separation system. [Pg.260]

To examine the situation with alloys in a little more detail, the Cu-Ni alloys will first be considered. Here the mutual solubility of the two oxides NiO and CU2O can probably be neglected, and these are the only two possible oxidation products. Assume for simplicity that the alloy is thermodynamically ideal, and let and Xn be the mole fractions in the alloy. Consider the reactions... [Pg.263]

For pure substances, n is usually held constant. We will usually be working with molar quantities so that n = 1. The number of moles n will become a variable when we work with solutions. Then, the number of moles will be used to express the effect of concentration (usually mole fraction, molality, or molarity) on the other thermodynamic properties. [Pg.9]

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. [Pg.383]

The thermodynamic activity equilibrium constant (Ka) is expressed in terms of mole fraction (X) and activity coefficient (y) by the following equation ... [Pg.385]

What Do We Need to Know Already This chapter develops the concepts of physical equilibria introduced in the context of thermodynamics (Chapters 6 and 7) and assumes a knowledge of intermolecular forces (Sections 5.1-5.5). The compositions of some of the solutions discussed are expressed in terms of mole fraction (Section 4.8). [Pg.430]

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. [Pg.18]

Equation 4.26 defines the relationship between the vapor and liquid mole fractions and provides the basis for vapor-liquid equilibrium calculations on the basis of equations of state. Thermodynamic models are required for (/) and [ from an equation of state. Alternatively, Equations 4.21, 4.22 and 4.25 can be combined to give... [Pg.60]

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) ... [Pg.266]

The observations on which thermodynamics is based refer to macroscopic properties only, and only those features of a system that appear to be temporally independent are therefore recorded. This limitation restricts a thermodynamic analysis to the static states of macrosystems. To facilitate the construction of a theoretical framework for thermodynamics [113] it is sufficient to consider only systems that are macroscopically homogeneous, isotropic, uncharged, and large enough so that surface effects can be neglected, and that are not acted on by electric, magnetic or gravitational fields. The only mechanical parameter to be retained is the volume V. For a mixed system the chemical composition is specified in terms of the mole numbers Ni, or the mole fractions [Ak — 1,2,..., r] of the chemically pure components of the system. The quantity V/(Y j=iNj) is called the molar... [Pg.408]

The simplest rules of thermodynamics suggest that energy must be expended to do work— You cannot get something for nothing, and that even if work is done some energy is forever lost to useful work— You cannot even get what you paid for . And that this entropy effect is such that the entropy of the universe is forever driving toward a maximum— Nature spontaneously falls into a mess Humor aside, the consequence is that any narrow packet as described above will spread over space in an attempt to make the local and universal mole fraction of A, B or C. .. the same everywhere. [Pg.406]

Species concentrations are shown in Figure 12. At 34 GPa (2.0g/cc), H2O is the predominant species, with H30+ and OH having mole fractions of ca. 5%. In addition, some aggregation has occurred in which neutral and ionic clusters containing up to six oxygens have formed. The concentrations of OH and H30+ are low for all densities investigated and nonexistent at 95 and 115 GPa (2.8 and 3.0g/cc, respectively). The calculated lifetimes for these species are well below 10 fs for the same thermodynamic conditions (less than 8 fs at 34 GPa). At pressures of 95 and 115 GPa, the increase in the O-H bond distance leads to the formation of extensive bond networks (Figure 13). These networks consist entirely of O-H bonds, whereas 0-0 and H-H bonds were not found to be present at any point. [Pg.178]


See other pages where Thermodynamic mole fractions is mentioned: [Pg.23]    [Pg.214]    [Pg.341]    [Pg.777]    [Pg.255]    [Pg.1102]    [Pg.660]    [Pg.246]    [Pg.33]    [Pg.270]    [Pg.16]    [Pg.134]    [Pg.372]    [Pg.17]    [Pg.64]    [Pg.243]    [Pg.63]    [Pg.77]    [Pg.267]    [Pg.403]    [Pg.299]    [Pg.250]   
See also in sourсe #XX -- [ Pg.397 ]




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