Fraction, mole

The other temperature-independent concentration unit introduced in this chapter is the mole fraction. The mole fraction of a substance in a solution is the ratio of the number of moles of that substance to the total number of moles in the solution. The symbol for mole fraction of A is usually Xa, although some texts use the symbol Na- Thus, for a solution containing x mol of A, y mol of B, and z mol of C, the mole fraction of A is 

EXAMPLE 15.7. What is the mole fraction of CH3OH in a solution of 12.0 g CH3OH and 25.0 g H2O  

Since the mole fraction is a ratio of moles (of one substance) to moles (total), the units cancel and mole fraction has no units. 

EXAM PLE 15.8. Show that the total of both mole fractions in a solution of two compounds is equal to 1. 

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In vapor-liquid equilibria, if one phase composition is given, there are basically four types of problems, characterized by those variables which are specified and those which are to be calculated. Let T stand for temperature, P for total pressure, for the mole fraction of component i in the liquid phase, and y for the mole fraction of component i in the vapor phase. For a mixture containing m components, the four types can be organized in this way ... 

In each of these problems, there are m unknowns either the pressure or the temperature is unknown and there are m - 1 unknown mole fractions. 

Equation (1) is of little practical use unless the fuga-cities can be related to the experimentally accessible quantities X, y, T, and P, where x stands for the composition (expressed in mole fraction) of the liquid phase, y for the composition (also expressed in mole fraction) of the vapor phase, T for the absolute temperature, and P for the total pressure, assumed to be the same for both phases. The desired relationship between fugacities and experimentally accessible quantities is facilitated by two auxiliary functions which are given the symbols (f... 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

According to Equation (14), the fugacity of component i becomes equal to the mole fraction multiplied by the standard-... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

The fugacity fT of a component i in the vapor phase is related to its mole fraction y in the vapor phase and the total pressure P by the fugacity coefficient ... 

In Equation (13), z and <(iT refer to the monomer of species i while y is the apparent mole fraction of component i, where apparent means that dimerization has been neglected. 

Next, and more difficult, is the calculation of the true mole fraction This calculation is achieved by simultaneous... 

By contrast, in the system propionic acid d) - methyl isobutyl ketone (2), (fi and are very much different when y 1, Propionic acid has a strong tendency to dimerize with itself and only a weak tendency to dimerize with ketone also,the ketone has only a weak tendency to dimerize with itself. At acid-rich compositions, therefore, many acid molecules have dimerized but most ketone molecules are monomers. Acid-acid dimerization lowers the fugacity of acid and thus is well below unity. Because of acid-acid dimerization, the true mole fraction of ketone is signi-... 

Vapor-Phase Mole Fraction Propionic Acid... 

Finally, Table 2 shows enthalpy calculations for the system nitrogen-water at 100 atm. in the range 313.5-584.7°K. [See also Figure (4-13).] The mole fraction of nitrogen in the liquid phase is small throughout, but that in the vapor phase varies from essentially unity at the low-temperature end to zero at the high-temperature end. In the liquid phase, the enthalpy is determined primarily by the temperature, but in the vapor phase it is determined by both temperature and composition. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

The sum of the squared differences between calculated and measures pressures is minimized as a function of model parameters. This method, often called Barker s method (Barker, 1953), ignores information contained in vapor-phase mole fraction measurements such information is normally only used for consistency tests, as discussed by Van Ness et al. (1973). Nevertheless, when high-quality experimental data are available. Barker s method often gives excellent results (Abbott and Van Ness, 1975). 

F, V, and L moles/unit time and their mole-fraction compositions are, respectively, w, y, and x. ... 

X and y are vectors containing the liquid mole fractions and vapor mole fractions respectively. 

Case Flash Number Type Components Mole Fraction Pressure (bar) Temperature (K) Pressure (bar) Temperature V (K) F Mole Fractions Liquid Vapor No. of Iterations... 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

The solvent components usually have a low mutual solubility and are present in reasonably large mole fractions in the system. If solvents are not so designated, we take as the "solvent components" those two components, present in significant mole fraction in the system, that have the lowest binary solubilities. ... 

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

Convergence of this iteration is influenced by initial estimates for the true mole fractions, zThe following rules have been found to lead to rapid convergence in all cases. 

These initial estimates are used in the iteration function. Equation (37), to obtain values of the 2 s that do not change significantly from one iteration to the next. These true mole fractions, with Equation (3-13), yield the desired fugacity... 

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

DX(2) cols 11-20 increment for derivative with respect to the liquid mole fraction, default value is 0.001. 

LIQUTD-PHASE MOLE FRACTION ACTIVITY COEFFICIENTS... 

TRUE VAPOR-PHASE MOLE FRACTION. CALCULATED WHEN THE CHEMICAL THEORY IS USED. 

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