Mole fraction relationships

It follows from rearranging the rate balance for a component / that 

It may be observed that the units for V are the same as for both V/A and P Py, where in its usage P refers to the overall permeability rather than the pointwise permeability that is, strictly speaking, the comparison 

Note in particular that the ratios of values as just represented is not the selectivity, as would be defined by the permeability ratios that is, by a- i = PJPj. Furthermore, the selectivity as used here is different than the concept of relative volatility, which would be the ratio of the X-values, one to another. Furthermore, the ratio of the /C-values so determined is lower than would be suspected from the ratio of the permeabilities. The implication is that the ensuing permeability separations are much less sharp than would be suspected from the permeability ratios or selectivities. 

In other words, the presence of the parameter or variable V , notably, affects these ratios of K-values that is, 

Furthermore, the relatively larger is the value of V , the more likely that the K ratio, as designated previously, approximates the permeability ratio. The relatively smaller is the value of V , the more likely that the K ratio approaches unity. Last, the K ratio must be greater than unity for a separation to occur. 

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The primary family of curves in Figure 2 shows a series of structures having the same cadmium activity-mole fraction relationships and separated by two-phase regions of equal width. Similar relationships are suggested for the secondary family also. 

Last, the relationship between n and "mole fraction" when two commercial surfactants are mffiid is always linear or very nearly so (10,21). This is never true for a mixture of two pure surfactants and a variety of different-shaped curves has been seen. Most often, the n. value of the mixture is biased towards that of the hiffeer n. component, which is usually that of the higher molecular weight component. This presumably reflects high interfacial activity for that surfactant. When four or more surfactants are mixed, a near linear n. /mole fraction relationship is once again observed. 

Later, this pressure effect became associated with the term Henry s law, the term being used in connection with the behavior of any gas and any liquid. However, it is necessary to point out that although certain chemists continued to look upon Henry s law as a mole ratio relationship, others, for reasons not apparent to the present writer, looked upon the law as a mole fraction relationship, = Kp, K being called the Henry s law constant. 

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... 

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. 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

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. 

A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

For polymer solutions we seek the relationship between mole fraction and volume fraction. Since 02/0i = nNj/Ni, Nj/Ni = (1/n) (02/01). Also,... 

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

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 ... 

In this equation, Pt is the vapor pressure of solvent over the solution, P° is the vapor pressure of the pure solvent at the same temperature, and Xj is the mole fraction of solvent. Note that because Xj in a solution must be less than 1, P must be less than P°. This relationship is called Raoult s law Francois Raoult (1830-1901) carried out a large number of careful experiments on vapor pressures and freezing point lowering. 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

By the solvent, we mean the component with the larger mole fraction, Mathematically, we can express this relationship as... 

When deviations from ideal solution behavior occur, the changes in the deviations with mole fraction for the two components are not independent, and the Duhem-Margules equation can be used to obtain this relationship. The allowed combinations"1 are shown in Figure 6.10 in which p /p and P2//>2 are... 

We used the system (.vic-Q,H 1CH3 +. vic-CeH ) as an example of a system that closely approximates ideal behavior. Figure 6.5 showed the linear relationship between vapor pressure and mole fraction for this system. In this Figure, vapor pressure could be substituted for vapor fugacity, since at the low pressure involved, the approximation of ideal gas behavior is a good one, and... 

Increasing the temperature increases the vapor pressures and moves the liquid and vapor curves to higher pressure. This effect can best be seen by referring to Figure 8.14, which is a schematic three-dimensional representation for a binary system that obeys Raoult s law, of the relationship between pressure, plotted as the ordinate, mole fraction plotted as abscissa, and temperature plotted as the third dimension perpendicular to the page. The liquid and vapor lines shown in Figure 8.13 in two dimensions (with Tconstant)... 

It can be immediately seen from the above relationship that when a component obeys Raoult s law, its activity is equal to its mole fraction... 

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... 

Ehase Inversion Temperatures It was possible to determine the Phase Inversion Temperature (PIT) for the system under study by reference to the conductivity/temperature profile obtained (Figure 2). Rapid declines were indicative of phase preference changes and mid-points were conveniently identified as the inversion point. The alkane series tended to yield PIT values within several degrees of each other but the estimation of the PIT for toluene occasionally proved difficult. Mole fraction mixing rules were employed to assist in the prediction of such PIT values. Toluene/decane blends were evaluated routinely for convenience, as shown in Figure 3. The construction of PIT/EACN profiles has yielded linear relationships, as did the mole fraction oil blends (Figures 4 and 5). The compilation and assessment of all experimental data enabled the significant parameters, attributable to such surfactant formulations, to be tabulated as in Table II. 

Incorporation of higher mole percent palmitic acid in films spread from both isomers causes an expansion of the films beyond that obtained from simple additivity relationships. At all mole fractions of palmitic acid, the... 

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