Ideal fractionation

The performance of an ideal screen in terms of the screen analysis of the feed is shown in Figure 2.13 (A). The cut point is the point C in the curve. Fraction A comprises all particles bigger than Dpc, and fraction B comprises all particles smaller than Dpc. The fractions A and B are the overflow and underflow respectively. Screen analyses of the ideal fractions A... 

A deviation from the ideal fractionation law will appear as a residual correlation between the ratios corrected for mass bias using the exponential law. It can be verified that even very small 8M/M in Equation (40) produces a potentially important isotopic effect on the order of 1 + P(5M,. - 5Mt)/Mj. The alignment of the correlation between the corrected ratios x = hir and y = In rj produced by sloping peaks in a log-log plot has a slope of 5M,/8M,. For a rormd peak, the second-order term should be included. 

Figure 7.10 Boiling-point diagram for a binary solution, illustrating two theoretical plates of an idealized fractional distillation process that progressively enriches the distillate in the low-boiling component A (see text).

Figure 1. The summative-fractionation parameter H (ideal-fractionation assumption) as a function of polydispersity K w/Kln for the Poisson, Schulz-Zimm exponential, Lansing-Kraemer logarithmic-normal, and rectangular...

Figure 2. The summative-fractionation parameter H as a function of polydispersity for the Poisson distribution. H is calculated for the ideal-fractionation assumption while H0.001 and H0 01 we calculated including the Flory-Huggins correction for imperfect fractionation, with the volume ratio R equal to 0.001 and 0.01, respectively.

The usefulness of a particular step may then be evaluated with reference to the increase in the specific activity of the enzyme and percent yield in the fractions of greatest enrichment. An increase in specific activity indicates a purification. An ideal fractionation would provide complete enrichment (pure enzyme) in 100% yield in practice, few pro-... 

An attempt to determine Ufr can be made as shown below, but first the principle of the method should be explained. Figure 2 shows part of a differential weight distribution curve which is divided into fractions. If we could obtain ideal fractions with an approximately rectangular shape, Ufr would be on the order of 10"4 and could be neglected. However the real fractions can have a nonuniformity comparable with that of the sample. Therefore it is necessary to determine, or at least to estimate, Ufr. This is possible by measuring the overlapping effect as shown in Figure 2. 

By solvent extracting the pitch with a solvent system having a solubility parameter of 8.63 a more or less ideal fraction for carbon fiber production is obtained. Extraction with solvent systems of different solubility parameter results in uniform changes in extracted product characteristics which can be tailored to other specific carbon product applications, such as carbon/carbon matrix materials, anodes, etc. 

Fractionation is called ideal when all macromolecules of length p above a certain fixed value Pi are in the precipitate phase (.sec the dashed area of f(p) in Figure 3.16). Ideal fractionation is depicted by lined in Figure 3.15. A problem how much the results of real and ideal fractionations differ is often solved in preparative and analytical fractionation. 

A theoretically ideal fractionation should give 20% of optically pure polymer and 80% of racemate. 

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 ... 

Raoult s law When a solute is dissolved in a solvent, the vapour pressure of the latter is lowered proportionally to the mole fraction of solute present. Since the lowering of vapour pressure causes an elevation of the boiling point and a depression of the freezing point, Raoult s law also applies and leads to the conclusion that the elevation of boiling point or depression of freezing point is proportional to the weight of the solute and inversely proportional to its molecular weight. Raoult s law is strictly only applicable to ideal solutions since it assumes that there is no chemical interaction between the solute and solvent molecules. 

C of the mixture in the ideal gas state Cpgp = Cp of the component i in the ideal gas state = weight fraction of component /... 

For petroleum fractions, there is a problem of coherence between the expression for liquid enthalpy and that of an ideal gas. When the reduced temperature is greater than 0.8, the liquid enthalpy is calculated starting with the enthalpy of the ideal gas. On the contrary, when the reduced temperature is less than 0.8, it is preferable to calculate the enthalpy of the ideal gas starting with the enthalpy of the liquid (... 

The data in Table III-2 have been determined for the surface tension of isooctane-benzene solutions at 30°C. Calculate Ff, F, F, and F for various concentrations and plot these quantities versus the mole fraction of the solution. Assume ideal solutions. 

It is sometimes convenient to retain the generality of the limiting ideal-gas equations by introducing the activity a, an effective pressure (or, as we shall see later in the case of solutions, an effective mole fraction. 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Figure A2.5.3. Typical liquid-gas phase diagram (temperature T versus mole fraction v at constant pressure) for a two-component system in which both the liquid and the gas are ideal mixtures. Note the extent of the two-phase liquid-gas region. The dashed vertical line is the direction x = 1/2) along which the fiinctions in figure A2.5.5 are detemiined.

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

At the outset it will be profitable to deal with an ideal solution possessing the following properties (i) there is no heat effect when the components are mixed (ii) there is no change in volume when the solution is formed from its components (iii) the vapour pressure of each component is equal to the vapour pressure of the pure substances multiplied by its mol fraction in the solution. The last-named property is merely an expression of Raoult s law, the vapour pressure of a substance is pro-... 

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. 

We define Fj to be the mole fraction of component 1 in the vapor phase and fi to be its mole fraction in the liquid solution. Here pj and p2 are the vapor pressures of components 1 and 2 in equihbrium with an ideal solution and Pi° and p2° are the vapor pressures of the two pure liquids. By Dalton s law, Plot Pi P2 Pi/Ptot these are ideal gases and p is propor-... 

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). 

This result should be compared with Eq. (8.28) for the case of the ideal mixture. It is reassuring to note that for n = 1, Eq. (8.36) reduces to Eq. (8.28). Next let us consider whether a change of notation will clarify Eq. (8.36) still more. Recognizing that the solvent, the repeat unit, and the lattice site all have the same volume, we see that Ni/N is the volume fraction occupied by the solvent in the mixture and nN2/N is the volume fraction of the polymer. Letting be the volume fraction of component i, we see that Eq. (8.36) becomes... 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

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