Fractions, evaluation

Equations (9.9), (9.10), (9.11), or (9.12) in conjunction with Equation (9.2) give expressions for the molar heat capacity of a saturated phase. However, each equation contains the quantity (dS/dXi)TtPtX, which in turn contains terms such as (H — Hk) when xk is taken to be the dependent mole fraction. Evaluation of such quantities requires the knowledge of the absolute values of the enthalpies. Therefore, such terms cannot be evaluated, and the values of the molar heat capacities cannot be calculated. The necessity of knowing the absolute values of the enthalpies arises from the fact that a number of moles of some components must be added to, and the same number of moles of other components must be removed from the 1 mole of saturated phase in order to change the mole fractions of the phase. However, if the saturated phase is pure, even though it is in equilibrium with other phases that are solutions, the molar enthalpy of the phase is not a function of the mole fractions and Equations (9.9)—(9.12) reduce to Equation (9.3). 

Nitroreductase activity has been demonstrated in liver homogenates as well as in the soluble fraction, whereas other studies have reported that nitroreductase activity has been found in all liver fractions evaluated. The reductase appears to be distributed in liver, kidney, lung, heart, and brain. The reaction utilizes both NADPH and NADH and requires anaerobic conditions. The reaction can be inhibited by the addition of oxygen. The reaction is stimulated by FMN and FAD, and at high flavin concentrations they can act simply as nonenzymatic electron donors. The reduction... 

The voids fraction, evaluated from the density of the composite and the components, is taken as a quality control. If the mass of voids is considered negligible and the volume they occupy is Vy, then... 

Now integrate the first equation by the method of partial fractions. Evaluate the integration constant for x = 0 when t — 0 and show that... 

FIGURE 10.10 Free-volume fractions evaluated from PALS and dilatometric data (f) and theoretical (h) (continuous Unes). Circles, spherical holes squares, anisotropic holes. 

RP-HPLC SEPARATION OF CASEIN FRACTIONS EVALUATION OF CHEESE PROTEOLYSIS... 

The composite curves (including utilities) are divided into enthalpy intervals. The minimum (fractional) number of shells for the temperatures of each interval k is evaluated using Eqs. (D.7) to (D.9). 

Although distillation and elemental analysis of the fractions provide a good evaluation of the qualities of a crude oil, they are nevertheless insufficient. Indeed, the numerous uses of petroleum demand a detailed molecular analysis. This is true for all distillation fractions, certain crude oils being valued essentially for their light fractions used in motor fuels, others because they make quality lubricating oils and still others because they make excellent base stocks for paving asphalt. 

MAV is expressed in mg of anhydride per gram of sample. It is still widely used to evaluate the quantity of conjugated, olefins in a fraction. This type of molecule is highly undesirable in a large number of end products because of its propensity to polymerize spontaneously and to form gums. 

The correction due to the temperature gradient in the capillary wave peak heights is the corresponding fractional difference, which can be obtained by evaluating A(<7, = w. The result is simple ... 

This hierarchical extrapolation procedure can save a significant amount of computer time as it avoids a large fraction of the most time consuming step, namely the exact evaluation of long range interactions. Here, computational... 

Ore samples are analyzed for %w/w Ni. A jaw crusher is used to break the original ore sample into smaller pieces that are then sieved into 5 size fractions. A portion of each fraction is reduced in size using a disk mill and samples taken for analysis by coning and quartering. The effect of particle size on the determination of %w/w Ni is evaluated. 

The first and second columns of Table 1.4 give the number of moles of polymer in six different molecular weight fractions. Calculate and for this polymer and evaluate a using both Eqs. (1.7) and (1.18). 

Apply Eq. (2.27) to some of the data points to evaluate the apparent viscosity at different 7 s. The first section of Table 2.2 shows the results of such calculations. Note that the calculated 17 s are constant at low 7 values, indicating Newtonian behavior. Table 2.2 also expresses all 17 values relative to the Newtonian limiting value 17 - Comparison of Eqs. (2.28) and (2.29) shows that t7/t7im values decrease from the Newtonian limit by the fraction sinh" (j37)/j37. 

An important application of Eq. (3.39) is the evaluation of M, . Flory et al.t measured the tensile force required for 100% elongation of synthetic rubber with variable crosslinking at 25°C. The molecular weight of the un-cross-linked polymer was 225,000, its density was 0.92 g cm , and the average molecular weight of a repeat unit was 68. Use Eq. (3.39) to estimate M. for each of the following samples and compare the calculated value with that obtained from the known fraction of repeat units cross-linked ... 

With these ideas in mind, we now turn to the question of evaluating the fraction of n-mers in a mixture as a function of p. The fraction of molecules of a particular type in a population is just another way of describing the probability of such a molecule. Hence our restated objective is to find the probability of an n-mer in terms of p. We symbolize this quantity P(n, p). Since the n-mer consists of n - 1 a s and 1 A, its probabiUty is the same as the probability of finding n - 1 a s and 1 A in the same molecule. Recalling from Chap. 1 how such probabilities are compounded, we write... 

The fraction of n-mers formed by combination may be evaluated by dividing d[Mjj-]/dt by 2j,d[Mj, ]/dt. Assuming that termination occurs exclusively by combination, then... 

The equations derived in Sec. 6.7 are based on the assumption that termination occurs exclusively by either disproportionation or combination. This is usually not the case Some proportion of each is the more common case. If A equals the fraction of termination occurring by disproportionation, we can write n = A[ 1/1 - p] + (1 - A)[2/(l - p)] and n /n = A(1 + p) + (1 - A)[(2 + p)/2]. From measurements of n and n /n it is possible in principle to evaluate A and p. May and Smith have done this for a number of polystyrene samples. A selection of their data for which this approach seems feasible is presented ... 

We recall that the fraction of times a particular outcome occurs is used to estimate probabilities. Therefore we could evaluate Ph/h t>y counting the number of times N j the first toss yielded a head and the number of times N j j two tosses yielded a head followed by a head and write... 

The copolymer composition equation relates the r s to either the ratio [Eq. (7.15)] or the mole fraction [Eq. (7.18)] of the monomers in the feedstock and repeat units in the copolymer. To use this equation to evaluate rj and V2, the composition of a copolymer resulting from a feedstock of known composition must be measured. The composition of the feedstock itself must be known also, but we assume this poses no problems. The copolymer specimen must be obtained by proper sampling procedures, and purified of extraneous materials. Remember that monomers, initiators, and possibly solvents are involved in these reactions also, even though we have been focusing attention on the copolymer alone. The proportions of the two kinds of repeat unit in the copolymer is then determined by either chemical or physical methods. Elemental analysis has been the chemical method most widely used, although analysis for functional groups is also employed. 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

This result enables us to convert mole fractions to volume fractions. Table 8.1 lists the corresponding values of 0j and Xj for n = 50, 100, and 500 as needed for the evaluation of ASj . With Xj s and the corresponding 0- s available, the required values of are calculated by Eq. (8.38) ... 

From plots of n/c2 versus C2, evaluate M for each of the four polymer fractions. Do the data collected from the two different solvents conform to expectations with respect to slope and intercept values ... 

Based on these ideas, the intrinsic viscosity (in 0 concentration units) has been evaluated for ellipsoids of revolution. Figure 9.3 shows [77] versus a/b for oblate and prolate ellipsoids according to the Simha theory. Note that the intrinsic viscosity of serum albumin from Example 9.1-3.7(1.34) = 4.96 in volume fraction units-is also consistent with, say, a nonsolvated oblate ellipsoid of axial ratio about 5. 

The first term reflects the fact that, in practice, volume fraction is not the concentration unit ordinarily used. Even for nonsolvated spheres, some factors will modify the Einstein 2.5 term merely as a result of reconciling practical concentration units with

[Pg.597]

Since the factor (1 - p/p2)co /2RT is common to both fractions, it can be evaluated separately ... 

Figure 9.14 Calibration curve for GPC as log M versus the retention volume Vj, showing how the location of the detector signal can be used to evaluate M. Also shown are the void volume Vy and the internal volume Vj in relation to Vj, and KVj as a fraction of Vj.

Use the molecular weights given to evaluate the factor 1 - p/p2 for each of the systems then use these factors to evaluate M for the other fractions. Compare the molecular weights obtained in the two solvents. 

Calculate Mn and Mvv the ratio M /Mn for the original polymer. Also evaluate the ratio Mw/Mn for the individual fractions. Comment on the significance of Mvv/Mn for both the fractionated and unfractionated polymer. 

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