Limiting expansion factor

Equation (10) directs attention to a number of important characteristics of the molecular expansion factor a. In the first place, it is predicted to increase slowly with molecular weight (assuming t/ i(1 — 0/T) >0) and without limit even when the molecular weight becomes very large. Thus, the root-mean-square end-to-end distance of the molecule should increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain configuration according to which the unperturbed root-mean-square end- o-end distance is proportional to (Chap. X), whereas /r = ay/rl. 

Qz = the volumetric flow rate in the reactor at length z x = the conversion of the limiting reactant at the same length eR = the expansion factor. 

All reactants and products are gases. If the limiting reactant, on which the expansion factor is based, is in very low concentration, i.e. the nonlimiting reactants are in great excess, then the gas volume cannot be changed considerably. This case is equivalent to the existence of a great excess of inerts, and the expansion could be taken as zero. 

This is why the limiting reactant is B and expansion factor should be based on B, but this is not a gaseous species, and thus does not contribute to the volume of the gas-phase. However, we can use the same equation in the following form ... 

If all products are liquids and the gas phase constitutes only one compound A, irrespective of whether it is the limiting reactant or not, its gas-phase concentration is constant (see Examples 4 and 5). However, in this case, the expansion factor is -1, and thus we cannot use eq. (3.365) to draw conclusions unless the conversion of A is very low. 

The limiting reactant is S02 and the expansion factor is zero. Then using eq. (5.125),... 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

At first it should be mentioned that due to the great excess of inerts (79%) and low concentration of CO (180 4000 ppm), the expansion factor could be taken equal to zero. Isothermal fixed bed. Since the phenomenon is rate-limited,... 

As already remarked in Section C, the theory presented here permits the Mark-Houwink-Sakurada index v to attain a limiting value of unity in extremely good solvents. Since v for cellulosic chains is frequently quite high, say between 0.8 and 1.0 [see Table 13 in the Appendix], it is clear that application of the new equations to these macromolecules would lead to a large expansion factor and hence to a relatively small unperturbed dimension. This is just contrary to a commonly held view, according to which certain cellulose derivatives are supposed to have abnormally extended unperturbed chains and a very small expansion factor even in good solvents. 

Equations (3.125) and (3.126) together with Eq. (3.120) show that the expansion factor depends significantly on two molecular parameters. The first is molecular weight. At conditions far removed from unperturbed conditions, a increases without limit as the square root of the molecular weight. The second parameter determining a is A2, which chracterizes polymer-solvent interaction. Under theta conditions at which z becomes zero, a becomes unity. Physical measurements made under these conditions will reflect the characteristics of the unperturbed molecule. The overall dimensions of such a molecule will be determined solely by bond lengths... 

The integral in the last expression above is not a simple form and is best evaluated by numerical means. Use of the expansion factor is limited to reactions where there is a linear relationship between conversion and volume. For reactions that have complex sequences of steps, a linear relationship may not be true. Then we must rewrite the rate definition for a batch reactor ... 

Note that for polymer in the bulk, is very large. This means that Cm 0 just as it is if X1 =i- Accordingly, a = 1 in the melt, a result that is identical with that for a 0-solvent. The expansion factor a increases slowly without limit with increasing molecular weight M. 

As mentioned repeatedly, according to the two-parameter theory, the expansion factor as> the penetration function and the hydrodynamic expansion factors a, and at in the non-draining limit should become universal functions of a single variable z. These non-dimensional quantities are experimentally determinable without any assumption. Thus, the validity of the two-parameter theory can be tested directly by looking at whether a single curve independent of polymer rind solvent condition (solvent species and temperature) is obtciined or not when any of them is plotted against the other. Such tests were made by many authors (for example, see Ref. [2] and [119]). Here we refer to a recent one by Miyaki and Fujita [49] (and also Miyaki [44]), who used the following criteria A and B. 

With eq 1.4 the end distance expansion factor and the radius expansion factor as can be calculated analytically (in the continuous chain limit). The results are as follows ... 

In the transition region, where x < 1, the expansion factor is close to unity a 1. The nature of the transition depends on the value of y. If y takes a value larger than the critical value yc = 0.0228, the transition from swollen coil to globule is a gradual crossover. If it is smaller than the critical value, the equation (1.86) has three solutions, so that the tfansition becomes discontinuous similarly to the first-order phase transition (Figure 1.8(a)). The transition temperature Xc lies I/a/m below the temperature 6. It approaches the temperature 0 in the limit of infinite molecular weight. 

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