Composition conversion equations

In order to determine the reactivity of pentachlorophenyl acrylate, 8, in radical initiated copolymerizations, its relative reactivity ratios were obtained with vinyl acetate (M2), ri=1.44 and r2=0.04 using 31 copolymerization experiments, and with ethyl acrylate (M2), ri=0.21 and r2=0.88 using 20 experiments.The composition conversion data was computer-fitted to the integrated form of the copolymer equation using the nonlinear least-squares method of Tidwell and Mortimer,which had been adapted to a computerized format earlier. 

It is often necessary to convert between the mole fractions (m), weight fractions (w) and volume fractions () of the components in dealing with multicomponent polymeric materials such as copolymers, blends and composites. The equations needed to make these conversions are listed below, for the i th component of an n-componcnt system. In these equations, p is the density, M is the molecular weight per mole, and V is the molar volume. 

Equation 7.36 is only valid for sharp split problems. This is quite a remarkable outcome, because it means at sharp split conditions, one has to operate on the A7 = 0 line in Figure 7.44 regardless of the value of R i or the feed composition. Conversely, it can be said that if the vapor and liquid flowrates in CS7 are not kept equal to each other, a sharp split will never be obtained. This is true regardless of the thermodynamic properties of the system. 

Knowledge of /ap is all that is required to calculate the composition of the copolymer formed, assuming that the reactivity ratios for copolymerization are known. In order to predict composition drift. Equations (7.20) and (7.22) must be solved to give /ap as a function of total comonomer conversion, which sinq)ly requires iterative solution of the equations for small increments of conversion over which the copolymer composition can be assumed constant. 

The instantaneous composition of the copolymer is equal to the change in monomer concentration, however, since the lost monomer molecules must be incorporated into the copolymer. Since this stipulation must also apply at every conversion, equation (22-12) may be revised to... 

However, there is one important difference between free-radical and living anionic polymerization, and this is the lifetime of the growing chain. This difference becomes important when considering the dependence of copolymer composition on conversion. Equation (42) gives the copolymer composition or mole fraction Fi of monomer Mi in the polymer as a function of mole fraction / in the monomer... 

Sometimes it is necessary to convert from one composition scheme to another—for example, from weight percent to atom percent. We next present equations for making these conversions in terms of the two hypothetical elements 1 and 2. Using the convention of the previous section (i.e., weight percents denoted by Cj and C2, atom percents by C l and C2, and atomic weights as Aj and A2), we express these conversion equations as follows ... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Substituting Equations 5-330 and 5-331 in the design Equations 5-328 and 5-329, values of exit conversions Xj, Xj, and the composition of die components are computed for various values of V/E. 

The computer program PLUG51 employing die Runge-Kutta fourdi order numerical mediod was used to determine die conversions and the compositions of die components. Applying die Runge-Kutta mediod, Equations 5-328 and 5-329 in differential forms are... 

The computer program PLUG51 used Equation 5-334 to determine the conversions and the compositions of the components. Table 5-13 illustrates the results of the computer program, and Figure 5-32 shows the plots of the rates of each reaction as a function of V/F. In both instances, the rates decrease toward zero as V/F increases. Figure 5-33 shows the plot of the total conversion versus V/F. 

With Freon 112 or 113 as a solvent, fluonnation of pnmary butyl halides with bromine trifluonde can give mixtures of primary and secondary fluorides When 1,4 dibromobutane is the substrate, 93% l-bromo-4-fluorobutane and 1% 1-bro-mo-3-fluorobutane is obtained, with 1,4 dichlorobutane, the product contains 65% l-chloro-3-fluorobutane and 35% 1-chloro 4 fluorobutane When 4-bromo- or 4-chlorobutyl trifluoroacetate is used, the ratio of 4-fluorobutyl tnfluoroacetate to 3 fluorobutyl trifluoroacetate is 1 4 The effect of solvent is measured in another set of experiments When the reaction of bromine trifluonde and l,3-dichloro-2-fluoropropane in either Freon 113 or hydrogen fluoride is allowed to proceed to 40% conversion, the product mixture has the composition shown m Table 1 [/O] When 1 chloro 2,3-dibromopropane is combined with one-third of a mole of bromine trifluonde, both 1 bromo 3 chloro-2-fluoropropane and l-chloro-2,3-di-fluoropropane are formed [//] (equation 10)... 

Figure 5 depicts the effect of calcination temperature on subsequent catalyst activity after reduction at 300°C (572°F). Activity was measured in laboratory tubular reactors operating at 1 atm with an inlet gas composition of 0.40% CO, 25% N2, and 74.6% H2, and an inlet temperature of 300°C. Conversion of CO is measured and catalyst activity is expressed as the activity coefficient k in the first order equation ... 

The existence of an azeotropic composition has some practical significance. By conducting a polymerization with the monomer feed ratio equal to the azeotropic composition, a high conversion batch copolymer can be prepared that has no compositional heterogeneity caused by drift in copolymer composition with conversion. Thus, the complex incremental addition protocols that arc otherwise required to achieve this end, are unnecessary. Composition equations and conditions for azeotropic compositions in ternary and quaternary eopolymerizations have also been defined.211,21... 

The traditional method for determining reactivity ratios involves determinations of the overall copolymer composition for a range of monomer feeds at zero conversion. Various methods have been applied to analyze this data. The Fineman-Ross equation (eq. 42) is based on a rearrangement of the copolymer composition equation (eq. 9). A plot of the quantity on the left hand side of eq. 9 v.v the coefficient of rAa will yield rAB as the slope and rUA as the intercept. 

It is also possible to derive reactivity ratios by analyzing the monomer (or polymer) feed composition v.v conversion and solving the integrated form of the Mayo Lewis equation.10 123 The following expression (eq. 44) was derived by Meyer and Lowry 12j... 

Numerical approaches for estimating reactivity ratios by solution of the integrated rate equation have been described.124 126 Potential difficulties associated with the application of these methods based on the integrated form of the Mayo-kewis equation have been discussed.124 127 One is that the expressions become undefined under certain conditions, for example, when rAo or rQA is close to unity or when the composition is close to the azeotropic composition. A further complication is that reactivity ratios may vary with conversion due to changes in the reaction medium. 

Thus, aout/cim = 0.432 for the series combination. A single CSTR with twice the volume has ki ajn = 1. Equation (4.16) gives Uoutlam =0.5 so that the composite reactor with two tanks in series gives the higher conversion. 

Solution The conversion is low so that the polymer composition is given by Equation 13.41 with the monomer concentrations at the initial values. There are five data and only two unknowns, so that nonlinear regression is appropriate. The sum-of-squares to be minimized is... 

Minimizing the cycle time in filament wound composites can be critical to the economic success of the process. The process parameters that influence the cycle time are winding speed, molding temperature and polymer formulation. To optimize the process, a finite element analysis (FEA) was used to characterize the effect of each process parameter on the cycle time. The FEA simultaneously solved equations of mass and energy which were coupled through the temperature and conversion dependent reaction rate. The rate expression accounting for polymer cure rate was derived from a mechanistic kinetic model. 

Equation (7) is a rigorous expression relating the instantaneous vapor and liquid compositions with respect to monomer 1. However, the liquid composition (X ) needs to be related to the polymerization conversion in order to complete the model. 

The estimation of the two parameters requires not only conversion and head space composition data but also physical properties of the monomers, e.g. reactivity ratios, vapor pressure equation, liquid phase activity coefficients and vapor phase fugacity coefficients. 

Figure 2 shows the conversions obtained with the three series studied, as a function of the mechanical mixtures composition, one hour after the beginning of the reaction and at the steady-state. Each series presents a maximum of activity, but at a different composition. SA6 series has a maximum between R , values of 50 and 75, whereas SA12 series has a maximum around = 50, and SA60 series near R , = 75. The dashed lines on the figures represent the sum of the individual contributions of the pure phases, calculated according to Equation 3. A very important synergetic effect is observed in all series, i.e., the activity of the mixtures is... 

Figure 2. DPM conversion as a function of the mechanical mixtures composition, after 1 h ( ) of reaction and at the steady-state ( ), compared to the theoretical values calculated by Equation 3 (dashed lines).

V. Copolvmerization Kinetics. Qassical copolymerization kinetics commonly provides equations for instantaneous property distributions (e.g. sequence length) and sometimes for accumulated instantaneous (i.e. for high conversion samples) as well (e.g. copolymer composition). These can serve as the basis upon whkh to derive nations which would reflect detector response for a GPC separation based upon properties other than molecular weight. The distributions can then serve as c bration standards analagous to the use of molecular weight standards. 

Because the dependence of probability P Uk on x should be established by means of the theory of Markov chains, in order to make such an averaging it is necessary to know how the monomer mixture composition drifts with conversion. This kind of information is available [2,27] from the solution of the following set of differential equations ... 

Let fractional conversion of CO to H2 be C. Then mols of CO reacted = 11.0 x C. From the stoichiometric equation and feed composition, the exit gas composition will be ... 

---