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utilization of the hydroxyl groups [Kienle et al., 1939]. This has been attributed to the lowered reactivity of the secondary hydroxyl group compared to the two primary hydroxyls.
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Kinetic analysis of a step polymerization becomes complicated when all functional groups in a reactant do not have the same reactivity. Consider the polymerization of A with B 0 A B where the reactivities of the two functional groups in the B 0 reactant are initially of dif B ferent reactivities and, further, the reactivities of B and B0 each change on reaction of the other group. Even if the reactivities of the two functional groups in the A reactant are A the same and independent of whether either group has reacted, the polymerization still involves four different rate constants. Any speci c-sized polymer species larger than dimer is formed by two simultaneous routes. For example, the trimer A B0A is formed by AB A
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The two routes (one is Eqs. 2-37b and 2-37c; the other is Eqs. 2-37a and 2-37d) together constitute a complex reaction system that consists simultaneously of competitive, consecutive and competitive, parallel reactions. Obtaining an expression for the concentration of A (or B or B0 ) groups or the extent of conversion or X n as a function of reaction time becomes more dif cult than for the case where the equal reactivity postulate holds, that is, where k1 k2 k3 k4 . As a general approach, one writes an expression for the total rate of disappearance of A groups in terms of the rates of the four reactions 2-37a, b, c, and d and then integrates that expression to nd the time-dependent change in [A]. The dif culty arises because the differential equations that must be integrated are not linear equations and do not have exact solutions except in very particular cases. Numerical methods are then needed to obtain an approximate solution. 2-2d-2-a Polymerization of A with B 0 . The kinetics of the reaction system A B described in Eq. 2-37 is relatively dif cult to treat because there are four different rate constants. Special (and simpler) cases involving only two rate constants can be more easily treated. One such case is where the two functional groups B and B0 have different reactivities but
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their individual reactivities are decoupled in that neither the reactivity of B nor of B0 changes on reaction of the other group:
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The reaction system converts from one (Eq. 2-37), that is, from the kinetic viewpoint, simultaneously competitive, consecutive (series) and competitive, simultaneous (parallel) to one (Eq. 2-38) that is only competitive, simultaneous. The polymerization consists of the B and B0 functional groups reacting independently with A groups. The rates of disappearance of A, B, and B0 functional groups are given by
d B k1 A B dt 0 d B k2 A B0 dt d A k1 A B k2 A B0 dt 2-39 2-40 2-41
The polymerization rate is synonymous with the rate of disappearance of A groups (or the sum of the rates of disappearance of B and B0 groups). In the typical polymerization one has equimolar concentrations of the A and B 0 A B reactants at the start of polymerization. The initial concentrations of A, B, and B0 groups are
A 0 2 B 0 2 B0 0 2-42
and the relationship between the concentrations of A, B, and B0 at any time during polymerization is
A B B0 2-43
Combination of Eqs. 2-41 and 2-43 yields the polymerization rate as
d A k1 k2 A B k2 A 2 dt 2-44
Introduction of the dimensionless variables a, b, g, and t and the parameter s de ned by
a b g A A 0 B B 0 B0 B0 0 2-45a 2-45b 2-45c 2-45d 2-45e
t B 0 k2 t k2 s k1
simpli es the mathematical solution of Eq. 2-44. a, b, and g are the fractions of A, B, and B0 groups, respectively, which remain unreacted at any time. t is dimensionless time and s is the ratio of two rate constants. It is then possible to solve the coupled system of Eqs. 2-39 through 2-44 to give
t 1 s db 2-46 2-47