R7.1 Polymerization

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Termination of R 1 Net rate of disappearance of radicals of chain length one Net rate of disappearance of radicals of chain length j 365 In general, R1 � Rj Chap. Consequently, the total loss of R1 radicals in the preceding reactions is found by adding the loss of R1 radicals in each reaction so that the rate of disappearance by termination addition is given by �r1t � k ta 2 R1 �r1t � ktaR1 ⎯⎯→ P j � ktaR1R2 � ktaR1R3 � �� ktaR1Rj � �� j�1 R j Free radicals usually have concentrations in the range 10�6 to 10�8 mol/dm3 . We can now proceed to write the net rate of disappearance of the free radical, R1. [R1 � (R1) � .] C R1 (R7.1-10) In general, the net rate of disappearance of live polymer chains with j monomer units (i.e., length j ) for ( j � 2) is (R7.1-11) At this point one could use the techniques developed in Chapter 6 on multiple reactions to follow polymerization process. However, by using the PSSH, we can manipulate the rate law into a form that allows closed-form solutions for a number of polymerization reactions. First, we let R � be the total concentration of the radicals R j: R � � (R7.1-12) and kt be the termination constant kt � (kta � ktd). Next we sum Equation (R7.1-11) over all free-radical chain lengths from j � 2 to j � �, and then add the result to Equation (R7.1-10) to get k ta �1 �r1 �ri k p R1 M kta R1 R j ktd R1 R � � � � � � � k m M R � j�2 j k c � � j�1 � � C R � j�2 j k s � � j�1 � � S R j�2 �r j k p MR ( j� R j�1) ( kta� ktd)Rj R � � � � i�1 �kMR �k�ksSRj m � � j�1 j R j cCR j i j j
Total free-radical concentration Long-chain approximation (LCA) Sec. R7.1 Polymerization The total rate of termination is just �rj � �ri � kt( R�) 2 Using the PSSH for all free radicals, that is, concentration solves to j�1 366 (R7.1-13) �rj � 0, the total free-radical (R7.1-14) We now use this result in writing the net rate of monomer consumption. As a first approximation we will neglect the monomer consumed by monomer chain transfer. The net rate of monomer consumption, �rM, is the rate of consumption by the initiator plus the rate of consumption by all the radicals Rj in each of the propagation steps ( r ). p �rM � �ri � �rp � �ri � kpM We now use the long-chain approximation (LCA). The LCA is that the rate of propagation is much greater than the rate of initiation: Substituting for r p and r i,we obtain � � rt kt R� ( ) 2 � �ri kt R� � -------- � �k p MR � r p ri ---- � 1 Consequently, we see that the LCA is valid when both the ratio of monomer concentration to initiator concentration and the ratio of k2 p to (k0 fkt) are high. Assume that the LCA gives � � j�1 2k0 ( I2)f -------------------- k t k p 2k 0 f I 2 � � j�1 R j r p ( ( ) � kt ) 1 2 ---- -------------------ri �ki MI � � � --------------------------------------------ki ( 2k0 f( I2) � Mki ) M k2 p � ------- --------------- 1�2 I 2 k 2 0 fkt
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R7.1 Polymerization

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