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R7.1 Polymerization

R7.1 Polymerization

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Termination of R 1<br />

Net rate of<br />

disappearance of<br />

radicals of chain<br />

length one<br />

Net rate of<br />

disappearance of<br />

radicals of chain<br />

length j<br />

365<br />

In general,<br />

R1<br />

� Rj<br />

Chap.<br />

Consequently, the total loss of R1<br />

radicals in the preceding reactions is found<br />

by adding the loss of R1<br />

radicals in each reaction so that the rate of disappearance<br />

by termination addition is given by<br />

�r1t<br />

� k<br />

ta<br />

2 R1 �r1t<br />

� ktaR1<br />

⎯⎯→ P j<br />

� ktaR1R2<br />

� ktaR1R3<br />

� ��� � ktaR1Rj<br />

� ���<br />

�<br />

�<br />

j�1<br />

R j<br />

Free radicals usually have concentrations in the range 10�6<br />

to 10�8<br />

mol/dm3 .<br />

We can now proceed to write the net rate of disappearance of the free radical,<br />

R1. [R1 � (R1) � .]<br />

C R1<br />

(<strong>R7.1</strong>-10)<br />

In general, the net rate of disappearance of live polymer chains with j<br />

monomer units (i.e., length j ) for ( j � 2) is<br />

(<strong>R7.1</strong>-11)<br />

At this point one could use the techniques developed in Chapter 6 on multiple<br />

reactions to follow polymerization process. However, by using the PSSH, we<br />

can manipulate the rate law into a form that allows closed-form solutions for a<br />

number of polymerization reactions.<br />

First, we let R � be the total concentration of the radicals R j:<br />

R � � (<strong>R7.1</strong>-12)<br />

and kt be the termination constant kt � (kta � ktd). Next we sum Equation<br />

(<strong>R7.1</strong>-11) over all free-radical chain lengths from j � 2 to j � �,<br />

and then add<br />

the result to Equation (<strong>R7.1</strong>-10) to get<br />

k<br />

ta<br />

�1<br />

�r1 �ri k p R1 M kta R1 R j ktd<br />

R1<br />

R<br />

�<br />

� � �<br />

�<br />

�<br />

� k m M R �<br />

j�2<br />

j<br />

k<br />

c<br />

�<br />

�<br />

j�1<br />

�<br />

�<br />

C R �<br />

j�2<br />

j<br />

k<br />

s<br />

�<br />

�<br />

j�1<br />

�<br />

�<br />

S R<br />

j�2<br />

�r j k p MR ( j� R j�1)<br />

( kta� ktd)Rj R<br />

�<br />

�<br />

�<br />

�<br />

i�1<br />

�kMR �k�ksSRj m<br />

�<br />

�<br />

j�1<br />

j<br />

R j<br />

cCR j<br />

i<br />

j<br />

j

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