Calculate the extent of reaction at which gelation

1. Calculate the extent of reaction at which gelation occurs for the following mixtures:

a. Phthalic anhydride and glycerol in stoichiometric amounts.
b. Phthalic anhydride and glycerol in the molar ratio 1.5 : 0.98
c. Phthalic anhydride, glycerol and ethylene glycol in the molar ratio 1.5:0.99:0.002
d. Phthalic anhydride, glycerol and ethylene glycol in the molar ratio 1.5 : 0.5 : 0.7

Compare the gel points calculated from the carothers equation (and its modifications)
with those using statistical approach. Describe the effect of unequal functional groups
reactivity (e.g., for the hydroxyl groups in glycerol) on the extent of reaction at the gel
point.



Solution:

a. Consider 2 moles of glycerol : 3 moles of phthalic anhydride


Carothers: f avg 
N f i i

N i



(2  3)  (3  2) 12
f avg    2.4
23 5

2
The critical extent of reaction at the gel point is given by, pc 
f avg

2
pc 
2.4

 pc  0.833
Thus, the glycerol-phthalic anhydride (2:3 molar ratio) system considered has critical
extent of reaction of 0.833 at the gel point.

, b. 1.5 moles of phthalic anhydride : 0.98 moles of glycerol
Carothers: glycerol is not in excess
2  0.98  3
f avg 
1.5  0.98
f avg  2.37097

2
pc 
f avg

2
pc 
2.37097
pc  0.844

Thus, the phthalic anhydride-glycerol (1.5:0.98 molar ratio) system considered has
critical extent of reaction of 0.844 at the gel point.



c. Phthalic anhydride, glycerol and ethylene glycol in the molar ratio (1.5:0.99:0.002)

2  N A f A  NC fC 
Carothers: f avg 
N A  NC  N B


2  0.002  2  0.99  3
f avg 
0.002  0.99  1.5

2
f avg  2.3868 , pc 
f avg


2
pc 
2.3868

pc  0.8379

Thus, the phthalic anhydride, glycerol and ethylene glycol (1.5:0.99:0.002 molar ratio)
system considered has critical extent of reaction of 0.8379 at the gel point.