Calculations using the general Carothers equation: For linear step polymerization using two bifunctional monomers A-R1-A and B-R2-B (A and B are mutually reactive function groups)
a. If 1 mol A-R1-A and 1 mol B-R2-B are used, and if the extent of reaction p is 0.99, calculate the degree of polymerization Xn
b. If 0.98 mol A-R1-A and 1.02 mol B-R2-B are used, and if the extent of reaction p is still 0.99, calculate the degree of polymerization Xn
c. By comparing the results in a and b, what is the effect of reactant ratio r on the degree of polymerization Xn? What is the reason for this large effect (please try to explain it using the reactions between the monomers and the chain end groups formed)?
The correct answer and explanation is :
To calculate the degree of polymerization ((X_n)) for a linear step polymerization using the general Carothers equation, we use the following relation:
[
X_n = \frac{1}{1 – p}
]
Where:
- (p) is the extent of reaction (fraction of reactive groups that have reacted).
Part a: 1 mol A-R1-A and 1 mol B-R2-B, with (p = 0.99)
In this case, the reactants are in a 1:1 ratio (1 mol A-R1-A and 1 mol B-R2-B), and the extent of reaction is 0.99.
Using the Carothers equation:
[
X_n = \frac{1}{1 – 0.99} = \frac{1}{0.01} = 100
]
So, the degree of polymerization (X_n) is 100.
Part b: 0.98 mol A-R1-A and 1.02 mol B-R2-B, with (p = 0.99)
In this case, we have a slight imbalance in the ratio of A-R1-A and B-R2-B. The extent of reaction is still (p = 0.99).
Again, using the Carothers equation:
[
X_n = \frac{1}{1 – 0.99} = \frac{1}{0.01} = 100
]
So, the degree of polymerization (X_n) is still 100.
Part c: Effect of reactant ratio (r) on (X_n)
From the results of part a and part b, we see that both scenarios give the same degree of polymerization, (X_n = 100), despite the slight imbalance in the molar ratio of the monomers (0.98 mol A-R1-A and 1.02 mol B-R2-B in part b vs. 1 mol of each in part a).
This result may seem surprising at first, since intuitively, we would expect the ratio of reactants to affect the polymerization process. However, the degree of polymerization in step-growth polymerization is primarily determined by the extent of reaction (p), which is the fraction of reactive groups that have reacted, rather than by the exact stoichiometric balance of the monomers. As long as the extent of reaction is high (close to 1), the difference in the amount of monomers doesn’t significantly impact the degree of polymerization.
However, the reactant ratio does play a critical role in the kinetics of the reaction. When the ratio of the monomers is not exactly 1:1, the unreacted functional groups of one type (say, A or B) could become excess, leading to a higher likelihood of termination or reduced efficiency of the polymerization. A slight excess of one monomer can cause a higher concentration of unreacted chain ends, potentially decreasing the overall polymerization rate. But once the reaction is close to completion (with a high extent of reaction, (p \approx 0.99)), this effect becomes less pronounced.
The reason the ratio has little effect in these cases is that, in both scenarios, the extent of reaction (p) has reached a very high value (0.99). This means that almost all of the available functional groups are reacting, and the polymer chains are nearly fully formed. Therefore, as long as the reaction is near completion, small deviations in the reactant ratio do not drastically change the degree of polymerization. However, if the extent of reaction were lower, the imbalance in reactant ratios could have a more pronounced effect on (X_n).