Question

The ‘unperturbed dimension’ of the polymer chain is represented as, \[ \left( \overline{r^2_0} \right)^{1/2} \propto \overline{\ell}(n)^{1/2} \] where, \(\left( \overline{r^2_0} \right)^{1/2}\) = root-mean square end-to-end distance
\(\overline{\ell}\) = average length of a segment
\(n\) = number of segments in the chain
Using the above information, root-mean square end-to-end distance of a branched polyethylene would be ............. when compared with that of the linear polyethylene of the same molecular weight and the same number of segments.

• A. Same
• B. Higher
• C. Lower
• D. Exactly \(\sqrt{2}\) times

Hint:

In polymer science, branching typically reduces the polymer chain's ability to extend, leading to a lower root-mean square end-to-end distance compared to linear polymers with similar properties.

Answer 1

The Correct Option is C

In polymer science, the root-mean square end-to-end distance \( \left( \overline{r^2_0} \right)^{1/2} \) is an important measure that gives an idea of the polymer's overall size or spatial extent. The equation provided in the question indicates that the end-to-end distance is proportional to the square root of the number of segments (\(n\)) in the polymer chain, and each segment has a length \(\overline{\ell}\).
For a linear polymer, the chain is extended and can take a relatively large end-to-end distance, given the linear arrangement of the polymer segments. As the number of segments increases, the chain becomes progressively longer, and the root-mean square end-to-end distance increases accordingly. This is because, in a linear chain, there is no branching, and the polymer can stretch out in space.
However, in the case of a branched polymer, the situation is quite different. When a polymer has branches, the polymer chains become more compact and less extended due to the branching points. The segments of a branched polymer are connected in such a way that they cannot stretch out as much as in a linear polymer. This leads to a lower overall end-to-end distance for a branched polymer compared to a linear one, even if they have the same molecular weight and the same number of segments. The branches restrict the extent to which the polymer can be stretched, effectively reducing the root-mean square end-to-end distance.
Therefore, the root-mean square end-to-end distance of a branched polyethylene will be lower than that of linear polyethylene with the same molecular weight and number of segments. Hence, the correct answer is (C) Lower.