Question

The ‘unperturbed dimension’ of the polymer chain is represented as, \[ \left( \overline{r_0^2} \right)^{1/2} \propto \bar{l}\,(n)^{1/2} \] where, \(\left( \overline{r_0^2} \right)^{1/2}\) = root-mean-square end-to-end distance
\(\bar{l}\) = 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:

Branched polymers are always more compact than their linear counterparts, giving lower root-mean-square end-to-end distances and smaller radius of gyration for the same molecular weight.

Answer 1

The Correct Option is C

Step 1: Recall the meaning of unperturbed dimension.
The expression \[ \left( \overline{r_0^2} \right)^{1/2} \propto \bar{l}\,(n)^{1/2} \] represents the root-mean-square end-to-end distance for an ideal (random coil) polymer chain. It depends on the number of segments \(n\) and their average length \(\bar{l}\).

Step 2: Compare linear vs. branched chains.
- For a linear polyethylene chain: segments are arranged in a straight sequence, giving a longer effective end-to-end distance.
- For a branched polyethylene chain: side chains reduce the effective extension along one direction. This makes the chain more compact and reduces the average end-to-end distance.

Step 3: Molecular weight condition.
- The problem states both polymers (branched and linear) have the same molecular weight and the same number of segments.
- Hence, differences in end-to-end distance arise only from branching (not from segment number).
- Branching increases compactness, leading to a lower RMS end-to-end distance.

Step 4: Eliminate wrong options.
- (A) Same → incorrect, branching changes chain conformation.
- (B) Higher → incorrect, branched chains are more compact, not extended.
- (D) Exactly \(\sqrt{2}\) times → not a general rule, so incorrect.
- (C) Lower → correct. Final Answer: \[ \boxed{\text{(C) Lower}} \]