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JEE Main levelSingle correct optionDifficulty 2 of 3pl044

For any polymer sample, which statement about $M_{w}$ and $M_{n}$ (as defined by $M_{n} = \sum N_{i} M_{i} / \sum N_{i}$ and $M_{w} = \sum N_{i} M_{i}^{2} / \sum N_{i} M_{i}$ ) is correct?

A $M_{w} \leq M_{n}$ , with equality only if all chains have the same molar mass
B $M_{w} = M_{n}$ for every sample
C $M_{w} > M_{n}$ only for natural polymers
D $M_{w} \geq M_{n}$ , with equality only if all chains have the same molar mass

Answer: D

M_{w}

weights each chain by its mass, so heavy chains count more and

M_{w} \geq M_{n}

(by the Cauchy-Schwarz inequality). Equality holds only for a monodisperse sample.

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