pi / RT = c/M
each species of the polydisperse solute makes a contribution to pi given by
pii / RT = ci/Mi
The measured osmotic pressure, pi, is the sum of the independent contributions from all species
pi / RT = Sumi pii / RT = Sumi (ci/Mi)
Using ci = MiNi/V,
pi / RT = Sumi (NiMi / VMi) = (1/V)Sumi Ni
Dividing both sides by c = Sumi ci = (1/V)Sumi NiMi,
pi / cRT = Sumi Ni / Sumi NiMi = 1/Mn
Therefore it is the number-average molecular weight that is obtained for the polydisperse sample.
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