2 Osmotic Pressure

Osmometry provides absolute measurements of molecular weight. It relies on Flory-Huggins lattice theory for a connection between the chemical potential of a solution and the polymer-solvent interaction along with the size of the polymer.

From thermodynamics, we know that the difference between the chemical potential of the solvent in the solution and that in the pure state is given as

where the molar volume of the solvent, ₁ is presumed to be independent of pressure. From F-H theory,

where N is the degree of polymerization, or the number of lattice sites occupied by the polymer and χ is related to the Gibbs free energy difference for the formation of a solvent-monomer contact, Δg₁₂ = g₁₂ - (1∕2)(g₁₁ + g₂₂) on a lattice of coordination z

This provides

where n₁ and n₂ are he number of solvent and polymer molecules respectively. The number density of polymer molecules is related to the mass concentration and molecular weight as n₂∕V = (m∕V )(n₂∕m) = c∕_n since the number average molecular weight is given as

We can now write:

From the lattice theory, N = V ₂∕V ₁ and so NV ₁∕_n = V ₂∕_n = 1∕ρ₂ so we arrive at Equation 8.

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Under theta conditions, χ = 1∕2 and so

In general, we can evaluate RT∕_n as the value of the reduced osmotic pressure in the zero concentration limit, so

The remaining term yields the second virial coefficient, providing information about the thermodynamics of the system. The theta temperature can be evaluated by plotting the value of the second virial coefficient and finding where it is equal to zero.

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