PS674

Osmometry

Osmometry is preferred over other colligative properties for measurement of M_n because the measurable effect is larger.

The Experiment

The experiment uses a semipermeable membrane between pure solvent on one side, and a dilute solution of the polymer in the same solvent on the other side, PC Fig. 10.1. The membrane is permeable to the solvent, but impermeable to the polymer. At constant T, the chemical potential of the solvent, mu, is

Reduced by the presence of a solute
Increased by the application of pressure

These two effects must balance if the system is at equilibrium. The necessary osmotic pressure, pi, measures the effect of the polymer on mu. Experimental difficulties include slow diffusion of the solvent through the membrane, and leakage of low molecular weight polymers through the membrane. Osmometers often apply a pressure that prevents diffusion of the solvent through the membrane, and identify pi with this applied pressure.

Relationship between pi and M

For equilibrium in a two-component ideal solution,

mu (pure solvent, pressure P) = mu (solution, pressure P + pi)

The right-hand side can be replaced by

mu (pure solvent, pressure P) + RT ln x₁ + integral_P^{P + pi} (d mu / d P)_T dP

where the second term accounts for the effect of concentration (ideality assumed, mole fraction of solvent used as the concentration variable) and the last term accounts for the effect of pressure. If the solution is incompressible,

V_m pi / RT = - ln x₁

where V_m is the molar volume of the solvent. Assuming dilute solution (so that x₁ is close to 1), the right hand side is well approximated as

- ln x₁ = x₂ = N₂/N₁

which leads to the van't Hoff equation

pi = m₂RT

Conversion to concentration in units of mass/volume, c, and introduction of solution nonideality via a virial equation, leads to

pi / c = RT/M + Bc + Cc² + ...

If the osmotic pressure is to be expressed in terms of the height, h, of a column of liquid with density rho, the conversion to pressure is via pi = rho g/h, where g is the acceleration due to gravity.

Data treatment

Polydisperse systems

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July 2, 1999

Wayne L. Mattice: wlm@polymer.uakron.edu