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Table 11.1 Terminology related to macromolecules in aqueous solution

Table 11.1 Terminology related to macromolecules in aqueous solution

Fig. 11.17 Devices producing osmotic pressure ( n above 1 atm). (a) standard osmometer; (b) constant-volume osmometer; (c) with a ligand that binds to the macromolecule; (d) dialysis; (e) charge effects: Donnan equilibrium

11.7.2 Osmotic second virial coefficient

The device in Fig. 11.17(a) functions because the macromolecule in the right-hand chamber lowers the activity of water, hence creating a driving force to move water from the left to the right through the membrane. At equilibrium, the chemical potentials of water on the two sides of the membrane are equal ( m w = mw ), or, in terms of the activity of water a ^ :

g w is the molar Gibbs free energy of liquid water at ambient temperature (usually 298 K) and the pressure rise P is called the osmotic pressure of the solution containing the macromolecule. It is easily measurable by the height difference between the liquid levels in the left and right compartments (a) or directly by the pressure rise (b). a ^ is the activity of water in the macromolecule solution on the right side of the osmometer. g w is the standard-state free energy of water, which is at 1 atm total pressure. Because the pressure in the right-hand compartment is greater than 1 atm, gw(1+P) is obtained by integrating the thermodynamic relation given by Eq (7.15) from 1 atm to 1 + P atm. This procedure yields:

Substituting Eq (11.38) into Eq (11.37) yields:

The size and complexity of huge macromolecules enhances interactions between them which are manifest as deviations from ideal behavior (ideality means macromolecule interaction only with solvent (water) molecules). To account for nonideality, the activity coefficient of the macromolecule in solution is represented in a virial expansion because of its similarity to the virial equation of state for gases (Ref. 3):

Terms containing [mm]2 and higher orders of [mm] are neglected. Equation (11.40) represents a first-order nonideality correction.

The activity of the water is obtained from the Gibbs-Duhem equation (Eq (8.33) with m = go + RTlna):

The molar concentration of water is [w] @ 1/vw. Substituting Eq (11.39) into the above and integrating from a lower limit of pure water yields:

ln a w = "vw |[mm] - 7^ln(1 + b mm [mm] ) I I bmm 0

In most applications, the concentration of the macromolecule is small enough to employ the approximation: ln(1 + x) = x - %x2, which reduces the above equation to:

lnaw = -vw([mm] + % bjmm]2) = -vw I ii™- + bmf cI

M mm ' 2Mm _mm where cmm is the mass concentration and Mmm is the molecular weight of the macromolecule. Substituting this equation into Eq (11.39) yields:

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