11.7.3 Osmosis in electrolytes
Sketch (e) of Fig. 11.17 illustrates an important variant of osmosis. The solutions contain low concentrations (< 0.1 M) of ionic species, either to buffer the pH or to mimic the liquid present inside cells of the human body. Typical salts (or electrolytes) are KCl, NaCl and KHPO4. Most ions can pass through the membrane separating the two solutions. In addition, the macromolecule is invariably charged, either negatively or positively. In the following, NaCl is used as a representative electrolyte. With [ ] denoting molarity and L and R indicating the left and right compartments of the osmometer, respectively:
[NaCl]o = initial sodium chloride in L or R
[MClz]o = initial macromolecule cation with its associated chloride anion in R
Transport of Na+ and Cl- ions through the membrane occurs in order to equalize the chemical potentials of NaCl on the two sides of the osmometer. At equilibrium, the concentrations are:
[Mz+] = [MClZ]o because the macromolecule cannot penetrate the membrane, its concentration remains the same.
Assuming that the volumes of the left and right compartments are equal, these final concentrations are related by species conservation and charge neutrality in the two compartments. Conservation of species gives:
Electrical neutrality requires:
Only three of these equations are independent, as can be seen by substituting (11.44) and (11.45) into (11.43); this procedure yields Eq (11.42). Therefore, Eqs (11.42), (11.44) and (11.45)
constitute a sufficient summary of the above set. One additional equation is needed. This is obtained from the condition of equal chemical potentials of NaCl on either side of the membrane.
Digression: activities of ions in solution.
Because strong electrolytes such as NaCl are completely dissociated in aqueous solution, the activity of NaCl is equal to the sum of the activities of the two ions:
Alternatively, the above equation is:
m NaCl = m NaCl + RT ln(a Na + a Cl" ) = mNaCl + RT ln a ±
where the product of the activities of the two ions is denoted by a± and m NaCl = m o + + m o ^ . a± can be divided into the usual product of an activity coefficient and a concentration:
where g± is the mean ionic activity coefficient. In pure water g± for NaCl, drops rapidly from unity at infinite dilution to ~ 0.7 at 0.3 M, and remains essentially constant to higher concentrations. However, g± is affected by even small molar concentrations of biological macromolecules in the solution.
For the present purpose, g± in the L and R solutions may be assumed to be equal, thereby reducing the equilibrium condition m NaCl = m NaCl to:
This equation is known as the Donnan equilibrium. It is the fourth equation needed to solve for the 4 concentrations.
Eliminating the Cl- concentrations in Eq (11.46) with the aid of Eqs (11.44) and (11.45) and the Na+ concentration in the left-hand compartment with Eq (11.42) gives the final result:
From which the remaining ion concentrations follow from Eqs (11.42), (11.44) and (11.45).
In order to complete the solution to this problem, the chemical potentials of water in the two compartments are set equal. However, in this case the activity of water in the left hand compartment is not unity because of the dissolved salt so Eq(11.39) is modified to:
Neglecting nonideal behavior, ln a W @ -vw ([Na+ ]L + [Cl ]L ) and ln a WW @ -v w ( [Na + ]R + [Cl - ]R + [M Z+ ]R ) . Substituting these into Eq (11.48) gives:
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