Chapter 6

Osmosis and filtration: passive solute and water transport

6-1. Introduction6-2. Osmotic phenomena6-3. The osmotic pressure, , across an ideal semipermeable membrane

6-3.1. Estimation via Vant Hoffs Law6-3.2. Experimental measurement of osmotic pressure

6-4. Effective osmotic pressures across membranes6-4.1. Reflection coefficient6-4.2. Transients in exerted pressures across membranes

6-5. Filtration across membranes6-5.1. Filtration as a separation process6-5.2. Porous membranes6-5.3. Freely permeable solutes ( = 0)6-5.4. Ultrafiltration with colloidal impermeants and freely permeable crystalloids6-5.5. Concentration polarization

6-6. General expressions for passive water and solute transport6-6.1. Flux6-6.2. Driving force6-6.3. Continuity expression6-6.4. Time to achieve steady state

6-7. Coupled fluxes of solute and solvent through aqueous channels6-8. A more general derivation of the Kedem and Katchalsky equations6-9. Patlak modification of the Kedem and Katchalsky equations6-10. Hydrodynamic theory of porous transport (Rewrite)

6-10.1. Determining channel characteristics from flux observations6-11. Summary6-12. Problems6-13. Further readings6-14. References

6-1. IntroductionIn this chapter the focus is on the interaction between fluxes due to combinations of hydrostatic pressures (hydrodynamic and osmotic) and chemical potentials driving net diffusive fluxes. Initially, consider independent fluxes of solute and solvent across a membrane, as would be the case if they traversed different pathways. When there are solutes which cannot penetrate the membrane, the system can reach an equilibrium where a pressure gradient opposes a concentration gradient. One interesting feature of the equilibration is the remarkably high pressure it takes to oppose a small concentration gradient.

Irreversible thermodynamic approaches, stemming from Kedem and Katchalsky, bring a constraint to impose on the system when water and solutes traverse the same channels or pores in

88 Osmosis and filtration: passive solute and water transport

a membrane. The drag of the water on the solute must be equal to the drag of the solute on the water. This mutual constraint is defined by Onsager reciprocity, which Onsager (1931a, 1931b) determined had to be obeyed to conserve energy. For a two component system composed of solute and water, this constraint requires that the cross-coefficients be equal in the 2 by 2 matrix expression, as seen in Section 6.7.

6-2. Osmotic phenomenaOsmosis. Osmosis is the transport of water due to a difference in solute activities across a membrane.

Osmotic pressure (OP). Osmotic pressure is defined (always with respect to a semipermeable membrane, that is, one permeable to solvent but not to the solute) as the attractive force of a solute in a solution that is equal to hydrostatic pressure on the solution that is required to prevent water from a pure water chamber from moving across an ideal semi-permeable membrane into the solution. An alternative definition is Osmotic pressure is the pressure required to raise the activity of the water in a solution so that activity equals the activity of pure water. These statements are thermodynamically equivalent. While the latter does not mention a membrane, an experiment to test whether or not the two definitions can be reconciled would involve a membrane.

Colligative properties of solutions. The colligative properties are closely related to osmotic pressure and are due to the chemical activity of the solutes. The properties include the lowering of the vapor pressure, depression of the freezing point, and raising the boiling point.

Additional terms. A solution which is hyperosmotic, iso-osmotic or hypo-osmotic relative to another solution has a greater OP, the same OP, or a lower OP, respectively. Tonicity concerns osmolarity of a solution with respect to a cell: hypertonic, isotonic, and hypotonic media are media in which cells shrink, maintain the same volume, and enlarge, respectively.

6-3. The osmotic pressure, , across an ideal semipermeable membrane

6-3.1. Estimation via Vant Hoffs LawIn 1887, Vant Hoff recommended a calculation based on first principles, namely that osmolarity can be estimated from the product of the concentration times the gas constant times the temperature in degree Kelvin:

(6-1)

An ideal solution is one in which the chemical potential or activity of a solute is equal to its concentration. Nowadays, recognizing that solutions are often not ideal, and that only very dilute solutions are close to ideal, Vant Hoffs Law has been modified to take into account an osmotic activity coefficient, ,

. (6-2)

approaches unity in dilute solutions.

Vant Hoffs Law: CRT= =

CRT=

08 February 2013, 9:17 am \\.host\Shared Folders\jbb On My Mac\writ-

Osmosis and filtration: passive solute and water transport 89

Examples. For 1 M sucrose, assuming = 1:

For 150 mM NaCl, complete ionization and an activity coefficient of 1.0 would give an osmolarity of 300 m Osm, but biological protein-free fluids have an osmolarity of only about 280 m Osm. The salt of a strong acid and a strong base are considered to be incompletely ionized, so this defines the activity coefficient to be 0.93. The effective osmotic pressure is that of the sum of the constituents, (2 0.93 + 0.07) 150 = 289.5 mM, given ideality and incomplete ionization.

which is almost as high as the tallest trees. This an alternative calculation considering the same solution of 150 mM NaCl now to consist of three osmotically active solutes, the ions Na+, Cl, and undissociated NaCl. With 93% dissociation the total number of the constituent molecules are less than would be estimated using complete ionization (300 mOsm rather than 289.5), and includes the activity of the undissociated 7% of the NaCl.

6-3.2. Experimental measurement of osmotic pressure

6-3.2.1. Hydrostatic In Figure 6-1 two versions of a hydrostatic osmometer are shown. The membrane must be as close to ideally semipermeable as possible. Since this is seldom achievable for small solute molecules the method is best for proteins or other large molecules. To reduce the time for measurement, one tends to use small rigid chambers equipped with capillary tubing to show the volume changes, and strain gauge manometers to indicate pressure.

6-3.2.2. Freezing point (fp) depression The measurement of the temperature at which a solution freezes is a standard technique for estimating total osmolarity. As the freezing occurs the rate of temperature drop in the solution temporarily plateaus at the freezing point. A weakness of the method is that it gives a measure of fp near 0 C, where solute activities are lower than in the usual biological situation.

6-3.2.3. Boiling point (bp) elevation The presence of solutes in water inhibits its vaporization and raises the boiling point. This is probably the poorest technique since many solutes are denatured, hydrolyzed, or undergo chemical change, and the conditions are far from biologically relevant.

CRT1 1 mole 1( ) 0.08208 1.atmos/mole degree( ) 310 (deg)25.4 atmospheres19 300 mm Hg (as 1 atmos. = 760 mm Hg) .,

====

CRT2 0.93 0.07+( ) 150 mM 0.08208 310

1.93 150mM 0.08208 310 7.367= atmos.5600 mm Hg7711cm H2O or 77 metres water

=====

\\.host\Shared Folders\jbb On My Mac\writing\903.13\903.13.06osm.fm 08 February 2013, 9:17 am

90 Osmosis and filtration: passive solute and water transport

6-3.2.4. Vapor pressure osmometry The vapor pressure is lowered because solvent activity is reduced by the presence of solute. Because it can be used at any temperature the method has the advantage over freezing point depression or boiling point elevation techniques, both of which may denature proteins, of being usable under the same conditions as one might desire for the experiments of interest.

6-4. Effective osmotic pressures across membranes

6-4.1. Reflection coefficient When solute penetrates the membrane, the p required to oppose an osmotic flux due to this solute is less than given by Vant Hoffs Law, causing the effective osmotic pressure, eff, across the membrane to be reduced to a fraction, , of that across an ideal semipermeable membrane (across which solute flux would be zero). Whenever solute can cross the membrane, the reflection coefficient must be less than 1:

, (6-3)

where is the reflection coefficient. Thus,

, (6-4)

Figure 6-1: Osmometers. (Left) The static type, in which solvent transfer occurs until ahydrostatic pressure difference equal to is developed. (Right) The dynamic type, inwhich the pressure difference is adjusted until the bubble, B, is stationary, showing thatthere is no solvent transfer occurring. This can be automated by a servomechanism.

2

1

B

h1

h2

h = h1 - h2

h

1 2

eff CRT=

eff

CRT---------------=

08 February 2013, 9:17 am \\.host\Shared Folders\jbb On My Mac\writ-

Osmosis and filtration: passive solute and water transport 91

as defined in Section 6-7, "Coupled fluxes of solute and solvent through aqueous channels". The corollary is that < 1 means that solute can permeate the membrane.

A crude calculation for for a spherical molecule of radius a in a cylindrical pore can be made as in Fig. 5-12.

,

. (6-5)

While this is not exact, it does illustrate a fundamental point, namely that when solute and solvent are constrained to use only the pore then when = 1 the permeability P is zero. Moreover, though the reflection coefficient is independent of th number of pores, the permeability is (1- ) Np, so that even with = 0.9999, the permeabil