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Quantitative Description of Osmotic Pressure

Appreciating the relationship between osmosis and solute concentration provides a basis for predicting whether a cell in any aqueous environment will swell or shrink and for measuring and regulating the effects of osmosis on cell volume. Often, however, measuring cell volume is difficult and it's more useful to consider the pressure produced by osmosis or alternatively, the osmotic pressure necessary to prevent such a change in volume.

What is the relationship between osmotic pressure and solute concentration?   Specifically, considering the simulation on the right, how might you derive an equation relating the pressure necessary to prevent the barrier movement  to the solute concentrations in the two compartments? 

Over 100 years ago, Jacob v'ant Hoff, who won the first Nobel Prize in Chemistry, derived a precise relationship between the osmotic pressure or potential of a solution (p, pi) and solute concentration (C), a proportionality constant (R) and the absolute temperature (T):

p = CRT

Moreover, he recognized this linear relationship between pressure and solute concentration (at constant temperature) resembled the behavior of gas molecules (Giese, 1979), whose pressure and volume are related by the gas law:

PV = nRT

where P is the pressure of a gas in atmospheres, V is its volume in liters, n is the number of moles of gas, R is the "gas constant" (0.082 liter-atmospheres per degree-mole) and T is is the absolute temperature. Rearranging this statement, segregating constants from variables and solving for pressure, produces:

P = (n/V) x RT

The expression (n/V) is more complex for a solution of a solute in a solvent than it is for a gas, and the behavior of the particles is equally more complex. At low concentrations of solute, however (in the range of cellular concentrations!), the expression simplifies to the osmotic concentration of the solute. Finally, substituting p for P produces the v'ant Hoff equation above. A 1 molar solution of an idealized, osmotically active non-electrolyte creates an osmotic pressure of 22.4 atmospheres! 

Are gaseous and osmotic pressures really equivalent?

The van't Hoff expression allows you to calculate the theoretical osmotic pressure of any single compartment (against a reference compartment containing distilled water).   How can you use the van't Hoff relationship to predict and calculate the osmotic pressure between two compartments (such as a cell and its environment)?  To address this question and explore the relationship between osmotic pressure and solute concentration further, consider the following simulation that has been expanded to include plotting and curve-fitting functions.