THE REALITY OF MOLECULES 87chemical equilibrium in solutions 'that there is a fundamental analogy, nay almost
an identity, with gases, more especially in their physical aspect, if only in solutions
we consider the so-called osmotic pressure.... We are not here dealing with a
fanciful analogy, but with one which is fundamental' [HI]. The experimental
basis for these discoveries was provided by the measurements on osmosis through
rigid membranes performed a decade earlier by Wilhelm Pfeffer, then an extraor-
dinarius in Bonn [P4].
Let us first recall what van 't Hoff meant by the osmotic pressure. Consider a
vessel filled with fluid, the solvent. A subvolume V of the fluid is enclosed by a
membrane that is fully permeable with respect to the solvent. Another species of
molecules, the solute, is inserted in V. If the membrane is fully impermeable to
the solute, solvent will stream into V until equilibrium is reached. In equilibrium,
the pressure on the membrane is an osmotic pressure. If the membrane has some
degree of elasticity, then this pressure will cause the membrane to dilate. For the
special case where the membrane is rigid and unyielding, the pressure exerted on
it is the osmotic pressure to which van 't Hoff referred and which we shall always
have in mind in what follows. (This pressure can be sizable; for example, a 1%
sugar solution exerts a pressure of % atm.)
It is one of the great merits of Pfeffer, renowned also for his work in botany
and plant physiology, that he was the first to prepare such rigid membranes. He
did this by placing unglazed, porous, porcelain pots filled with an aqueous solution
of K 3 Fe(CN) 6 in a bath filled with copper sulfate. The resulting precipitate of
Gu 2 Fe(CN) 6 in the pores of the porcelain pots constituted the rigid membrane.
Pfeffer performed elaborate measurements with his new tool. His results led him
to suspect that 'evidently there had to exist some connection between osmotic [pres-
sure] on the one hand and the size and number of molecules on the other' [C4].
The connection conjectured by Pfeffer was found by Einstein and reported in his
doctoral thesis, with the help of the laws found by van 't Hoff. In turn, van 't
Hoff's purely phenomenological discovery was based exclusively on the analysis
of data obtained by Pfeffer.
Van 't Hoff's laws apply to ideal solutions, 'solutions which are diluted to such
an extent that they are comparable to ideal gases' [HI].* For such ideal solutions,
his laws can be phrased as follows (it is assumed that no electrolytic dissociation
takes place):
- In equilibrium, one has
independent of the nature of the solvent. In this analog of the Boyle-Gay-Lus-
sac law, p is the osmotic pressure, V the volume enclosed by the rigid mem-
brane, T the temperature, and R' a constant."Van 't Hoff noted that a negligible heat of dilution is a practical criterion for solutions to be ideal.