In my previous post *JH van ‘t Hoff and the Gaseous Theory of Solutions*, I related how van ‘t Hoff deduced a thermodynamically exact relation between osmotic pressure and the vapor pressures of pure solvent and solvent in solution, and then abandoned it in favor of an erroneous idea which seemed to possess greater aesthetic appeal, on account of a chance encounter with a colleague in an Amsterdam street.

Rendered in modern notation, the thermodynamically exact equation van ‘t Hoff deduced in his *Studies in Chemical Dynamics* (1884), was

Following his flawed moment of inspiration upon learning the results of osmotic experiments conducted by Wilhelm Pfeffer, he leaped to the conclusion that the law of dilute solutions was formally identical with the ideal gas law

It would seem van ‘t Hoff was so enamored with the idea of solutions and gases obeying the same fundamental law, that he failed to notice that the latter equation is actually a special case of the former. Viewed from this perspective, the latter’s resemblance to the gas law is entirely coincidental; it arises solely from a sequence of approximations applied to the original equation.

As a footnote to history, CarnotCycle lays out the path by which the latter equation can be reached from the former, and shows how accuracy reduces commensurately with simplification.

**– – – –**

We begin with van ‘t Hoff’s thermodynamically exact equation from the *Studies in Chemical Dynamics*

where Π is the osmotic pressure, V_{1} is the partial molal volume of the solvent in the solution, p^{0} is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution.

Assuming an ideal solution, in the sense that Raoult’s law is obeyed, then

where x_{1} and x_{2} are the mole fractions of solvent and solute respectively. So for an ideal solution, equation 1 becomes

If the ideal solution is also dilute, the mole fraction of the solute is small and hence

so that

For a dilute solution x_{2} approximates to n_{2}/n_{1}, where n_{2} and n_{1} are the moles of solute and solvent, respectively, in the solution. The above equation may therefore be written

In dilute solution, the partial molal volume of the solvent V_{1} is generally identical with the ordinary molar volume of the solvent. The product V_{1}n_{1} is then the total volume of solvent in the solution, and V_{1}n_{1}/n_{2} is the volume of solvent per mole of solute. Representing this quantity by V’, the above equation becomes

which is identical with the empirical equation proposed by HN Morse in 1905. For an extremely dilute solution the volume V’ may be replaced by the volume V of the solution containing 1 mole of solute; under these conditions we have

which is the van ‘t Hoff equation.

**– – – –**

It is instructive to compare the osmotic pressures calculated from the numbered equations shown above and those obtained by experiment. It is seen that Eq.1, which involves measured vapor pressures, is in good agreement with experiment at all concentrations. Eq.3 fails in all but the most dilute solutions, while Eq.2 represents only a modest improvement.

These figures give a measure of van ‘t Hoff’s talent as a theoretician in deducing Eq.1, and the error into which he fell when abandoning it in favor of Eq.3.

**– – – –**

**Mouse-over links to works referred to in this post**

Jacobus Henricus van ‘t Hoff **Studies in Chemical Dynamics**

Nice work laying out the steps. I find these disturbingly more interesting when they show a mistake than the more correct derivations.