Annus MirabilisEinstein in 1905
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brownian motion

The equation explained

pV∗=RTz.p V^* = R T z.
In wordsThe osmotic pressure times the part of the liquid's volume that holds the solution equals the gas constant times the absolute temperature times the number of gram-molecules dissolved.OverviewA dissolved substance, held in part of a liquid by a wall that lets only the solvent through, presses on that wall as a gas would: pressure times that volume equals R times the temperature times the gram-molecules dissolved. The paper states it for a non-electrolyte, when the volume per gram-molecule is large enough.Full explanationThe paper sets up the case it will compare with suspended particles. In a part V∗V^* of a liquid, zz gram-molecules of a non-electrolyte are dissolved, and a wall that lets the solvent through but not the dissolved substance separates V∗V^* from pure solvent. The wall then feels the osmotic pressure pp, and for large enough values of V∗/zV^*/z it satisfies pV∗=RTzp V^* = R T z, with RR the gas constant and TT the absolute temperature. That is the form of the gas law for zz gram-molecules, and the paper states it without derivation, as known. Its two conditions matter: a non-electrolyte, and a dilute solution, since V∗/zV^*/z large means much volume for each gram-molecule dissolved. The next display asks what the same wall feels when the part V∗V^* holds suspended bodies instead.Every steppV∗=RTzp V^* = R T zThe law the paper states for zz gram-molecules of a non-electrolyte dissolved in the part V∗V^*, when V∗/zV^*/z is large enough. It is not derived in the paper.p=RTzV∗p = R T \frac{z}{V^*}Divide both sides by V∗V^*. The pressure grows with the gram-molecules dissolved per unit of that volume, and with the absolute temperature.p=RTV∗/zp = \frac{R T}{V^* / z}The same law with the volume per gram-molecule dissolved, V∗/zV^*/z, in the denominator. The paper asks for V∗/zV^*/z to be large enough: the law holds for a dilute solution, where the pressure is correspondingly small. What it counts is gram-molecules; the kind of dissolved substance does not enter.Historian's marginThe law is van 't Hoff's (Z. physik. Chem. 1, 1887, 481). He printed it in his own letters as ‘PV = RT’ (p. 491), valid for solutions with the restriction that holds for gases too: the dilution must be large enough. Most salts, and the strong acids and bases, give a larger pressure in water, by a factor he writes i, which he ascribed on Arrhenius's suggestion to a splitting into ions (p. 501). The paper names neither him nor his paper, and asks for the same two conditions: a non-electrolyte, and large enough V∗/zV^*/z.Also

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