Annus MirabilisEinstein in 1905
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brownian motion

The equation explained

p=RTV∗nN=RTN⋅ν,p = \frac{R T}{V^*} \frac{n}{N} = \frac{R T}{N} \cdot \nu,
In wordsThe osmotic pressure of the suspended bodies equals the gas constant times the absolute temperature over the part of the liquid's volume that holds them, times the number of bodies over the number of real molecules in a gram-molecule; that is, R T over N times the number of bodies in unit volume.OverviewIf a suspended body is counted as one molecule, n of them in the volume V* press on the wall as n/N gram-molecules of a solute would, p = (RT/N)ν, with ν their number per unit volume and N the number of real molecules in a gram-molecule. The paper proposes it as the molecular-kinetic view, for bodies far enough apart, and promises to show in § 2 that the theory leads to it.Full explanationPut small suspended bodies in the part V∗V^* instead of a dissolved substance, bodies that cannot pass the wall either. Classical thermodynamics, the paper says, does not lead one to expect a force on the wall: the free energy of the system seems not to depend on where the wall and the bodies are, only on the total masses and properties of the suspended substance, the liquid and the wall, and on the pressure and temperature. Gravity is neglected, and so are the energy and entropy of the surfaces of contact, since the changes of position considered are taken to leave those surfaces unchanged. The molecular-kinetic theory of heat leads elsewhere. On it, a dissolved molecule differs from a suspended body only in size, and the paper sees no reason why a number of suspended bodies should not give the same osmotic pressure as the same number of dissolved molecules. The bodies must perform a slow, disordered motion in the liquid, driven by its molecular motion, and if the wall keeps them in V∗V^* they will push on it as dissolved molecules do. So with nn bodies in V∗V^*, that is ν=n/V∗\nu = n/V^* in unit volume, and neighbours far enough apart, the pressure is that of n/Nn/N gram-molecules: p=RTV∗nN=RTNνp = \frac{RT}{V^*}\frac{n}{N} = \frac{RT}{N}\nu, where NN is the number of real molecules in a gram-molecule. The paper calls this an extended view of osmotic pressure and defers its justification to § 2.Every steppV∗=RTzp V^* = R T zThe law for a dissolved substance, the display before: zz gram-molecules in the part V∗V^*.z=nNz = \frac{n}{N}Count each of the nn suspended bodies as one molecule. With NN real molecules in a gram-molecule, nn molecules make n/Nn/N gram-molecules. This is the paper's premise, not a result: suspended bodies differ from dissolved molecules only in size.pV∗=RTnNp V^* = R T \frac{n}{N}Put n/Nn/N for zz in the law.p=RTV∗nNp = \frac{R T}{V^*} \frac{n}{N}Divide both sides by V∗V^*: the first form the paper prints.ν=nV∗\nu = \frac{n}{V^*}The paper's number of bodies in unit volume.p=RTN⋅nV∗=RTNνp = \frac{R T}{N} \cdot \frac{n}{V^*} = \frac{R T}{N} \nuTake 1/V∗1/V^* over to nn and write ν\nu for n/V∗n/V^*: the second form printed. It holds only while neighbouring bodies are far enough apart, as the paper stipulates.Historian's marginIn modern notation R/NR/N is Boltzmann's constant kBk_B, so the display reads p=νkBTp = \nu k_B T, the pressure of an ideal gas of ν\nu particles in unit volume. The paper writes R/NR/N throughout, and treats NN as a quantity still to be measured: § 5 turns the result round to estimate it from observed displacements.Also

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