Let us consider in the first instance the case of a jet of sand or water of unit cross section which is playing against a surface. Suppose for the present that all the molecules which strike the surface have the same velocity.
Then the number of molecules which strike the surface per second, will be proportional to this velocity. If the particles are moving quickly they can reach the surface in one second from a greater distance than is possible if they be moving slowly. Again, the number reaching the surface will be proportional to the number of molecules per unit of volume. Hence, if we call v the velocity of each particle, and N the number of particles per unit of volume, the number which strike the surface in one second will be N v; if m be the mass of each molecule, the mass which strikes the surface per second is N m v; the velocity of each particle of this mass is v, therefore the momentum destroyed per second by the impact is N m v × v, or N m v², and this measures the pressure.
Hence in this case if p be the pressure
p = N m v².
In the above we assume that all the molecules in the jet are moving with velocity v perpendicular to the surface. In the case of a crowd of molecules flying about in a closed space this is clearly not true. The molecules may strike the surface in any direction; they will not all be moving normal to the surface. To simplify the case, consider a cubical box filled with gas. The box has three pairs of equal faces at right angles. We may suppose one-third of the particles to be moving at right angles to each face, and in this case the number per unit volume which we have to consider is not N, but ⅓ N. Hence the formula becomes p = ⅓ N m v².
Moreover, if ρ be the density of the gas—that is, the mass of unit volume—then Nm is equal to ρ, for m is the mass of each particle, and there are N particles in a unit of volume.
Hence, finally, p = ⅓ ρ v².
Or, again, if V be the volume of unit mass of the gas, then ρ V is unity, or ρ is equal to 1/V.
Hence pV = ⅓v².
Formulæ equivalent to these appear first to have been obtained by Herapath about the year 1816 (Thomson’s “Annals of Philosophy,” 1816). The results only, however, were stated in that year. A paper which attempted to establish them was presented to the Royal Society in 1820. It gave rise to very considerable correspondence, and was withdrawn by the author before being read. It is printed in full in Thomson’s “Annals of Philosophy” for 1821, vol. i., pp. 273, 340, 401. The arguments of the author are no doubt open to criticism, and are in many points far from sound. Still, by considering the problem of the impact of a large number of hard bodies, he arrived at a formula connecting the pressure and volume of a given mass of gas equivalent to that just given. These results are contained in Propositions viii. and ix. of Herapath’s paper.