LAW OF THE VERTICAL DISTRIBUTION OF A GAS We all know that the air is rarer at the top of a mountain than at sea-level, and in general the atmospheric pressure must dimin ish as one mounts, because it is caused by a shorter and less dense column of the atmosphere than at sea-level. Each horizontal stra tum of a gas in equilibrium is treated as if it were in a large ver tical cylinder and imprisoned between two rigid pistons, which prevent any exchange of molecules between this stratum and neighbouring strata of the gas, and thus conserve equilibrium. The pistons are supposed to exert respectively the pressures pre vailing at the lower and upper surface of the stratum; thus, per unit surface area, the difference between these pressures is equal to the weight of gas supported. If the thickness dh of the stratum be so small that there is only an infinitesimal difference between the molecular concentration n near the upper face and the concen tration near the lower face, then dp, the difference between the pressure at these faces will be equal to n w dh, where w repre sents the mass of a molecule, therefore dp = n Co dh.
This very simple equation expresses two important facts: first, since the concentration n of molecules is proportional to the pres sure p at a given temperature, we see that in a column of some given gas (i.e., Co is given) at uniform temperature, the relative decrease of pressure dp/p, or alternatively, the relative lowering of concentration which may be said to measure the degree of rarefaction, has always the same value for the same difference dh in level, whatever the level may be. For example, when climb ing a staircase, the air pressure (or molecular concentration) falls by 4 0 0 of of its value, for each step. Summing these effects step by step, we can see that from whatever level we start, in air at uniform temperature, whenever we ascend by the same degree, the pressure (or the density) always becomes divided by the same number; thus in oxygen at o° C, the rarefaction will be doubled for each vertical rise of five kilometres.
The second point immediately resulting from our equation con cerns the mass Co of the molecule ; for a given level dh, the rare faction dp/p (or dn/n)varies inversely as the molecular weight. Here again, integrating the effect, we see that, in two different gases at the same temperature, elevations which cause the same degree of rarefaction are inversely proportional to the molecular weights. Thus the oxygen molecule weighs 16 times as much as the hydrogen molecule; in order to double the degree of rarefaction, therefore, it is necessary to rise 16 times as high in hydrogen as in oxygen, viz., 8o kilometres. We may assume that Avogadro's law holds good for short columns of emulsions, as it does for gases, if the particles are comparatively large.
AN EMULSION DISTRIBUTES ITSELF LIKE A GAS Suppose a stable emulsion with comparatively large particles of uniform size has been made and left to stand at constant tem perature solely under the influence of gravity. For a small depth we may apply the preceding argument without any appreciable change, other than that resulting from the fact that the particles are now separated, not by a vacuum, but by a liquid which exerts a thrust on each particle in the opposite direction to its weight, in accordance with Archimedes' principle. Hence, the effective weight of the particle, to which the argument applies, is its real weight less this thrust. If, then, our generalization is permissible, once the emulsion is in equilibrium it will act as a miniature at mosphere with visible molecules, in which equal increases of alti tude will be accompanied by equal rarefaction. If, for instance, it is necessary to ascend i,000 million times further in oxygen than in the emulsion in order to double rarefaction, it is because the effective mass of a particle in the emulsion is i,000 million times the mass of the molecule of oxygen. It will, therefore, suffice to determine the effective mass of the visible particles (which form our link between ordinary and molecular dimensions) in order to find, by simple proportion, the mass of any molecule and hence Avogadro's number.
It is easy to confirm that the distribution of the particles attains a permanent condition. All that is required is to note the value of the ratio, of the concentrations at two levels, from hour to hour. This ratio, at first almost unity, increases and tends to reach a limit. In a height of one-tenth millimetre, with water as the con tinuous medium, the distribution limit was practically attained in one hour. The values of the ratio actually observed after three hours, and again at the end of a fortnight, were exactly the same. The distribution limit is a distribution in reversible equilibrium, for if it be exceeded the system automatically reverts to it. One method of exceeding this limit, that is, to cause too many particles to accumulate in the lower layers, is to cool the emulsion—this causes a higher concentration in these layers—and then to raise the temperature to its original figure, when the origi nal disposition will be regained. The most careful measurements gave 68 X as the value of Avogadro's number N. (See ELEC TRON, THE, for Millikan's value for N.)