Page:Scientific Papers of Josiah Willard Gibbs.djvu/347

existing in excess at the surface), is volatile, the effect of evaporation and condensation may be considerable, even when the mean value of the potential for that component is the same in the film as in the surrounding atmosphere. To illustrate this, let us take the simple case of two components $S_{1}$ and $S_{2}$, as before. (See page 301.) It appears from equation (508) that the potentials must vary in the film with the height $z$, since the tension does, and from (98) that these variations must (very nearly) satisfy the relation

$\gamma _{1}{\frac {d\mu _{1}}{dz}}+\gamma _{2}{\frac {d\mu _{2}}{dz}}=0,$

(658)

$\gamma _{1}$ and $\gamma _{2}$ denoting the densities of $S_{1}$ and S_{2} in the interior of the film. The variation of the potential of $S_{2}$ as we pass from one level to another is therefore as much more rapid than that of $S_{1}$, as its density in the interior of the film is less. If then the resistances restraining the evaporation, transmission through the atmosphere, and condensation of the two substances are the same, these processes will go on much more rapidly with respect to $S_{2}$. It will be observed that the values of ${\frac {d\mu _{1}}{dz}}$ and ${\frac {d\mu _{2}}{dz}}$ will have opposite signs, the tendency of $S_{1}$ being to pass down through the atmosphere, and that of $S_{2}$ to pass up. Moreover, it may easily be shown that the evaporation or condensation of $S_{2}$ will produce a very much greater effect than the evaporation or condensation of the same quantity of $S_{1}$. These effects are really of the same kind. For if condensation of $S_{2}$ takes place at the top of the film, it will cause a diminution of tension, and thus occasion an extension of this part of the film, by which its thickness will be reduced, as it would be by evaporation of $S_{1}$. We may infer that it is a general condition of the persistence of liquid films, that the substance which causes the diminution of tension in the lower parts of the film must not be volatile.

But apart from any action of the atmosphere, we have seen that a film which is truly fluid in its interior is in general subject to a continual diminution of thickness by the internal currents due to gravity and the suction at its edge. Sooner or later, the interior will somewhere cease to have the properties of matter in mass. The film will then probably become unstable with respect to a flux of the interior (see page 305), the thinnest parts tending to become still more thin (apart from any external cause) very much as if there were an attraction between the surfaces of the film, insensible at greater distances, but becoming sensible when the thickness of the film is sufficiently reduced. We should expect this to determine the rupture of the film, and such is doubtless the case with most liquids. In a film of soap-water, however, the rupture does not take place, and the processes