EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 325 



cannot properly represent the true value of the quantities intended 

 unless SN is equal to the distance between two successive layers of 

 molecules in the crystal, or a multiple of that distance. Since this 

 can hardly be treated as an infinitesimal, we can only conclude with 

 certainty that sensible changes cannot take place for which the 

 expression (664) would have a positive value.* 



* That it is necessary that certain relations shall be precisely satisfied in order that 

 equilibrium may subsist between a liquid and gas with respect to evaporation, is 

 explained (see Clausius, "Ueber die Art der Bewegung, welche wir Warme nennen," 

 Pogg. Ann., Bd. o, S. 353 ; or Abhand. iiber die median. Warmetheorie, XIV) by suppos- 

 ing that a passage of individual molecules from the one mass to the other is continually 

 taking place, so that the slightest circumstance may give the preponderance to the 

 passage of matter in either direction. The same supposition may be applied, at least 

 in many cases, to the equilibrium between amorphous solids and fluids. Also in the 

 case of crystals in equilibrium with fluids, there may be a passage of individual mole- 

 cules from one mass to the other, so as to cause insensible fluctuations in the mass of 

 the solid. If these fluctuations are such as to cause the occasional deposit or removal 

 of a whole layer of particles, the least cause would be sufficient to make the probability 

 of one kind of change prevail over that of the other, and it would be necessary for 

 equilibrium that the theoretical conditions deduced above should be precisely satisfied. 

 But this supposition seems quite improbable, except with respect to a very small side. 



The following view of the molecular state of a crystal when in equilibrium with 

 respect to growth or dissolution appears as probable as any. Since the molecules at 

 the corners and edges of a perfect crystal would be less firmly held in their places 

 than those in the middle of a side, we may suppose that when the condition of 

 theoretical equilibrium (665) is satisfied several of the outermost layers of molecules 

 on each side of the crystal are incomplete toward the edges. The boundaries of these 

 imperfect layers probably fluctuate, as individual molecules attach themselves to the 

 crystal or detach themselves, but not so that a layer is entirely removed (on any side 

 of considerable size), to be restored again simply by the irregularities of the motions 

 of the individual molecules. Single molecules or small groups of molecules may 

 indeed attach themselves to the side of the crystal but they will speedily be dislodged, 

 and if any molecules are thrown out from the middle of a surface, these deficiencies 

 will also soon be made good ; nor will the frequency of these occurrences be such as 

 greatly to affect the general smoothness of the surfaces, except near the edges where 

 the surfaces fall off somewhat, as before described. Now a continued growth on any 

 side of a crystal is impossible unless new layers can be formed. This will require a 

 value of fa" which may exceed that given by equation (665) by a finite quantity. 

 Since the difficulty in the formation of a new layer is at or near the commencement 

 of the formation, the necessary value of p." may be independent of the area of the 

 side, except when the side is very small. The value of fa" which is necessary for the 

 growth of the crystal will however be different for different kinds of surfaces, and 

 probably will generally be greatest for the surfaces for which a- is least. 



On the whole, it seems not improbable that the form of very minute crystals in 

 equilibrium with solvents is principally determined by equation (665), (i.e., by the 

 condition that 2(<r) shall be a minimum for the volume of the crystal except so far 

 as the case is modified by gravity or the contact of other bodies), but as they grow 

 larger (in a solvent no more supersaturated than is necessary to make them grow at 

 all), the deposition of new matter on the different surfaces will be determined more by 

 the nature (orientation) of the surfaces and less by their size and relations to the 

 surrounding surfaces. As a final result, a large crystal, thus formed, will generally 

 be bounded by those surfaces alone on which the deposit of new matter takes place 

 least readily, with small, perhaps insensible truncations. If one kind of surfaces 



