556 PROCEEDINGS OF THE AMERICAN ACADEMY. 



That is, at the high pressures there is no indication that the change of 

 volume will ever become zero- This inflection in the volume curve is 

 also mirrored by a corresponding inflection in the latent heat curve, 

 which rises more and more rapidly at the upper end. Also this change 

 of direction of the volume curve occupies the same general locality as 

 the region on the compressibility curves for the liquid where the com- 

 pressibility ceases to decrease as fast as one would expect from the be- 

 havior at low pressures. This behavior of the volume curve, together 

 with that of the latent heat curve, shows in the first place that the 

 latent heat and the change of volume do not vanish together, so that 

 there can be no critical point, and in the second place, that the change 

 of volume apparently will not vanish at any finite temperature, so that 

 we will not have a maximum as supposed by Tammann, but the curve 

 will rise instead to infinite pressures and temperatures. 



Recently J. J. van Laar^i has been developing a theory of the solid 

 state which is more far reaching than that of Tammann, in that it at- 

 tempts to show the actual mechanism which makes a liquid pass to the 

 solid. This theory explains the solid state by the association of the 

 simple molecules to molecular complexes. For the sake of simplicity, 

 the theory has been developed for the case where the complexes are 

 double molecules, although this restriction is not necessary. Given, 

 then, a liquid in which both single and double molecules may exist, 

 van Laar has found, by writing down the thermodynamic potential of 

 the two kinds of molecules, how the dissociation of the double mole- 

 cules into single molecules varies with pressure, volume, and tempera- 

 ture. Accompanying the dissociation is a change of volume, for the 

 volume of the double molecule is not in general twice that of the two 

 single molecules from which it comes. This change of volume, due to 

 dissociation, is found to so modify van der Waals' equation, which is 

 still supposed to hold for either kind of molecule separately, that an 

 isotherm now has two maxima and two minima, instead of the single 

 maximum and minimum of van der Waals' original equation. This 

 evidently means the existence of a new phase, the solid, the equilib- 

 rium conditions of which are determined in the same way as the equilib- 

 rium conditions liquid-gas of the ordinary equation, by applying the 

 condition that the work done in a reversible isothermal cycle is zero. 



By detailed numerical computation, van Laar has shown how on this 

 theory the equilibrium pressure solid-liquid changes with increasing 

 temperature. The results are similar to those of Tammann in that a 



" van Laar, Proc. Amster., 11, 765-780 (1909); 12, 120-132, 133-141 

 (1909); 13, 454-475, 636-649 (1910). 



