January 28, 1898.] 



SCIENCE. 



139 



concentration are all fixed ; a change in any one 

 of these conditions results in the elimination of 

 one phase ; or, if both temperature and pressure 

 are made to change arbitrarily, equilibrium 

 cannot be restored without the loss of two 

 phases from the system. When two kinds of 

 change may thus be made at pleasure, the sys- 

 tem is said to be divariant and to have two de- 

 grees of freedom. It is here assumed that any 

 disturbing effects due to gravity, electricity, 

 capillarity or the distortion of solid masses are 

 avoided, pressure and temperature being uni- 

 form throughout the system. The absolute and 

 relative masses of the several phases have no 

 effect upon equilibrium, except as some phase 

 disappears entirely. The number of independ- 

 ent variables (including temperature and pres- 

 sure) is two more than the number of compo- 

 nents; and Gibbs' phase rule asserts that in the 

 equilibrium of heterogeneous substances the 

 number of degrees of freedom is equal to two 

 more than the number of components, dimin- 

 ished by the number of phases. Thus, a non- 

 saturated solution of a salt, in contact with 

 vapor, may be altered at pleasure in regard to 

 the concentration of the salt and either the 

 temperature or the pressure ; with two com- 

 ponents in two phases, two arbitrary changes 

 are subject to the will of the operator. 



Le Chatelier's theorem asserts that "any 

 change in the factors of equilibrium from out- 

 side is followed by a reverse change within the 

 system." Thus, if a small amount of salt be 

 added to the non-saturated solution, without 

 change of temperature, this increase of concen- 

 tration is offset in part by condensation of 

 vapor and the pressure is therefore diminished. 

 The system tends to return to its former condi- 

 tion of equilibrium by elimination of the dis- 

 turbing element. Thus the sense of change re- 

 sulting from the disturbance from conditions of 

 equilibrium can be predicted, but the amount 

 or rate of change involves quantitative relations 

 which lie beyond the scope of the volume under 

 review. 



Experimental data are discussed, as indicated 

 above, with regard to systems of one, two, three 

 and four components. The]relations of temper- 

 ature, pressure and concentration are repre- 

 sented graphically. Much ingenuity has been 



shown in devising triangular diagrams to repre- 

 sent the relative masses of three substances by 

 coordinates on a single plane. Melting and 

 boiling points, critical temperature, allotropj"^, 

 cryohydrates, solubility of anhydrous and hy- 

 drated salts, double salts, efflorescence, dissocia- 

 tion, supersaturation, volatile solutes, partially 

 miscible liquids, eutectic mixtures and tempera- 

 tures, fractional distillation, solid solutions, oc- 

 clusion, alloys and fractional crystallization 

 are among the subjects discussed, with numer- 

 ous concrete examples. 



The author's distinction between solvent and 

 solute, or between solubility curve and fusion 

 curve (as on pp. 36, 45, 95, etc.), does not find 

 general acceptance. An attempt to determine 

 the ' hypothetical line ' of demarkation (page 

 158) may help to decide the point at issue. 



"While physical chemistry is rapidly gaining 

 importance, many are deterred by the mathe- 

 matical difficulties. To such, this work will 

 give a welcome clew to the import of differen- 

 tial equations. Students of physical chemistry 

 will here find a considerable field brought undei 

 review and duly systematized. Numerous in- 

 dications are given of the present limitations of 

 science and the open fields for profitable in- 

 vestigation. 



ROBT. B. Wakder. 



The Principles of Mathematical Chemistry ; The 

 Energetics of Chemical Phenomena. By Dr. 

 Georg Helm. Authorized Translation by 

 J. Livingstone R. Morgan, Ph. D. New 

 York John Wiley & Sons ; London, Chapman 

 & Hall, Limited. 1897. 12mo. Pp. viii-F 

 228. Price, $1.50. 



The original German edition has been recog- 

 nized for three years or more as a work of 

 value, and this translation will doubtless find 

 a welcome. The principles of thermodynamics 

 (including the conservation of energy) are 

 asumed as the basis for the discussion. 

 The intensity and quantity factors are distin- 

 guished in the various forms of energy, and the 

 principle of constant or increasing entropy is 

 applied to various reversible and non-reversible 

 changes. Equations for chemical intensity are 

 applied to electrolysis, simple chemical reac- 

 tions, chemical equilibrium, freezing and boiling 



