414 



NATURE 



[March i, 1900 



scheme with full instructions for the collection of the material 

 desired, and for the counting, measuring, &c., of fishes, fry and 

 eggs. If these schemes are approved by the General Council, 

 the president or secretary of the Central Bureau is authorised 

 to recommend the leaders of the fishery experiments in all 

 parts of the seas concerned to select the material desired from 

 every catch, and to measure, mark and register it in the manner 

 prescribed. Messrs. A. and B. will thus receive preparations 

 or specimens of cod, halibut, &c., of such size and stage of 

 development as they wish to study from every corner of the 

 area investigated. Likewise, they will get analytical data of 

 the salinity, temperature of the water or samples of plankton, 

 stomach contents, gases contained in the water or in the 

 bladder, &c. 



The field of research of each specialist will thus be immensely 

 enlarged. Another advantage is that material of purely 

 scientific value can be distributed to public and private institu- 

 tions, museums, &c., in the different countries. 



Otto Pettersson. 



Hydrografiska Kommissionen, Stockholm. 



Gibbs's Thermodynamical Model. 



In Maxwell's "Theory of Heat " (p. 207) is a drawing show- 

 ing some of the principal lines on a thermodynamical model 

 suggested by Prof. J. Willard Gibbs, of Yale University. I have 

 been told that Prof. Maxwell had two of these models con- 

 structed, one of which remained at Cambridge, England, the 

 other being sent to Prof. Gibbs at Yale, There is also a copy 

 of this model at Clark University, Worcester, the only one 

 which I have seen. While there may be others in existence, 

 these are the only ones which I have known of, and I suspect 

 that very few have ever been constructed. 



This year, in connection with a course in thermodynamics, 

 two of my pupils are attempting to construct one of these models, 

 but are met by various serious difficulties, which may interest 

 others. In the diagram to which I refer, the directions chosen 

 for the different co-ordinates are not immediately evident. Even 

 by the aid of the description in the text, I have not been able to 

 locate them satisfactorily. In the attempt so to do, I have been 

 guided by the following general considerations. Using Max- 

 well's notation, in which 2^ = volume, / = pressure, / = absolute 

 temperature, <? = energy, 4) = entropy, the equation connecting 

 these quantities is 



td<^ = de +pdv, 

 which, transposed, gives 



de = td<p -pdv = ^d<p -f ^dv, 



the differential equation of the thermodynamical surface of 

 which the co-ordinates are the entropy, volume and energy, 

 and the slope of which at each point in the principal directions 

 gives the temperature and pressure, by the identities 



t=— • fi= - — 



These are subject to the conditions that i is always positive, 

 and/ is usually positive, always so for the gaseous state, usually 

 for the liquid and solid states. 



If, then, e is taken vertically downward, and v and tp hori- 

 zontal, passing along a section of the surface by a plane of 

 constant volume, in the direction of increasing entropy, the 

 slope will always be downward, and generally convex, as the 

 addition of heat, that is, energy to a substance at constant 

 volume increases its entropy, and generally its temperature, 

 never decreasing it. A section by a plane of constant entropy 

 will have a slope in the direction of increasing volume, which is 

 in general upward, corresponding to a positive pressure, and in 

 all parts of the model referring to stable states of the substance 

 this will be convex, since increase of volume is then accompanied 

 by decrease of pressure. 



I have attempted to determine the choice of co-ordinates by 

 the properties of the critical state. In the two diagrams the 

 broken line separates the parts representing stable or homo- 

 geneous states from parts representing unstable or non-homo- 

 geneous states. In the pressure-volume diagram, lines of con- 

 stant pressure, volume, entropy and temperature are drawn. 

 On the other are drawn lines of constant pressure and temper- 

 ature, taken from Maxwell. In both diagrams these lines are 

 tangent to the broken line. In Fig. I the line z/= const, cuts 

 sharply through the broken line. 



NO. 1583 VOL. 61] 



I have attempted to find the behaviour of a line of constant 

 entropy in the following way. 



For a substance following van der Waals's equation 



the equation of an isentropic can be shown to be 



(p+-^{v-i)^= const. 



where k is the ratio of the specific heats at constant pressure 

 and constant volume. The slope of this curve is then found 

 to be 



dp_2a _ k[p + alv'^) 

 dv z'* • v-b 



which becomes, substituting the values of the critical pressure 

 and volume 



\dv)<. 



2a{i- k) 



27(^3 ' 



which is negative for real positive values of a and b. Hence 

 the isentropic appears also to cut through the broken line, but 

 less sharply. 



Still further, we believe that the line of constant volume does 

 not again pass out of this non-homogeneous or unstable area, 

 while the isentropic may. Hence it has seemed to me necessary 



to consider the vertical line of Fig. 2 a line of constant volume, 

 and the horizontal line an isentropic, while the critical point 

 lies a little to the left of the vertex of the curve, so that the 

 isentropic slightly cuts through the broken line. 



The choice of co-ordinates will then be : energy, vertically down- 

 ward, in the three-dimensional model, volume, measured to the 

 right, in Maxwell's diagram, and entropy vertically downward 

 in the same diagram. This choice is not inconsistent with the 

 arrows in the upper left hand corner of the diagram. The 

 model, which has been constructed in accordance with these 

 considerations, is shown in the accompanying figure [(Fig. 3). 

 It satisfies the general requirements as to slope and convexity. 

 It represents the gaseous or vapour state, as having in general 

 the greatest volume and a great range of pressure, &c. 



One property, however, does not seem to be indicated by this 

 model, nor do I see how to satisfactorily change it so that this can 

 be done. It has been deduced mathematically and shown ex- 

 perimentally that if a saturated vapour be expanded adiabatically, 

 or isentropically, it may either become superheated, or partially 

 condense to liquid, in fact both phenomena can be shown with 

 one substance, for instance, chloroform above 127° C. becomes 

 superheated, and below this temperature no visible effect is 

 produced by either expansion or compression. That is, there is 

 an isentropic which is at a particular point tangent to the 

 " steam line," those on one side of it not touching it at all, 



