The American Journal of Science
describe the closed path AEBD. The amount of this work can be easily computed from the experimental data and the area of the quadrilateral can also be obtained analytically from the equations of its sides:
4 E: 98 = KT E
DB: s=KTE
AD: Q* -Kg?-?K= N.
EB: 9°-Kg»--?K= N Since the last equation involves the unknown constant N, the area of the quadrilateral will be obtained in terms of N, and by equating said area to the external work previously computed in terms of V and T, we shall obtain an equation giving the unknown constant N in terms of the experimental data.
In short N can be considered as known, as soon as the volume V is given, so that the two equations :
99-K50-?K= N
7 98 = KTE can be regarded as giving the value of the coördinates and s in terms of the experimental data V and T. These equations enable us to find the position of the point corresponding to any given physical state of the substance, without having to know its thermodynamic function, provided only that its specific heat be known.
The equation to the curves of constant volume can also be made use of, for the determination of the thermodynamic function in terms of y and s; for, since we know how to find the value of the constant N corresponding to any given value of the volume V, it may be possible to express V in terms of
V by an empirical function : N = Ý (V). In this case, the general equation to the curves of constant volume can be written :
Y(V)=ye-Kge--?K This equation gives V in terms of y and $; hence it is one of the three equations involved in the thermodynamic function.
All that has been said in this paragraph concerning the volume V, applies also to the pressure P, provided the specific heat at constant volume c, be replaced by the specific leat at constant pressure C. The general equations to the curves of constant pressure would be:
x(P) = 4C-K.C-K (2(P) being a function obtained empirically), and the complete thermodynamic function of the substance would then be:
1 X(P) = 9C-K,C-OK
7(V)=q KTE=0
This form of the thermodynamic function applies only to the case in which the specific heats are constant.
In the general case, the specific heats vary with the physical state of the substance; but as this variation is always compara-. tively small, the sheet of paper can be divided into a sufficient number of regions, so that either one of the specific heats of the substance may be regarded as constant inside of any one of these regions. Then the equations given above will hold true, provided that the specific heat be given a special value for each region.
Another method consists in tinding first the value of the specific heat in terms of the volume and the temperature ; since the variation of c is slow, it will be sufficient to take only in consideration the first and second powers of V and T, so that it can always be assumed that:
c= a +bV+kT+dT? +eVT+f V? a, b, k, etc., being constant coefficients determined empirically.
The differential equation to the curves of constant volume being still :
dᎢ (c – K)
T or, by replacing c by its value: (a+V+f V-K
ds +al
T We shall obtain by integration, V being constant : (a+bV+f V-K) log T+(k+eV)T+$dT? = K log 8+ constant. Replacing T by its value the result will be the equation
KE' of the curves of constant volume, involving only one unknown constant, which can be determined by the method given above.
=R
Conclusion. 15. The object we have been trying to fulfill in this short study, was to determine which is the best system of coördinates to adopt in graphical representations of thermodynamical phenomena. Evidently, the best system is the one in which the value of each variable depending upon the phenomenon can be obtained graphically; hence, each one of these variables must be represented by a geometrical magnitude depending only upon the form and position of the path described by the substance with respect to said system of coördinates.
The system of coördinates o, & seems to possess this property to a higher degree than the system P, V, which is usually adopted and known as Clapeyron's system.
It is true that in the latter system, the location of the point representing any state of the substance, is obtained directly from the experimental data P and V, while in the system Q, 8 · the point has to be located by a more or less indirect way ; but when the path of the substance has been traced in Clapeyron's system, it does not give any information relating to the phenomena (except the value of the external work, but this work can usually be computed without difficulty); if it is desired to know more about the transformation, it is necessary to trace first all the isothermals and adiabatics ; this is a long work, these curves being irregular curves in the system P, V. On the contrary, when y and s are chosen as coördinates, the path itself is sufficient to give any information concerning the phenomenon, because the various physical variables depending upon the transformation are given by geometrical magnitudes determined by the path itself, and also because the isothermals
and adiabatics, instead of being irregu
lar, are geometrical curves whose equa+ tions are known, being the same for
any substance; so that the curves : os constant, and : yso= constant, need to be traced but once for all ; they can be reproduced as many times as necessary by the blue-print process or by any other printing process. The use of the coördinates
and s S simplifies also the demonstration of
most of the theorems of thermodynamics ; let us take, as an example, Carnot's theorem.
If a substance describes a closed path formed by two adiabatics AD, CB (fig. 8) and by two isothermals DC, BA, and if the amount of heat absorbed and abandoned by the substance be denoted respectively by II and II, the expression H.-II.
which is called the economical coefficient of the cycle, is independent of the nature of the substance, and is equal to T-T
T, and T, being the temperatures corresponding to T. the isothermals.
Since there is no transmission of heat along the adiabatics AD and CB, the amounts of heat II, and Il correspond respectively to the isothermals DC and BA. We have seen that:
H,
the equation to the isothermals being: qs = KTE, the integration gives :
EH = KTE(log 4+2 log 8) Hence:
EH, = KT E(log 2,5*, – log 2.8*)
- EH,= KT E(log 4,8, - log 2.8.) But the equation to the adiabatics being : oso= constant, we have also : 98*c = 4 **g and ym**, = 4482so that by dividing :
H H
T. which can be written :
H.-H, T.-T.
Thus, the demonstration of the theorem consists simply in the analytical measurement of areas, and the reason of it is that the heat of transformation II, which is the unknown quantity, is represented graphically by areas. This is not the case in the usual graphical methods, and Carnot's Theorem can only be demonstrated indirectly, by showing first that the economical coefficient of Carnot's cycle is independent of the nature of the substance, then computing its value for a perfect gas. One can object to this method that a perfect gas is only a theoretical substance, having no real existence.
The use of the coördinates $ and s, does not imply any hypothesis upon the nature of the motion constituting heat, as
and can be regarded simply as two auxiliary variables, without attributing to them any special significance.
Another advantage of expressing the thermodynamic function in terms of $ and s, is that each one of the specific heats of the substance can be obtained separately from said function, while the ordinary form of the thermodynamic function F(P, V, T)=0 furnishes only the difference between the two specific heats (C-c). Finally, if heat is really a vibratory motion of the particles of matter, the state of this vibration at any time is obtained from the experimental data, by solving the thermodynamic function with respect to © and s.
Art. IV.-On the Application of the Schroeder-Le Chatelier
Law of Solubility to Solutions of Salts in Organic Liquids ; by C. E. LINEBARGER.
It has been found by two scientists, working independently, that, by means of thermodynamical considerations, the solubility of a substance may be shown to be the same in all solvents; the mode of deduction employed by each as well as the nature of the experimental proof offered is quite different. Schroeder* established these two equations :
pT-T 1
(1) ,
2 TT and
=constant,
(1 bis) T T. in which s represents the solubility (defined by the ratio of the number of the molecules of the dissolved substance to the total number of molecules making up the saturated solution); P, the latent heat of fusion of a kilogram of the dissolved (solid) substance; To, the absolute point of fusion of the dissolved substance; T, the temperature at which saturation takes place; and Q, the heat of solution of a kilogram-molecule of the solid substance in alınost saturated solution. Experiments carried out with solutions of para-di-brom-benzene in carbon disulphide, benzene, and nono-brom-benzene; of naphthaline in benzene, mono-chlor-benzene, and carbon tetrachloride; of meta-di-nitro-benzene in benzene, mono-brom-benzene, and chloroform : corroborated fully the statement that "the solubilities at equal intervals from the temperatures of fusion for different solid bodies and in different solvents are the same."
For alcoholic solutions, however, this law was found not to obtain even approximately; this is undoubtedly due to the circumstance that the alcohols are made up of associated molecules, although Schroeder did not take this view of the matter, perhaps because, at the time of the appearance of his paper, our knowledge of the molecular state of liquids was exceedingly slight.
The law was further tested by Schroeder, by determining the solubility of para-di-brom-benzene in mixtures of chloroform and benzene, which was found to be very nearly the same as in either of the pure solvents. The formula developed by Le Chateliert runs thus :
* Zeitschr. f. phys. Chem., xi, 449, 1893. + Comptes rend., cxiii, 638, 1894.