ENERGETICS 



33 



quantity, called the " gas constant," whose value depends on the units in which 

 the other factors are expressed. 



This same law was shown by van't Hoff (1885) to apply to dilute solutions, 

 and the theory of solutions based on the fact has had great effect on the progress 

 of science. A portrait of van't Hoff in the year 1889 will be found in Fig. 23, 

 in the year 1899 in Fig. 24. These portraits are given by the kindness of Prof. 

 Ernst Cohen, of Utrecht. 



When gases approaching their liquefying point, or concentrated solutions, are 

 dealt with, the formula becomes 

 more complex, since factors must 

 be introduced on account of the 

 molecules coming close together, 

 so that their influence on one 

 another, as well as the actual 

 space they occupy, have to be 

 taken into account. This ques- 

 tion will be discussed in Chapter 

 VT. In the present place, we 

 will merely direct our attention 

 to the expression which gives 

 us the work done in compressing 

 a perfect gas, or, by van't Hoff's 

 theory, that done in concentrat- 

 ing a dilute solution. For 

 simplicity, the temperature is 

 supposed to be kept constant. 

 This general equation will be 

 found to turn up repeatedly in 

 calculations involving considera- 

 tions of osmotic pressure, such 

 as the electromotive force of 

 batteries, or the work done by 

 the kidney. 



Suppose, then, that we take a 

 volume, v, of a gas at a pressure, p, 

 and that we compress it so that its 

 volume is diminished by a minute 

 fraction of its original volume, that 

 is by dv. The work done is pdr. 



Further, if we diminish the volume r. 2 , which is occupied by one gram-molecule, to ??,, 

 the total work done (A) is the sum of all the minute portions, pdv, between the limits of 

 these two volumes.- In the notation of the infinitesimal calculus : 



FIG. 23. PORTRAIT OF VAN'T HOFF IN 1889. 



(Jorissen and Reicher, 1912, p. 35. Re- 

 produced by the kindness of Prof. 

 Ernst Cohen, Utrecht.) 



A = 



pdt: 



(Note that As a lengthened s, the first letter of sum, and is used to indicate the totality 



of a process. ) 



' = RT, hence 



- RT and 



(Note here that R and T, being constants, are not subject to integration, which of course 

 applies only to variables.) 



The value of this last integral is 



RT log, - 2 . 

 *'i 



For the complete solution, the textbooks must be consulted, e.g., that of Nernst and 

 Schonflies (1904, pp. Ill and 143) or of Mellor (1909, p. 254). A few words may perhaps be 

 useful in enabling the reader to appreciate the meaning of the formula. The appearance of 



