114 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



practical interest that Helmholtz next directed his attention. He 

 investigated how large a part of the heat developed in a galvanic cell 

 by chemical processes appears in the work done by the current, and 

 endeavored to arrange the different forms of energy in' the order in 

 which they may more or less completely be transformed into work. 

 The previous experiments on the work equivalent of chemical processes 

 had been concerned almost exclusively with the evolution or disap- 

 pearance of heat accompanying the formation or solution of a com- 

 pound, though in most chemical changes there are changes also in the 

 condition of aggregation and density of the bodies, for which, also, 

 work is performed or required. Since in most chemical processes the 

 changes of melting, evaporation, etc., abstract heat from the surround- 

 ings, it becomes necessary to inquire what the work equivalents of 

 these changes are. When one further considers that the chemical 

 forces may produce not simply heat, but also other forms of energy 

 without requiring that any of the change of temperature corresponding 

 to the operation should enter into their production, it appears neces- 

 sary that in the chemical processes a separation should be made 

 between the parts of the forces of affinity which are directly changed 

 to other forms of energy and those which generate heat. These two 

 parts of the inner energy were designated by Helmholtz as free and 

 combined; and he found that a chemical reaction proceeding from a 

 state of rest, and at a constant temperature without the application of 

 external work, can only go in such a direction that the free energy 

 decreases. Thus, under the assumption of the universal application of 

 the laws of the mechanical theory of heat, the value of the free energy 

 decides in what sense chemical affinity shall act. 



The calculation is only possible when the changes supposed are in 

 the thermodynamic sense reversible. Helmholtz was led to consider 

 under what circumstances, if at all, the latent heat of the gases set 

 free by the decomposition of water would exert influence on the electro- 

 motive force of cells, but required, in order to pursue this inquiry, to 

 first give analytical expression to the principles of thermodynamics. 

 In previous applications of the conception of potential energy, changes 

 of temperature had not in general been taken into account, either 

 because the forces entering into the energy changes under evaluation 

 did not materially depend on the temperature, such, for example, as 

 gravitation, or else because the temperature remained constant during 

 the cycle of events considered, or was a function entering into a mechani- 

 cal change fully determined, as, for example, in the motion of sound 

 waves the temperature may be considered as a function of the density 

 of the gas. But when, as might be in the last case, the density is a 

 function of the temperature, the arbitrary constant must be redeter- 

 mined for each new temperature, and one can not make a transforma- 

 tion from one temperature to another. 



Helmholtz showed that the thermodynamic equations require for 



