284 Thermodynamic Theorems of Carnot 



loliicli is a conseqiieyice of the first laic and of the iact that the energy of a gas 

 is a function of temperature alone. 



Finally, if one takes as a working substance a system of a single component 

 in two phases, a and fB, in equilibrium, one obtains, corresponding to eq. (5.1) 

 (7.1) (8.1), for/ 



1 dP ^ 1 dP _ \ dP 

 ~ '^) ddi ~ ij.{t) dt ~ T dT 



while equations corresponding to (5.2) (7.2) (8.2) for h do not exist. In this 

 case however eq. (4. 1) can be integrated directly to give for / 



/= « 



y& — J/a 



where V°- and V^ are the molar volumes of the component in the phases a and /? 

 respectively, and Q is the heat absorbed per mole by the isothermal transition 

 a-^fi. One accordingly obtains 



1 dP _ I dP \ dP Q 



lJLi{9i) dd, fi(t) dt T dT V^ - V- 



of which the equation in t was first deduced, in slightly different form, and 

 for the special case of a liquid in equilibrium with its vapor, by Clapeyron 



(.834). 



