228 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The first task in preparing the heat diagram is to define mathemat- 

 ically the unit of heat extent, so that it may be deduced from known 

 properties of the medium dealt with. 



Taking first the case of perfect gases : 



If heat is applied to a gas, the results of the transfer may be ex- 

 pressed mathematically by the equation — 



dQ^CydT+Apdv, . (1) 



where Q represents the heat supplied, Cy the specific heat at con- 

 stant volume, dT the increase in temperature and p the pressure 

 during the infinitesimal change in volume dv, and A the heat equiva- 

 lent of work, or ^- CvC^T" represents the heat involved in the tempera- 

 ture change, and Apdv the heat equivalent of the external work. 



The combined laws of Boyle and Charles for gases are expressed 

 mathematically by the equation pv^=RT, in which j9 is the pressure, 

 V the volume, T the. absolute temperature, and R a constant depend- 

 ing on the medium. From this equation we have: 



p = ^ (2) 



Substituting this value of p in equation (1), we have: 



dQ = CydT-\-ART'^  (3) 



By definition, entropy or <^ is equal to j 4^ Hence, dividing equa- 

 tion (3) by T, we have: 



d9 = ^ = e.^+AR'^ (4) 



The heat absorbed at constant pressure is CpdT, while that re- 

 quired to raise tiie temperature at constant volume is Cxd T. The dif- 

 ference (cp — C\)dT must be equal to the amount of heat necessary 

 to effect expansion at constant pressure, namely, Apdv, or 



(cp — Cv)dT=Ap^^dT (5) 



(cp — Cv)=^p|J (6) 



If we consider pressure to be constant, we may differentiate the 

 equation ^v =/? T, writing it pdv^=^RdT, and substituting this value 

 for pdv in equation (6), we have : 



Cy> — c^=AR (7) 



Substituting this value for AR in equation (4), we have : 



rf<^ = c,f+(cp-cv)f (8) 



The general law of expansion of a perfect gas may be expressed by 

 the equation 



pivi^ ^= piVi"^ ^= pV" (9) 



or, BincQ pv=RT, we may write equation (9) as: 



RTv^-^=piVi'^ (10) 



