54 BELL SYSTEM TECHNICAL JOURNAL 



formerly empty. During the entire latter process, if it be gradual enough, 

 the pressure of the gas is definite, and related to V and T by the equation (1). 

 The former is an irreversible way of effecting the transition. The latter is 

 the reversible way of effecting the transition. 



As the heat flows into the gas, its entropy mounts up. Since the tem- 

 perature is meanwhile changing, we must mentally subdivide the incoming 

 heat into driblets dQ, during the inflow of each of which T remains about the 

 same. The corresponding driblets of energy are given by dQ/T, the total 

 change in entropy due to inflow of heat is given by y dQ/T integrated from 

 beginning to end of the inflow. ^ By the former way the integral is 

 y {Cv/T)dT or Cv ln{Ti/To), by the latter way the integral is y {Cp/T)dT 

 or Cp IniTi/To). The two are not the same. Now we see why the phrase 

 "in a reversible way" was necessary in the definition of change-of-entropy. 

 But for that phrase or something similar, we should now have no definition. 

 But having accepted the phrase, we are invited and required to write, 



^S 



= / {CJT) dT (2) 



for all transitions of a single substance within a single phase; A5 signifying 

 the change of entropy, and the use of Cp implying that the conditions are 

 those under which specific heat at constant pressure is properly measured : 

 the pressure of the substance being definite, nothing turbulent or gusty or 

 explosive happening within the substance, and an equal pressure bearing 

 down upon it from the outer world. 



We have now the necessary and complete statement for the variation of 

 entropy with temperature, pressure remaining the same; but it has to be 

 supplemented with a statement for the dependence of entropy on pressure, 

 T remaining the same. For this and other purposes, let us return to the 

 irreversible passage between {P,To) and (P,Ti)~~the passage, rather, of 

 which one stage is reversible but the other not. During the reversible 

 stage, the gain of entropy is y {Cv/T)dT. This falls short of the gain of 

 entropy incurred along the other route as given by (2). However the other 

 route leads reversibly all the way to the goal, while the reversible part of 

 this one brings us only a part of the way, leaving us with the irreversible 

 expansion still before us. By assuming that the remainder of the total gain 

 of entropy demanded by (2) is made up during the irreversible expansion, 

 we rescue the concept of entropy. Now to save and establish the concept 



1 It is often said or implied that this formula should not be used unless the process is 

 fully reversible, in the sense that the inflow of heat occurs from a reservoir of temperature 

 identical with that of the gas. As no such precaution is taken when specific heats are 

 measured, and as measurements of specific heat are commonly used for establishing values 

 of entropy, I assume that the Hmitation is needless. 



