56 BELL SYSTEM TECHNICAL JOURNAL 



denote by P and V. Remembering (1) the equation of state of the ideal 

 gas, and applying it to the reversible heating at constant volume which 

 preceded this expansion, we find 



T,/To = P'/P (4) 



and so, 



AS = (Cp - C.) In (P'/P) (5) 



for the entropy-change incurred when one mole of ideal gas expands at 

 uniform temperature so that its pressure falls from P' to P. I do not have 

 to state the temperature, since it has vanished from the equation. I do not 

 have to say that the expansion is irreversible, for if there be such a thing 

 as entropy at all, its alteration depends only upon its values in the initial 

 and in the final state, and not on the manner in which the system has made 

 the passage from initial to final. I ought, however, to recall that we are 

 still supposing a gas of which the specific heats do not depend upon tempera- 

 ture, nor in fact upon any variable whatsoever. 



Now we have an expression — equation (5) — for the way in which entropy 

 changes with pressure at constant temperature, and another — equation 

 (2) — for the way in which entropy changes with temperature at constant 

 pressure. We may combine them to get the change of entropy occurring 

 when the gas proceeds, by whatever route, from an initial state (Pi, Ti) to 

 a final state (^'2, ^2). This is, 



S2- Si = Cp ln{T2/T,) - (Q, - C.) IniPi/Pi) (6) 



which, by stripping off the subscripts 2, and gathering into one term / all 

 of the terms containing the subscripts 1, may be written, 



5 = Cp InT - {Cp - C) hiP -f / (7) 



the expression, for a gas of the special type stated, of entropy as a function 

 of absolute temperature and of pressure. The quantity / is an "additive 

 constant" which will turn out to be one of the major topics of this paper. 



All this way we have come without invoking the First Law of thermo- 

 dynamics, and we might even go further without its aid! But there is no 

 point in deferring it longer, and I wish to be able to convert (7) into a more 

 familiar form by replacing (Cp — C^) with the constant R of the equation- 

 of-state of the ideal gas. To do this I return to the irreversible process 

 which has so long engaged our attention; the free expansion of the gas from 

 the higher pressure P' to the lower pressure P, its temperature remaining 

 the same. I seek a reversible way of conducting the expansion from the 

 same beginning to the same ending under the same condition of steady 

 temperature. A reversible way of going between the extremes we have 



