58 BELL SYSTEM TECHNICAL JOURNAL 



the unprimed symbols referring to the final state of the expanding gas and 

 the primed symbols to the initial state. Now identifying AW with AQ, 

 and dividing it by T to get the change-of-entropy, we have for the very 

 quantity A5 which figured in equation (5) a second expression, viz. 



AS = R ln{P'/P) (10) 



where now R stands in place of {Cp — Cv). We make the substitution into 

 (7), and arrive at our next-to-final form for the entropy of the ideal gas of 

 constant specific heats: 



S = CphiT - RlnP + I Cone mole) (11) 



I pause to mention that since for any gas PV/T is measurable and so is 

 also (Cp — Cv), the Tightness of our assumptions may be tested by ascer- 

 taining whether for gases nearly ideal, the one — which for an ideal gas is 

 R — and the other are nearly equal. This is so; and if we wish to pick out 

 a special one among the assumptions for which this shall constitute a 

 special test, then we may set down as having been particularly proved the 

 assertion, that the energy of an ideal gas depends upon its temperature 

 alone and not upon its pressure nor its volume. 



For quite some time I have been referring to a single mole of the gas in 

 question; but for an odd and probably an unexpected reason, it is going to 

 be desirable to make an explicit broadening of these equations to the general 

 case of any number n of moles. The broadening required for (11) is so 

 simple and trite as to seem not worth the doing: we have simply to multiply 

 every term of (11) by n, and so obtain 



S = nCplnT - nR InP -f nl {n moles) (12) 



But now let us translate this into an expression for 5 as a function of volume 

 and temperature, by use of the fact that P is nRT/V, and the further fact 

 that i? is Cp — Cv. We come upon the astonishing equation, 



S = nCJnT -{- nR InV - nR ln{nR) -f w/ (13) 



astonishing because the terms to the right of nR InV are not reducible to n 

 multiplied into a constant, but involve a more intricate function of n. 

 Perhaps, after all, we ought not to have taken the simple recourse of broad- 

 ening (12) by multiplying n into every term of (11) including the last one? 

 Actually it is quite all right; the additive constant in (12) is truly propor- 

 tionate to the number n of moles; that in (13) is the more intricate function 

 of n which we have just derived. 



In Fig. 1 there is shown a "phase-diagram" appropriate to a substance 

 of a single kind, capable of existing as a gas and as a liquid and as a solid. 

 Pressure and temperature are the coordinates along the vertical and hori- 



