ENTROPY 57 



indeed already found, but it involves a retracing of our steps: the cooling 

 of the gas to a lower temperature at a constant volume, followed by the 

 lieating of the gas to the original temperature at the constant pressure. 

 We wish a forward-going reversible way, and such a one can be found. It 

 is necessary to have the container built with a frictionless piston-head for 

 one wall, and to bear down upon this from without with a pressure always 

 nicely calculated to be equal to the internal pressure of the gas. If under 

 this condition the piston-head is gliding slowly outward it will continue so 

 to glide, and the gas will expand in the gradual, languid, crawling manner, 

 with its internal pressure always definite — the manner which we call 

 "reversible." All that is now required is to know the amount of heat 

 which enters the gas during this process, so that we may divide it by T 

 and so assess the entropy-change. It is from the First Law that we get 

 this information. 

 The First Law is to be spoken in the form 



energy-gain — ^^heat in" less "work out" 



and written in the form 



AU = AQ - AW (8) 



the symbols fitting the words in the way which is obvious. 



For an ideal gas, the energy U is independent of pressure or volume, de- 

 pending on the temperature alone. The reader may or may not take this 

 as a matter of course, but it will be proved later on. We are considering 

 an isothermal expansion, and therefore AU is zero, and the problem of 

 evaluating AQ is that of evaluating AW. Now the "work out" — the work 

 done by the gas upon the outer world — is equal to the pressure bearing 

 down upon the piston-head from the outer world, multiplied by the area of 

 the piston-head and the distance through which it advances. The last two 

 factors multiplied together give the gain in volume of the gas, AV; and 

 without a moment's hesitation one usually puts for the first factor the 

 symbol P signifying the pressure of the gas. However, it is wise to hesitate 

 just long enough to realize that by so doing one assumes the reversible 

 expansion with all the attributes set forth above. For an irreversible 

 expansion All' would not be equal to PA]', but less. But assuming the 

 reversible expansion, and remembering that P is not independent of volume 

 as in these last few lines I have tacitly assumed, we find 



Air = f P'^^' = ^'^' [ < VF) dV 



= RT In (V/V) = RT In {P'/P) (9) 



