ENTROPY 53 



connected among themselves by what is called an "equation of state." 

 This takes for granted that one knows what kind of substance one is speak- 

 ing of, how many moles there are of it, and which phase it is in; for the 

 statement is valid not only for gases but also for liquids, and even (under 

 certain restrictions) for solids. If two or three phases are coexisting, 

 precautions must be taken. The statement is not valid for mixtures of 

 substances which differ; but there is doubt as to just how much it takes to 

 constitute a difference, and isotopes of a single element are usually consid- 

 ered to be the same. 



Now let us envisage two states which differ in temperature but are the 

 same in pressure. P is to stand for the common value of pressure, To for 

 the lower and Ti for the higher of the temperatures. For the utmost in 

 simplicity at the beginning, the substance shall be an ideal gas, and more- 

 over a gas of which the specific heats do not depend upon the temperature; 

 any noble gas under ordinary conditions approaches very closely to these 

 stipulations. 



One may operate the transition from {P,Tq) to {P,Ti) in the manner 

 which follows: Let the gas be heated from To to Ti while the volume of its 

 container remains the same. Its pressure will have gone up meanwhile; 

 let this situation be remedied by piercing a hole through the container wall, 

 into an evacuated chamber prepared in advance of just the right dimensions, 

 so that after the "free expansion" the pressure of the gas will be down again 

 to P. The temperature will still be Ti, for that is one of the qualities of the 

 ideal gas. 



Or one may operate the transition in the manner which follows: Let the 

 container be made in advance with a movable wall, say a piston-head against 

 which an outer pressure P — it might be that of the atmosphere — is steadily 

 pressing. Let the gas be gradually heated: as its temperature rises, the 

 piston-head glides gradually outward, increasing the volume of the con- 

 tainer at such a rate that the pressure of the gas always remains the same 

 and equal to P. The process is to be stopped when the temperature 

 reaches Ti. 



These, then, are two very different ways of carrying through the transition; 

 and the reason for bringing them in is to elucidate the words "in a reversible 

 way" which occur in the phrase defining entropy. The former way includes 

 a process which is spontaneous, rapid, turbulent, explosive; nothing can 

 inhibit it when the piercing of the hole creates the opportunity. The latter 

 way is gradual, languid, crawling, always on the verge of stopping, would 

 instantly stop if the inflow of heat should cease, would reverse its trend if 

 the inflow should be turned into outflow. The former way includes a 

 process during which the pressure of the gas is ill-defined or not defined at 

 all, for the gas is spurting through the hole and reverberating in the chamber 



