60 BELL SYSTEM TECHNICAL JOURNAL 



absorbed: denote it by L for one mole. If the crossing of the divide is a 

 reversible process, the entropy of the substance goes up by L/T as it turns 

 from solid to gas. 



Is the crossing of the divide a reversible process? It is indeed, and that 

 is why phase-diagrams are constructed. Think of the substance enclosed 

 in a container with frictionless piston-head, equal pressures P bearing upon 

 the piston from the world without and from the substance within, and the 

 whole affair at temperature T. If the point defined by P and T lies exactly 

 on one of the curves which partition the diagram off, then the substance 

 within the container may be a mixture of the two adjoining phases in any 

 ratio imaginable. If the piston-head is gliding slowly inward it continues 

 so to glide, P and T remaining steadily the same while all that there is of 

 the phase of greater specific volume- is converted gradually into the other. 

 If the piston-head is gliding slowly outward it continues so to glide, P and 

 T remaining constant while the latter phase is converted into the former. 

 All of the attributes of the reversible process are here: gradualness, lack 

 of turbulence and of explosiveness, willingness to go in either sense, 

 willingness to stand indefinitely still at any partway stage. For ease of 

 formulation it is the prince of reversible processes, since P and T are both 

 unchanging while it is going on.^ 



Accepting then the crossing of the divide as a reversible process, we have 

 our second expression for the entropy at the point of interest: 



S(P, T) = S{P, 0) + f {C]:'/T) dT + L/T (14) 



>) o 



I pause to quiet the fear that the expression here written down is necessarily 

 infinite, because of T standing in the denominator of the integrand on the 

 right! It is a fact of experience that as T approaches absolute zero, Cp 

 approaches zero for all solids, and with such rapidity that (if extrapolation 

 is safe) the integrand tends to zero and not to infinity. Now comes the 

 climax, which consists in equating the right-hand members of (12) and (14). 

 Referring to a single mole of the substance, and rearranging the terms, we 

 find: 



^ Volume per unit mass. 



^ Irreversible transitions from "supercooled" liquid to solid may occur. In such a case 

 the area called "liquid" may be regarded as spreading somewhat over the divide and 

 overlapping a portion of the area called "solid." The values of P and T for any point 

 in this region of overlapping may characterize either a solid or a liquid, though what is 

 called a "liquid" for this purpose may be so stiff and tough as to deserve and bear the name 

 of "glass." If the solid and the liquid are brought into contact the latter may pass into 

 the former (but never the former into the latter) in a manner so nearly explosive as to be 

 clearly not reversible. I should be less than frank were I to imply that this is always so. 

 With the transition from "supersaturated gas" to liquid, the irreversibility is manifest. 



