POLYMORPHISM AT HIGH PRESSURES. 177 



stants. An easy thermodynamic transformation gives the relation 



Cp = 71^ — — t''. Now if we assume that each phase separately 



satisfies these conditions, except with different values of the constants, 

 and if we write down the condition of equality of the thermodynamic 

 potentials of the two phases, we shall find for the equation of the 

 transition line near the absolute zero, 



af + fepr^ + 07"" + (^i9 + cr* + / = 0, 

 a conic in p and r^, where, 



Q 1 r/9 2 /3_2~1 



d = t\ — Vi, 



El and E2 are the internal energies of the two phases at 0° abs. and 

 zero pressure. 



It may be shown immediately by differentiation that the curve 

 approaches the pressure axis perpendicularly, a fact which, of course, 

 has been known for some time. It is easy to see by solving the equa- 

 tion for 2> at r = 0, that the following conditions must hold if there is a 

 transition at absolute zero at some positive pressure. The existence 

 of such a transition point is compatible with either a positive or a 

 negative value for a. If the phase of larger volume is more compres- 

 sible, no restriction is placed thereby on h, and no restriction placed on 

 El — E2 ( = /) as to sign, but there is a restriction as to magnitude 

 to insure that p shall be real. If the phase of larger volume is less 

 compressible, no restriction is thereby placed on h, but Ei — E2 must 

 be negative and not too large. That is, the internal energy of the 

 phase of smaller volume must be greater in this case. ^Yhether the 

 transition line, which is vertical at the axis, bends to the right or the 

 left at higher temperatures depends on the sign of 6 ; it curves toward 

 the temperature axis if the phase of larger volume becomes more 

 expansible at higher temperatures, and conversely. 



The only result of this analysis of immediate interest is that quantum 

 hypothesis still leaves open the possibility of various types of behavior 

 at absolute zero, and in so far does not demand phenomena different 

 in kind from those which we have found within our range. 



