Change of State : Solid-Liquid. 37 



Now introduce a particle of ice at —t° into the cylinder, 

 and condensation into ice will go on till all the vapour has dis- 

 appeared. If the ultimate volume of the ice is v', the work 

 done on the substance is ^(V— i/). 



Increase the pressure from *r' to v? f +p till the melting- 

 point is lowered to —t°. If k' is the coefficient of cubic com- 



possibility of ice, ^-idxf is the work done in the compression. 



Introducing a drop of water, allow the whole to melt into 

 water under the pressure o/+p, the work done during the 

 melting being 



(©' +p){v'(l -pic f ) -v(l -pic)} , 



where k is the coefficient of cubic compressibility for water. 



Now let the water expand to its original volume v by gra- 

 dually reducing the pressure to «r. The external work done 

 . p 2 ' 



IS ^r- fCV. 

 a 



We now have the substance in its original state ; and the 

 cycle through which it has been taken was reversible at every 

 step ; therefore 



But T is constant; therefore 



Then the total external work is zero, or 

 m(V-v) + w'V log J -*/( V'-t/) 



-K + ^(i-^)-.(i-f)}=«. 



By means of the equation 



and neglecting products of -sr and /c, this reduces to 



0\og^ = p{rf{l~Pf)~ v {\-Pl) }"+(•-•> (1) 



Neglecting the term (<a — a>')v, and putting for sr'V &> V aT, 

 where co V are the pressure and volume at 0° C, and T the 

 absolute temperature, we have 



For temperatures near 0° C. we may neglect products ot' p 



