Page 131 - Modern physical chemistry

122 Relationships between Phases

At afirst order transition, these increments differ from zero. Introducing relation (6.21)
into (6.29) gives us the formula
dP L [6.30]
dT = T~V'

which is known as the Clapeyron equation.
At a second order transition, volume V and entropy S are continuous and the right
side of (6.29) is indeterminate. From the continuity of S, we have

[6.31 ]
for movements along the boundary between the phases. But since

dS=[as) dT+[as) dP = C dT-aV dP, [6.32]
p
aT p ap T T
from formulas (6.23), (5.86), (6.24), equation (6.31) yields

[6.33]

This rearranges to
dP Cp(2) -Cp(l) ~Cp
dT = TV( a(2) -a(l)) = TV~a' [6.34]

where ~Cp is the increment in energy capacity and ~a the increment in expansion coef-
ficient associated with the transition.
From the continuity of V, we have
[6.35]

for movements along the boundary between the phases. But since

dV =(avJ dT+(aVJ dP = aV dT- f3V dP, [6.36]
aT p ap T
from formulas (6.24) and (6.25), equation (6.35) yields

a(l)V dT -f3(1)V dP = a(2)V dT -f3(2)V dP, [6.37]

whence
dP a(2)-a(1) ~a
-= =- [6.38]
dT 13(2) - 13(1) ~f3
Here M is the increment in expansion coefficient while M3 is the increment in com-
pressibility occurring during the transition.
Relationships (6.34) and (6.38) are known as the Ehrenjest equations.

Example 6.3

Calculate the change in melting point of ice produced by a decrease in pressure from
760 torr to 4.6 torr, the pressure at the triple point. The specific volumes of liquid water and
of ice are 1.0001 and 1.0907 cm 3 gl , respectively, at 0° C. The heat of fusion is 333.5 J gl .