5.4 The Gibbs Energy and Phase Transitions 217
Consider the second derivatives of the Gibbs energy in the vicinity of a first-order
phase transition. Equations (4.2-20) and (4.3-15) give(
∂^2 Gm
∂T^2)
P−
(
∂Sm
∂T)
P−
CP, m
T(5.4-4)
Equations (4.2-21) and (1.2-14) give(
∂^2 Gm
∂P^2)
T(
∂Vm
∂P)
T−VmκT (5.4-5)whereCP, mis the molar heat capacity at constant pressure andκTis the isothermal
compressibility. IfSmhas a discontinuity, thenCP, mmust have asingularity(a point
at which it becomes infinite) at the phase transition. IfVmhas a discontinuity, thenκT
must have a singularity. Figure 5.9 schematically showsCP, mas a function ofTin the
vicinity of a first-order phase transition, and Figure 5.10 showsκTas a function ofP
in the vicinity of a first-order phase transition. The upward-pointing arrow indicates an
infinite value at a single point. The infinite value of the heat capacity at the first-order
phase transition corresponds to the fact that a nonzero amount of heat produces no
change in the temperature as one phase is converted to the other. The infinite value of
the compressibility corresponds to the fact that a finite volume change occurs as one
phase is converted to the other with no change in the pressure.C
P,mT`Figure 5.9 The Constant-Pres-
sure Heat Capacity as a Function
of Temperature at a First-Order
Phase Transition.Because of the
discontinuity in the entropy as a
function of temperature, there is an
infinite spike in the heat capacity at
the phase transition.P`kTFigure 5.10 The Isothermal Com-
pressibility as a Function of
Pressure at a First-Order Phase
Transition (Schematic).Because of
the discontinuity in the molar volume
as a function of pressure, there is an
infinite spike in the compressibility at
the phase transition.
There are phase transitions that are not first-order transitions. Asecond-order phase
transitionis one in which both of the first derivatives of the Gibbs energy are continuous
but at least one of the second derivatives is discontinuous. Figure 5.11 schematically
showsCP, mas a function ofTin the vicinity of a second order phase transition, and
Figure 5.12 showsκTas a function ofPin the vicinity of a second-order phase transi-
tion. Neither the compressibility nor the heat capacity becomes infinite, although both
quantities have discontinuities in these figures. The molar entropies of the two phases
must be equal to each other and the molar volumes of the two phases must be equal to
each other.
The order of a phase transition must be determined experimentally. To establish
whether a phase transition is second order, careful measurements of the compressibility
and the heat capacity must be made in order to determine whether these quantities
diverge (become infinite) at the phase transition. Second-order phase transitions are
not common. The transition between normal and superconducting states is said to be
the only well-established second-order transition.
There are some phase transitions that do not fall either into the first-order or
second-order category. These include paramagnetic-to-ferromagnetic transitions in
some magnetic materials and a type of transition that occurs in certain solid metal
alloys that is called anorder–disorder transition. Beta brass, which is a nearly equimo-
lar mixture of copper and zinc, has a low-temperature equilibrium state in which every
copper atom in the crystal lattice is located at the center of a cubic unit cell, surrounded
by eight zinc atoms at the corners of the cell. At 742 K, an order-disorder transition
occurs from the ordered low-temperature state to a disordered high-temperature state
in which the atoms are randomly mixed in a single crystal lattice. The phase transition
between normal liquid helium and liquid helium II was once said to be a second-
order transition. Later experiments indicated that the heat capacity of liquid helium
appears to approach infinity at the transition, so that the transition is not second