## 3.C. Classification of Phase Transitions

For a system of *m* kinds of particles,

G=
∑
j=1
m
n
j
μ
j
(3.7)

Since

dG=
(
∂G
∂T
)
Y{
n
j
}
dT+
(
∂G
∂Y
)
T{
n
j
}
dY+
∑
j=1
m
(
∂G
∂
n
j
)
TY{
n
i
| i≠j
}
d
n
j

we have, for constant *T* and *Y*,

(
dG
)
YT
=
∑
j=1
m
(
∂G
∂
n
j
)
TY{
n
i≠j
}
d
n
j
=
∑
j=1
m
μ
j
d
n
j
(3.8)

At a transition point, 2 or more phases can
coexist in equilibrium. The absence of matter
exchange means that, for every kind of particles, the values of its chemical
potential in the coexisting phases are equal, i.e.,

μ
i
r
=
μ
i
s
for
i=1,⋯,m
(a)

where
r,s
denotes the coexisting phases.

Without loss of generality, we shall
restrict our discussion to cases where the entire system is always in a single
homogeneous phase.

Eqs(a) and (3.7) mean that, at a transition
point, *G* of all coexisting phases are
equal.

In other words, the function
G(
Y,T,x
)
is continuous at a transition point.

The manifest differences in the properties
of the phases must then appear as discontinuities in some derivatives of *G*. If the 1^{st} order derivatives are
discontinuous, the transition is called **1**^{st}
order. Otherwise, it is called **continuous**.

In an older scheme introduced by Ehrenfest,
the **order of the transition** is taken
to be the lowest order of the derivatives that are discontinuous.

Typical behavior of *G* of a *PVT* system near a
1^{st} order transition is shown in Fig.3.2.

Discontinuity in
(
∂G
∂P
)
T{
n
j
}
=V
means a finite difference of volumes of the 2
phases,

ΔV=
V
I
−
V
II
=
(
∂G
∂P
)
T{
n
j
}
I
−
(
∂G
∂P
)
T{
n
j
}
II
(3.9)

Discontinuity in
(
∂G
∂T
)
P{
n
j
}
=−S
means a finite difference of entropy of the 2
phases,

ΔS=
S
I
−
S
II
=
(
∂G
∂T
)
P{
n
j
}
II
−
(
∂G
∂T
)
P{
n
j
}
I
(3.10)

which means the heat capacities are not
defined.

Since *G*
is continuous, the difference in enthalpy (**latent
heat**) is given by

ΔH=Δ(
G+TS
)=TΔS
(3.11)

The case for a continuous transition is
shown in Fig.3.3.

The distinguishing feature is the peak in
the heat capacity at the transition point.