Chemistry 360
Spring 2015
Dr. Jean M. Standard
April 1, 2015
The Clapeyron Equation: Solid H2O Floats, Solid CO2 Sinks
Two Forms of the Clapeyron Equation
The Clapeyron Equation gives the slope dP / dT of the coexistence curves of a one-component phase diagram. One
form of the Clapeyron Equation is
dP
ΔSm
=
.
dT
ΔVm
€
(1)
In this equation, the molar entropy change ΔSm along a coexistence curve separating phases A and B is
€
ΔSm = SB,m – SA,m ,
(2)
where SB,m and SA,m are the absolute molar entropies of phases B and A, respectively. Similarly, the molar volume
change is
ΔVm = VB,m – VA,m ,
(3)
where VB,m and VA,m are the molar volumes of phases B and A, respectively. Another form of the Clapeyron
Equation is obtained by using the relation for the molar entropy of a phase transition,
ΔSm = ΔH m
,
T
(4)
where ΔH m is the molar enthalpy and T is the temperature of the phase transition. Substituting, the second form of
the Clapeyron Equation is
dP
ΔH m
= .
dT
TΔVm
(5)
Coexistence Curve Slopes
The Clapeyron Equation provides a relation for the slope of the coexistence curve corresponding to two phases in
equilibrium in a one-component system. This includes equations of the slopes of the solid-liquid, liquid-vapor, and
solid-vapor coexistence curves. For example, the slope dP/dT of the solid-liquid coexistence curve has the form
ΔH fus,m
dP
= ,
dT
T fus ΔVm
(6)
where ΔH fus,m is the molar enthalpy change for fusion (the solid-liquid phase transition), T fus is the temperature of
the fusion phase transition, and ΔVm is the difference between the liquid and solid molar volumes,
ΔVm = VL,m – VS,m .
The Solid-Liquid Coexistence Curve for Water
A sketch of the phase diagram of water is shown in Figure 1. Note that this phase diagram does not show the
various solid phases that are observed for water at higher pressures; however, the essential details of the phase
diagram in the region of interest are correct.
(7)
2
L
P
S
V
T
Figure 1. Sketch of the P–T phase diagram of water.
In particular, if we focus on the solid-liquid coexistence curve, we see that for water the slope is negative,
dP
< 0 .
dT
(8)
Substituting from the Clapyeron Equation, we have
ΔH fus,m
T fus ΔVm
< 0 ,
(9)
The molar enthalpy of fusion is always positive (it takes energy to melt a substance),
ΔH fus,m > 0 ,
(10)
and the temperature of fusion T fus (in degrees K) is positive. Therefore, we can use the Clapeyron Equation to
determine the sign of the molar volume change, which in this case must be negative because the slope dP/dT is
negative,
ΔVm = VL,m – VS,m < 0 ,
(11)
VL,m < VS,m . (12)
or
Here, we see for water that the molar volume of the liquid is lower than the molar volume of the solid; this implies
that when water undergoes a phase change from solid to liquid it contracts. In other words, the Clapeyron Equation
allows us to predict that water contracts upon melting (and therefore expands upon freezing).
Furthermore, the density d of a substance can be expressed in terms of the molar volume Vm as
d = M
,
Vm
(13)
where M is the molar mass. Because the density is inversely proportional to the molar volume, we have for water,
dL > d S . (14)
Since the density of liquid water is greater than the density of solid water, we have the prediction that solid water
(ice) will float on top of the liquid.
3
The Solid-Liquid Coexistence Curve for Carbon Dioxide
A sketch of the phase diagram of carbon dioxide is shown in Figure 2.
L
S
P
V
T
Figure 2. Sketch of P–T phase diagram of carbon dioxide.
For the solid-liquid coexistence curve of carbon dioxide, the slope is positive,
dP
> 0 .
dT
(15)
Substituting from the Clapyeron Equation, we have
ΔH fus,m
T fus ΔVm
> 0 ,
(16)
Again we know that the molar enthalpy of fusion is always positive (it takes energy to melt a substance) and the
temperature of fusion T fus is positive. Therefore from the Clapeyron Equation, the sign of the molar volume change
in this case must be positive,
ΔVm = VL,m – VS,m > 0 ,
(17)
VL,m > VS,m . (18)
or
This time we see for carbon dioxide that the molar volume of the liquid is greater than the molar volume of the solid,
so when carbon dioxide undergoes a phase change from solid to liquid it expands. Therefore, the Clapeyron
Equation allows us to predict in this case that carbon dioxide expands upon melting (and thus contracts upon
freezing).
Finally, since the density is inversely proportional to the molar volume, for carbon dioxide we have that
dL < d S . This result predicts that solid carbon dioxide sinks in the liquid.
(19)