Phases of matter and phase diagrams
Transition to Supercritical CO2
Water
Ice
Vapor Pressure and Boiling Point
• Liquids boil when the external pressure equals the
vapor pressure.
• Temperature of boiling point increases as pressure
increases.
• Two ways to get a liquid to boil: increase
temperature or decrease pressure.
– Pressure cookers operate at high pressure. At
high pressure the boiling point of water is higher
than at 1 atm. Therefore, there is a higher
temperature at which the food is cooked,
reducing the cooking time required.
• Normal boiling point is the boiling point at 1 atm.
Critical Temperature, Tc
Critical Points
• The critical temperature Tc of a gas is the
highest temperature at which the gas can be
liquified.
• The critical pressure Pc is the pressure
required to liquify a gas at its critical
temperature.
• The critical molar volume Vm,c is the molar
volume of the gas at its critical temperature
and pressure.
• The critical isotherm has an inflection point.
Critical Constants
pc
(atm)
Vm,c
(cm3)
57.76
Tc
(K)
He
2.26
5.2
Ar
48.00
75.25 150.7
N2
33.54
90.10 126.3
O2
50.14
78.00 154.8
CO2
72.85
94.0 304.2
Phase Diagrams (P,T)
Equilibrium can exist not only between the liquid and vapor phase
of a substance but also between the solid and liquid phases, and
the solid and gas phases of a substance.
A phase diagram is a graphical way to depict the effects of pressure
and temperature on the phase of a substance:
The curves indicate the conditions of temperature and pressure
under which equilibrium between different phases of a substance
can exist
The vapor pressure curve is the border between the liquid and
gaseous states of the substance. For a given temperature, it tells us the
vapor pressure of the substance.
The vapor pressure curve ends at the critical point.
The line between the gas and solid phase indicates the vapor pressure
of the solid as it sublimes at different temperatures.
The line between the solid and liquid phases indicates the melting
temperature of the solid as a function of pressure.
For most substances the solid is denser than the liquid. An increase in
pressure usually favors the more dense solid phase.
Usually higher temperatures are required to melt the solid phase at
higher pressures
The temperature above which the gas cannot be liquefied no
matter how much pressure is applied (the kinetic energy
simply is too great for attractive forces to overcome,
regardless of the applied pressure) is called critical
temperature
The "triple point" is the particular condition of temperature
and pressure where all three physical states are in
equilibrium.
Regions not on a line represent conditions of temperature and
pressure where only one particular phase is present.
Gases are most likely under conditions of high temperature.
Solids are most likely under conditions of high pressure.
Phase Diagram for Water
The frozen state of water (ice) is actually less dense than the liquid state,
thus, the liquid state is more compact than the solid state.
Increasing pressure, which will favor compactness of the molecules, will
thus favor the liquid state.
Increasing pressure will thus lower the temperature at which the solid
will melt
van der Waals Isotherms - Ar
200
500K
150
p/atm
200K
100
50
150K
0
-50
100K
-100
0.00
0.10
0.20
Vm/L
0.30
0.40
The van der Waals Isotherms (P,V)
Nk BT
aN 2
P=
− 2
(V − Nb ) V
Nk BT  2 aN 2
abN 3

V −  Nb +
V−
=0
V +
P 
P
P

3
0
N·b
At high T, the vdW isotherms appear similar to those of an ideal gas.
The black isotherm exhibits an unusual feature - a small region
where the curve is essentially horizontal (flat) with no curvature.
Below this critical temperature TC, the vdW isotherms start to exhibit
unphysical behavior : there are regions where P decreases with
decreasing V and regions of negative pressure.
The van der Waals Isotherms
Experimentally, below TC, the system becomes unstable against the phase
separation (gas↔liquid) within a certain range V(P,T).
The black isotherm represents a boundary between those isotherms along
which no such phase transition occurs and those that exhibit phase
transitions.
For this reason, the black isotherm is called the critical isotherm, and the
point at which the isotherm is flat and has zero curvature
(∂P/∂V= ∂2P/∂V2=0) is called a critical point.
Maxwell Construction
• Below the critical temperature, the van der Waals
equation exhibits unphysical behavior, the so-called van
der Waals loops.
P
T = const (< TC)
V1
V2
V
• These loops may be eliminated using the Maxwell
construction in which the oscillating region is replace by a
horizontal line for which the areas above and below the
line are equal.This horizontal line connects the liquid and
vapor phases that coexist at equilibrium.
Maxwell Construction
100
600 K
400 K
80
p / atm
308 K
60
280 K
40
250 K
20
200 K
0
0.0
0.2
0.4
Vm / L
0.6
0.8
CO2 Critical Isotherm
304 K
600
p/atm
500
400
300
Ideal Gas
Real Gas
200
100
0
0.0
0.1
0.2
Vm /L
0.3
0.4
Inflection Points
1500
y (x)
1000
500
y'
0
y"
-500
-10
-5
0
x
5
10
Inflection Points
At the inflection point x0
2
dy
d y
= 0 and 2
=0
dx x=x0
dx x=x0
van der Waals Inflection Point
dp
RT
2a
=−
+ 3 =0
2
dVm
(Vm − b) Vm
2
d p
2RT
6a
=
− 4 =0
2
3
dVm (Vm − b) Vm
solution: Vm,c = 3b and pc = a / 27b
2
Close-Packed Structures are the most efficient way to fill space with spheres
Features of Close-Packing:
• Coordination Number = 12
• 74% of space is occupied
Estimating melting and sublimation energies
There is the Avogadro number NA of atoms in the mole of a solid.
We assume that each atom has n nearest neighbours and the strength of
the pair-wise interaction between atoms is equal to ε.
Then the energy required to melt one mole (latent heat of melting) is
approximately equal:
L ≈ ½ NA×ε×?n,
where ?n is the change of the number of nearest neighbours from
solid to liquid or vapour and ½ stands to avoid the double counting.
We can then use n = 12 for a solid and n ≈ 10 for a melt. Then
Lmelt ≈ ½NA∆nε, where ∆n = 2,
change of the coordination number from crystal to vapour ∆n = 12