Phases of matter and phase diagrams

Transition to Supercritical CO2

Water Ice

• 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.

Vapor Pressure and Boiling Point

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) Tc (K)

He 2.26 57.76 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

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:

Phase Diagrams (P,T)

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

-100

-50

0

50

100

150

200

0.00 0.10 0.20 0.30 0.40

Vm/L

p/at

m

100K

150K

200K

500K

The van der Waals Isotherms (P,V)

( ) 2

2

VaN

NbVTNk

P B −−

=

032

23 =−+

+−

PabN

VP

aNV

PTNk

NbV B

0

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.

N·b

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.

The van der Waals Isotherms

Maxwell Construction• Below the critical temperature, the van der Waals

equation exhibits unphysical behavior, the so-called van der Waals loops.

• 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.

V

P

V1 V2

T = const (< TC)

Maxwell Construction

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8Vm / L

p / a

tm

200 K

250 K

280 K

308 K

400 K600 K

CO2 Critical Isotherm

0

100

200

300

400

500

600

0.0 0.1 0.2 0.3 0.4

Ideal GasReal Gas

304 K

Vm /L

p/at

m

Inflection Points

-500

0

500

1000

1500

-10 -5 0 5 10

x

y"

y'

y (x)

Inflection Points

At the inflection point x

and

0

dydx

d ydxx x x x= =

= =0 0

0 02

2

van der Waals Inflection Point

dpdV

RTV b

aV

d pdV

RTV b

aV

V b p a b

m m m

m m m

m c c

= −−

+ =

=−

− =

= =

( )

( )

/,

2 3

2

2 3 4

2

20

2 60

3 27solution: and

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

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

Estimating melting and sublimation energies