Gibbs phase ruleF = C - Φ + 2

F - number of independently variations

C - number of components

Φ- number of phases

2 – T, P If C=1

F Φ

0 3

1 2

2 1

P

T

Phase 1

Phase 2Phase 3

5.3 Thermal analysis approach

If observe by seeing is impossible or not convenient

C=1, Φ=1(L), F=1 (T)

( )P, F=C- Φ+1

Unviewable

Thermal jacket

Sensor

C=1, Φ=2(L,S), F=0

C=1, Φ=1(S), F=1

The cooling curve

For example

T

t/min

Vapour

The cooling curve

vapour

Vapour+water

water

water+ice

ice

Tb

Tf

T

t/min

The heating curve

The measurement of phase diagram

T

t/min

The locations of phase boundary -----Clapeyron equation

( , ) ( , )T p T p

, ,

, ,

m m

m m

d S dT V dP

d S dT V dP

m md S dT V dP

, , , ,( ) ( )m m m mS S dT V V dP

, ,

, ,

m m m

m m m

V V VdT

dP S S S

m m

m m

S HdP

dT V T V

The solid-liquid boundaryfus m

fus m

HdP

dT T V

1

212 ln

T

T

V

Hpp

mfus

mfus

The integration equation

The gas-liquid boundary

vap m2

1 1 2

1 1ln ( )

Hp

p R T T

vap m -1 1

b

85 J K molH

T

)/(g)(d

d mvap

m

mvap

pRTT

H

TV

H

T

p

vap m

2

d ln

d

Hp

T RT

Trouton’s Rule

ln vapHP constRT

The gas-solid boundary--sublimation

KRT

Hp msub

ln

2

ln

RT

H

dT

pd msub

sub m

sub m

HdP

dT T V

Larger slope than vapour/liquid boundary

Usage of phase diagram

The critical conditions: (T,P) phase transition

Phase diagram of water

pt mHCalculate

pt mVPredict

Density

Example

Estimate the vapour pressure of a liquid at any temperature from its normal boiling point.

Benzene, normal boiling point 80 (353K),℃ 30.8vap mH kJ

P=12kPa

vap m

2

d ln

d

Hp

T RT

vap m2

1 1 2

1 1ln ( )

Hp

p R T T

Q: P=? at 20℃(293K)

Triple points MPa °C D2O

liquid gas Ih 0.0006116 0.010 661 Pa, 3.82°C

liquid gas XI 0 -201.0 0 MPa, -197°C

liquid Ih III 207.5 -22.0 220 MPa, -18.8°C

Ih II III 212.9 -34.7 225 MPa, -31.0°C

II III V 344.3 -24.3 347 MPa, -21.5°C

liquid III V 346.3 -17.0 348 MPa. -14.5°C

II V VI ~620 ~-55  

liquid V VI 625.9 0.16 629 MPa, 2.4°C

VI VII VIII 2,100 ~5 1950 MPa, ~0°C

liquid VI VII 2,200 81.6 2060 MPa, 78°C

VII VIII X 62,000 -173  

liquid VII X 43,000 >700  

Ice polymorphDensity, g cm-3

a Protonsf Crystalh SymmetryDielectric con

stant, Si

Hexagonal ice, Ih

0.92 disordered Hexagonal one C6 97.5

Cubic ice, Ic 0.92 disordered Cubic four C3  

LDA b 0.94 disordered Non-crystalline    

HDA c 1.17 disordered Non-crystalline    

VHDA d 1.25 disordered Non-crystalline    

II, Ice-two 1.17 ordered Rhombohedral one C3 3.66

III, Ice-three 1.14 disordered Tetragonal one C4 117

IV, Ice-four 1.27 disordered Rhombohedral one C3  

V, Ice-five 1.23 disordered Monoclinic one C2 144

VI, Ice-six 1.31 disordered Tetragonale one C4 193

VII, Ice-seven 1.50 disordered Cubice four C3 150

VIII, Ice-eight 1.46 ordered Tetragonale one C4 4

IX, Ice-nine 1.16 ordered Tetragonal one C4 3.74

X, Ice-ten 2.51 symmetric Cubice four C3  

XI, Ice-eleven 0.92 ordered Orthorhombic three C2  

XI, Ice-eleven >2.51 symmetric Hexagonale distorted  

XII, Ice-twelve 1.29 disordered Tetragonal one C4  

Chemistry of water [www.nsf.gov]

One of the most familiar substances Unique chemistry Mysterious, complicated, weird:

Its solid form is less dense than the liquid Boils at a very high temperature

Hydrogen bonds

Dissolving substances

The most common solvent on earth

The mystery of water

Three models of water molecular structures

One linked to four neighbours(left); One to two: rings(middle) ; chains(right)

Anders Nilsson, Stanford University, 2004, high-powered x-rays

Salt contained in sea spray droplets may play an important role in atmospheric chemistry

At least some ions are present in the surface layers of water particles

Interesting and impacting problems about water Life Science; Environment; Materials; society:

Kitchen chemistry

Acid mine drainage and precipitates

Reusing wastewater

Stream restoration

Phase diagram of Carbon dioxide CO2

Can not be liquid at atmosphere pressure---dry ice: solid-gas

The melting temperature of solid rises as the pressure is increased

The triple point lies above 1 atm

The pressure should be at least 5.11 atm to obtain the liquid

Cools by Joule-Tomson effect

Phase diagram of Helium He

The solid and gas phases are never in equilibrium

The solid an be obtained only by applying pressure

He-II behave as superfluid: flow with no viscosity

The Ehrenfest classification of phase transitions

mP

ST

, ,m m mTT

V V VP P

mT

VP

, ,m mPP

HS S

T T T

The first order phase transition: A transition for which the first derivative of the chemical potential with respect to temperature is discontinuous

First order

Second order

The second order transition: the first derivative of chemical potential with respect to temperature is continuous but its second derivative is discontinuous

Homeworks

P, T phase diagram of carbon

Y: P147: 5 P151: 12 P152: 13,14,16

Preview: Y: 5.4

A: 8.3; 8.4