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 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
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