1. CHEMISTRY E182019 Phase equilibria G=0 Clausius Clapeyron CH7
This course is approximately at this level
Phase equilibria G=0 Clausius Clapeyron
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. T-S diagrams T-s diagrams Gas steam Liquid L+G CH7
p=1000 bar technically realizable maximum
Saturated liquid curve
Gas steam
Saturated vapour curve
Liquid
L+G

3. T-S diagrams evaporation
CH7
T-s diagrams
Saturated liquid
s’=2 kJ/kgK
Saturated steam
s’’=6 kJ/kgK
Enthalpy of evaporation hLG=T(s’’-s’)=5004=2MJ/kg

4. Solid-Liquid-Gas Phase diagrams L-liquid G-gas L-liquid S-solid
CH7
Phase diagrams
G-gas
L-liquid
S-solid
S+G
L+G v p
S-solid
L-liquid
G-gas T p
The reason why the regions L+G, S+G appear in the p-v diagram is that the specific volume v (unlike T,p) varies during phase transformations.

5. liquid-like hydrogen-bonded clusters dispersed within a gas-like phase
Solid-Liquid-Gas
CH7
Phase diagram ice-water-steam
TRIPLE POINT
CRITICAL POINT
liquid-like hydrogen-bonded clusters dispersed within a gas-like phase
Cubic ice
Hexagonal ice

6. Solid-Liquid-Gas L-liquid S-solid G-gas CH7 p T
Melting hSL>0, sSL>0, GSL=0, dp=dT=0
S-solid
L-liquid
G-gas T p
Evaporation hLG>0, sLG>0, GLG=0, dp=dT=0
Sublimation hSG>0, sSG>0, GSG=0, dp=dT=0
During phase transitions the pressure and temperature are constant. Also Gibbs energy remains constant as follows from its definition g=h-Ts=0. Only specific volume increases or decreases.

7. Clausius Clapeyron Solid-Liquid-Gas
CH7
S-solid
L-liquid
G-gas T p
Slopes dp/dT are given by Clausius Clapeyron equation
Phase transition lines in the p-T diagram are described by the Clausius Clapeyron equation
Enthalpy of phase changes, e.g. hLG
Specific volume changes, e.g. vG-vL

8. Clausius Clapeyron Solid-Liquid-Gas
CH7
S-solid
L-liquid
G-gas T p
dp/dt>0 because hLG>0 vLG>0 (volume of steam is greater than volume of liquid)
The slope dp/dT is negative because specific volume of ice is greater than volume of liquid
Melting point temperature of ice decreases with pressure – therefore ice under skates melts and forms a liquid film

9. Closed loop (evaporation, expansion, condensation, compression)
Clausius Clapeyron derivation
CH7
T+dT T
s=s’’-s’
Clausius Clapeyron equation can be derived from energy balance of a closed cycle in Ts diagram:
Heat added-difference between evaporation enthalpy at T+dT and condensation enthalpy at temperature T
Mechanical work-received in one cycle
Closed loop (evaporation, expansion, condensation, compression)

10. Clausius Clapeyron and hLG
CH7
Clausius Clapeyron equation is exact, because follows from thermodynamic principles. Individual terms (dp/dT,v’’) can be approximated by semiempirical equations (different state equations, Antoine’s equation)
State equation
Antoine’s equation
Result can be improved when using Van der Waals
Giving expression for ΔhLG
or Redlich Kwong state equation

11. Multicomponent equilibrium
CH7
HEAVY
LIGHT A B
xA=
yA=
Liquid phase
Gaseous phase
Question: Is there a relationship between composition of binary mixture in the liquid phase xA and gaseous phase yA?
Binary mixture
pA=pA"xA
Answer: Yes, Raoult’s law applicable to ideal liquids

12. Raoult’s law
CH7
HEAVY B
Fact: It does not matter, how much liquid is in the vessel, pressure of vapours is the same, and given by Antoine’s equation
Therefore also the molar volume nB/V is independent of amount of liquid.

13. Raoult’s law LIGHT HEAVY A B xA CH7 Volume xAV Volume V
after expansion of nA molecules to volume V xA
Volume V
Volume xAV
giving Raoult’s law
…and also answer to the previous question

14. HEAVY
LIGHT A B
Raoult’s law
CH7 xA
Volume V
Volume xAV
1-xA=xB

15. Distillation Liquid mixture enriched by heavy component
Cooling
condensing
Liquid mixture enriched by heavy component
Liquid mixture enriched by light component
Initial composition of Liquid Mixture

16. Henry’s constant can be found in tables
Liquid – noncondensable gas
CH7
Given temperature T and molar fraction of dissolved CO2
H2O+CO2
yCO2
What to do if T > Tcrit = 31 oC ?
Henry’s law
H2O+CO2
xCO2
Henry’s constant can be found in tables

17. Tutorial LIGHT HEAVY A B CH7
Given total pressure p and molar fraction of liquid phase xA calculate
Equilibrium temperature T
Molar composition of vapours
Nonlinear equation for T (Excel solution)
p=const
Repeated distillation yA xA

18. Tutorial SYRINGE alcohol
CH7
Initial state: Syringe filled by liquid mixture H2O (B) + CH3OH (A) (methylalcohol). Initial volume V0, molar fraction of methylalcohol xA, number of moles nA, nB are given (approximated from density).
Final state: Volume is increased to V1. Temperature is constant (room temperature).
Calculate pressure p, molar fraction of methylalcohol in liquid and vapour phase. V1 VL VG V0

19. Tutorial Syringe alcohol
CH7
Unknown 9 variables
xA=? yA=? p=? VL=? VG=? nAL=? nAG=? nBL=? nBG=?
Equations
LINEAR NONLINEAR