Chemical Engineering Thermodynamics
CHMT3041 2022
Lecture 8
Lecture 8 Modelling Phase Equilibrium for
Pure Components
- Equilibrium criterion for phase equilibrium
- Estimation of P and T for phase equilibrium
- Clausius and Clausius-Clapeyron
- Shortcut equation
- Antoine equation
- Using fugacity to model phase equilibrium
- Calculating changes in G
- Poynting method for liquids and solids
Video: Modelling Pure
Component Phase
Equilibrium Part 1
Video: Modelling Pure
Component Phase
Equilibrium Part 2
(Moran: 11.4.1 and 14.5, Dahm*: Chapter 7.2.6 and Chapter 8)
*Dahm K.D., Visco D.P., 2015. Fundamentals of Chemical Engineering Thermodynamics. SI edition, CENGAGE Learning.
Modelling Phase Equilibrium
for Pure Components
Part 1
- Equilibrium criterion for phase equilibrium
- Estimation of P and T for phase equilibrium
- Clausius and Clausius-Clapeyron eq
- Shortcut equation
- Antoine equation
Equilibrium criterion - revisit
(14.1)
(14.2)
(14.6)
Constant T and P
(14.11)
or
0
0
0
Multicomponent
Pure compound
Irreversible process
Note alternative symbols
Equilibrium between two phases of pure substance
A system consisting of two phases ‘ and “
Note alternative symbols
At equilibrium
Also at equilibrium chemical potential and fugacity of phases equal
𝜇′ = 𝜇′′
𝑓′ = 𝑓′′
μL = μV
f L= f V
Note alternative symbols
Clapeyron equation - revisit
Special form for liquid or solid and vapour (Clausius Clapeyron)
Phase equilibrium
Note similarity to van’t Hoff equation
Chemical equilibrium
Clapeyron equation - alternative derivation
Note that
For two phases in equilibrium
=>
=>
(11.30)
(11.31)
(Moran 14.61)
Clausius-Clapeyron equation
For a system at equilibrium with
liquid/solid and vapour
negligible compared to
and ideal gas
Note alternative symbols
(Dahm 8.16)
reasonable assumptions at low pressure
(Dahm 8.20)
Note when integrating enthalpy of phase
change is assumed constant over T range
Shortcut Equation
Clausius-Clapeyron only for modelling at low pressure
Note alternative symbols will now be used as we switch to our other recommended
textbook (Dahm)
Clapeyron equation depends on no assumptions
(Dahm 8.23)
Goal: integrate into an expression relate Psat to T
=>
Shortcut Equation - 2
Assumption
=>
Now use critical point and acentric
factor ω
=>
Shortcut Equation - 3
Alternatively
Acentric factor
(Dahm 8.41)
Antoine equation
An empirical “version” of Clausius-Clapeyron equation
(Dahm 8.23)
Table in Appendix E in Dahm
Acentric factor
(Dahm Chapter 7 paragraph 7.2.6)
- A measure of non-sphericity (centricity) of molecules
- Introduced by Kenneth Pitzer in 1955, also known as Pitzer
acentric factor
- For spherical molecules ω is almost exactly zero.
- From definition
Acentric
factor - 2
(Dahm par 7.2.6)
Acentric factor - 3
THE END
Modelling Phase Equilibrium
for Pure Components
Part 2
Using fugacity to model phase equilibrium
- Calculating changes in G
- Poynting method for liquids and solids
Calculating changes in Gibbs Energy
-Changes in S and H can be modelled using an equation of
state
- Changes in G can be computed using an EOS
- Entire VLE curve can be predicted using an EOS
-Compute G at any T and P for liquid or vapour using an
EOS, can identify P’s and T’s where
Example Calculating Gibbs on isothermal path
(Dahm Example 8-4)
Solution Calculating Gibbs on isothermal path
1
Solution Calculating Gibbs on isothermal path - 2
2
Solution Calculating Gibbs on isothermal path - 3
3
G (μ) as a function of P – inconvenient function!
-G/n (μ) is an awkward
mathematical function to use
- GV approaches -∞ as
P goes to 0.
- Concern when mixtures
are modelled as partial
pressure could be in order
of 1 kPa
(reference: Dahm Example 8-4 p 371)
G (μ) as a function of P – inconvenient function!
Fugacity – recall
real gas
ideal gas
=>
and
into
=>
𝑓
𝜑=
𝑝
Complete
fugacity
function
Fugacity in terms of residual/excess μ or G
Note alternative symbols
remember:
and
𝑓
𝜇𝑅 = 𝜇 − 𝜇 = 𝑅𝑇 (𝑙𝑛𝑓 − ln 𝑝) = 𝑅𝑇 ln
= 𝑅𝑇𝑙𝑛𝜑
𝑝
∗
∗
𝑓 𝜇−𝜇
ln =
= 𝜇𝑅/𝑅𝑇
𝑝
𝑅𝑇
Dahm
eq (8.57)
or
Definition of fugacity
Fugacity in terms of Z (EOS) – recall
Equation of state
or
(Dahm eq 8.62)
(Dahm eq 8.63)
Fugacity chart
(Moran eq 11.124)
Example Estimating vapour pressure using f and EOS
(Dahm Example 8-5)
The boiler of a refrigerator system is designed to operate at T = 263.71 K.
Assuming Freon 22 in the liquid and vapour phases can be described by
the Van der Waals equation, compute the vapour pressure of Freon 22
at T = 263.71 K.
Van der Waals constants
a = 4.888 x 106 Pa m6 / kmol2 and b = 0.09988 m2/kmol
Solution Estimating vapour pressure using f and EOS
1
The Gibbs phase rule
(Dahm p 56 and Moran 14.6.2)
F = C–π+2
= 2–2+1=1
2
Substitute
into
Solution Estimating vapour pressure using f and EOS
Integrate to find:
(Dahm eq 8-66)
Note: go through steps on your own
3
Solution Estimating vapour pressure using f and EOS
Discussion
Poynting method
-Reliable EOS less frequently available for liquids and solids
-Poynting correction is an alternative approach used for liquid
and solid fugacity at elevated pressures
- Developed by John Henry Poynting (1852-1914)
- Poynting equation can be applied between any two pressures
- Allows f to be calculated using Psat data
Poynting
factor (Dahm eq 8.83)
Example Estimating f of compressedwater
(Dahm Example 8-6)
Poynting derivation in this example
Solution Estimating f of compressedwater
(Dahm Example 8-6)
1
Solution Estimating f of compressedwater
(Dahm Example 8-6)
2
=>
Solution Estimating f of compressedwater
(Dahm Example 8-6)
3
=>
Solution Estimating f of compressedwater
(Dahm Example 8-6)
4
=>
Solution Estimating f of compressedwater
(Dahm Example 8-6)
5
=>
Solution Estimating f of compressedwater
(Dahm Example 8-6)
Poynting method continued
-Common simplifications when used:
- P low – ideal gas assumed and Poynting factor negligible
and f = f sat = Psat
- At high P and Psat low enough also f sat = Psat
-At high Psat the f sat can be obtained from and EOS for vapour
phase.
THE END