Chemical & Biomolecule Separation
EN.540.306. SP26
Lecture 3: Thermodynamics Review
SHR Ch. 2
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Concept question
Magnesium sulfate is added to an organic mixture containing a
polymer of interest to remove water. This is an example of a(n):
A. Crystallization process
B. Energy-separating agent
C. Mass-separating agent
D. Leaching process
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•
•
•
T
𝐺 = 𝐺{𝑇, 𝑃, 𝑁1 , 𝑁2 , … , 𝑁𝐶 }
𝑇
P
𝑃
𝑁𝑖
𝑖
(Eq. 1)
𝜇𝑖
(thermodynamic property;
partial molar Gibbs free energy)
𝐺𝑠𝑦𝑠𝑡𝑒𝑚
=0
(Eq. 2)
Try the algebra
on your own!
=0
(Eq. 3)
(1)
𝑑𝑁𝑖
(𝑝)
(Eq. 3)
𝑑𝑁𝑖
(𝑝)
𝑑𝑁𝑖
(Eq. 3)
𝑓ഥ𝑖 = 𝜅 exp(
𝜇𝑖
)
𝑅𝑇
𝑓𝑖
𝜅
ҧ = 𝑃𝑖 = 𝑦𝑖 𝑃
𝑓𝑖𝑉
𝑦𝑖
𝑖
𝑓𝑖𝐿ҧ = 𝑥𝑖 𝑃𝑖𝑠
𝑥𝑖
𝑃𝑖𝑠
𝑖
𝑖
For phase equilibrium, the chemical potential 𝝁𝒊 ,
fugacity 𝒇ത 𝒊 , and activity 𝜶𝒊 (not fugacity coefficient or
activity coefficient) of any species is equal in all phases.
𝑓𝑖
𝑃
𝜙
𝑎𝑖
𝛾𝑖
•
•
•
𝑖
𝑖
𝑗
This ratio tells us how easy it is to separate i and j through equilibrium-based separations
easy
IMPOSSIBLE
easy
ҧ = 𝜙ത𝑖𝑉 𝑦𝑖 𝑃
𝑓𝑖𝑉
𝑓𝑖𝐿ҧ = 𝜙ത𝑖𝐿 𝑥𝑖 𝑃
𝑦𝑖
𝜙ത𝑖𝐿
𝐾𝑖 =
=
𝑥𝑖
𝜙ത𝑖𝑉
ҧ = 𝑓𝑖𝐿ҧ
𝑓𝑖𝑉
𝑓𝑖𝐿ҧ = 𝛾𝑖𝐿 𝑥𝑖 𝑓𝑖𝐿0
ҧ = 𝜙ത𝑖𝑉 𝑦𝑖 𝑃
𝑓𝑖𝑉
𝛾𝑖𝐿 𝑓𝑖𝐿0 𝛾𝑖𝐿 𝜙𝑖𝐿
𝐾𝑖 =
=
ത
𝜙𝑖𝑉 𝑃
𝜙ത𝑖𝑉
𝑦𝑖
𝜙ത𝑖𝐿
𝐾𝑖 =
=
𝑥𝑖
𝜙ത𝑖𝑉
𝑠
𝑃
𝜙ത𝑖𝐿 = 𝑃𝑖
𝜙ത𝑖𝑉 = 1
𝛾𝑖𝐿 𝑓𝑖𝐿0 𝛾𝑖𝐿 𝜙𝑖𝐿
𝐾𝑖 =
=
𝜙ത𝑖𝑉 𝑃
𝜙ത𝑖𝑉
𝑦𝑖
𝜙ത𝑖𝐿
𝐾𝑖 =
=
𝑥𝑖
𝜙ത𝑖𝑉
𝜙𝑖𝐿
𝛾𝑖𝐿 𝑓𝑖𝐿0 𝛾𝑖𝐿 𝜙𝑖𝐿
𝐾𝑖 =
=
𝜙ത𝑖𝑉 𝑃
𝜙ത𝑖𝑉
𝑠
1 𝑃
𝑠 𝑃𝑖
𝜙𝑖𝐿 = 𝜙𝑖𝑉 exp( න 𝑉𝑖𝐿 𝑑𝑃)
𝑃
𝑅𝑇 𝑃𝑠
𝑖
𝑃𝑖𝑠
𝑠
𝛾𝑖𝐿 𝜙𝑖𝑉
𝑃𝑖𝑠
1 𝑃
𝐾𝑖 =
exp
න 𝑉 𝑑𝑃
ഥ 𝑖𝑉 𝑃
𝑅𝑇 𝑃𝑠 𝑖𝐿
Φ
𝑖
𝑠
𝑃
1
𝑠 𝑃𝑖
= 𝛾𝑖𝐿 𝜙𝑖𝑉
exp( න 𝑉𝑖𝐿 𝑑𝑃)
𝑃
𝑅𝑇 𝑃𝑠
𝑠
𝜙𝑖𝑉
𝑖
𝑃
lim = 𝐻𝑖
𝑥→0 𝑥
𝑃𝑖
𝐻𝑖 =
𝑥𝑖
𝑃𝑖
𝑦𝑖
𝐻𝑖
𝐾𝑖 =
= 𝑃 =
𝑥𝑖 𝑃𝑖
𝑃
𝐻𝑖
𝑃
= 𝑃𝑖𝑠
𝑥→1 𝑥
lim
§
Tc : critical temperature
Pc : critical pressure
Tr : reduced temperature (T/ Tc)
ω: acentric factor
Z: compressibility factor
(1949)
(1972)
(1976)
(6) van der Waals (1873)
a = 0.421875 R2Tc2/Pc
b = 0.125 RTc/Pc
van der Waals (1873)
a = 0.421875 R2Tc2/Pc
b = 0.125 RTc/Pc
Integral Departure functions for:
Fugacity coefficient (pure species)
Partial fugacity coefficient
Solving these departure functions with R-K / S-R-K EOS give us
𝑎𝑃
𝐴= 2 2
𝑅 𝑇
Can be used in vap or liq
phases because cubic EOS can
approximate 2-phase behavior!
⟹ substitute to get yi and xi:
𝑏𝑃
𝐵=
𝑅𝑇