Thermodynamics of separation
Pure
Component 1
Mixture 12
Pure
Component 2
W in
Q out
What is the minimum work to separate a mixture into
it’s pure components? Ex. Mining, Desalination, Material
Purification, Recycling.
Balance Eq’ns for Mass, Energy & Entropy
dN i,sys
Pure
Component 1
Mixture 12
Sirr
Pure
Component 2
W in
Q out
dt
 N i,in  N i,out
dE
 Qout  Win  H 12  H 1  H 2
dt
dS
Qout

 S12  S1  S2  Sirr
dt
T0
Win  ((H1  H 2 )  H12 )  To ((S1  S2 )  S12 )  ToSirr
Win  N12 (hmix  T0 smix )  T0 Sirr
Minimum Work of Separation
Win  N12 (hmix  T0 smix )  T0 Sirr
Win  N12 g
wmin
o
mix
 T0 Sirr
Wmin


 g mix
N12
Gibbs Free Energy of Mixing*
gomix = homix –T0 somix.
gomix  –T0 smix =
–T0 (s12 –x1s1 – x2s2)
For non-interacting molecules entropy can dominate
often resulting in a negative Gibbs Free Energy and hence
spontaneous mixing. I.e. gomix < 0
* at standard conditions
Boltzmann’s entropy equation
S = k ln 
How many ways can “r”
atoms be positioned in
a lattice with “n” locations?
n!

r!(n  r)!
Ex. 4 atoms in 8 locations
n!
8!
70
12 


r!(n  r)! 4!4! 1
wmin = T0smix = k T0 (ln 12)
Using Stirling’s Approximation
ln N! = N ln N - N
wmin  T0 R(x ln x  (1  x)ln(1  x))
Where x is mol fraction r/n, and R = k Navo
Multi-component System
n!

n1 !n2 !.....n j !
j
wmin  T0 R xi ln xi
i 1
“Separation”
n
wmin  T0 R xi ln xi
i 1
“Extraction”
W min i  T0 R( N 1 ln x  N 2 ln( 1  x ))
(N )
( N1 1)
Wmin
 T0 R(( N1  1) ln x  N 2 ln( 1  x))
1
wmin, 1  T0 R(ln )
x1
Separation Examples
• From the atmosphere
• From the Ocean
• Solutions
– Polymer
– Water based
– Liquid metals (activity coef)
The minimum work to separate
O2 from the atmosphere
Table from the
EngineeringToolBox.com
o
ex,O
 T0 R(ln
2
1
)  298(K)  8.314(J / molK)ln(0.212)  3.84(kJ / mol)
xO2
In wet air you get 3.97 kJ/mol : compare with Szargut
Energy
kg (processed)
Energy

kg(target)
kg (target)
kg (processed)
1
Energy
~
g kg (processed)
energy requirements for mining
and milling, possible future trends
underground ~ 1000/g (MJ/t metal)
open pit ~ 400/g (MJ/t metal)
Chapman and Roberts p 113 & 116
Sherwood plot showing the relationship between the concentration
of a target material in a feed stream and the market value of (or cost to remove)
the target material [Grübler 1998].
Exergy of a Mixture
e
o
x, mixture
 x e
o
i x, i
 RT0  xi ln xi
Pure metal
Exergy
Recycle to
pure metal
Metal alloy
Mixing in product
Mixing in waste stream
Pure ore (e.g. Fe2O3)
Further mixing and
corrosion
Ore value at mine
CRUST at To, po
Purification
Stages
Theoretical Exergy Values for a metal extracted from the earth’s
crust shown at various stages of a product life cycle (not to scale)