Transcript Solution Thermodynamics: Applications

Solution Thermodynamics:
Applications
Chapter 12-Part I
EQUILIBRIUM CONDITIONS for VLE
TL = TV
PL =PV
miL = miV for all i =1, 2, …k
Equality of chemical potentials is
equivalent to equality of fugacities
ˆf V  fˆ L
i
i
For all components, i = 1, 2, …k
How do we model fugacities in real systems?
ˆf V  y ˆV P
i
i i
What happens at low pressures (P < 1 bar)?
How about the liquid phase
First approximation
ˆf L  fˆ id  x f
i
i
i i
Let’s see how it works
Fugacities
(from data at constant T)
Graph at constant
T and P
As a general rule, fugacities
follow Henry’s law at xi0
and Lewis-Randall’s rule
at xi 1
What happens at intermediate
concentrations
• Actual behavior in the liquid phase:
ˆf L  x  f
i
i i i
Therefore:
f i  Pi
sat
yi P  xi Pi  i
sat
How fugacity of a given component
depends on the nature of the
second component
In both cases, the system is
non-ideal
The activity coefficients are related
to GE
• By the additivity rule:
E
G
 x1 ln  1  x2 ln  2
RT
So, if i are obtained experimentally, so is GE
Calculated from
yi P  xi Pi  i
sat
From additivity rule
Lim x10 of GE/x1x2RT
Lim x10 of GE/x1x2RT
lim x1 0
GE
G E / RT
dGE / RT
 lim x1 0
 lim x1 0
x1 x2 RT
x1
dx1
dGE / RT
d ln  1
d ln  2
 x1
 ln  1  x2
 ln  2
dx1
dx1
dx1
From the Gibbs-Duhem equation:
d ln  1
d ln  2
x1
 x2
0
dx1
dx1
Then, Lim x10 of GE/x1x2RT
G
1
 limx1 0 ln 
x1 x2 RT
2
E
limx1 0
Data Reduction
• We measure VLE experimentally and we
want to obtain analytical expressions to
calculate VLE at other conditions
• For example, from Figure 12.5(b) we see
that GE/x1x2RT follows a straight line
  (nG E / RT ) 
ln  1  

n1

 P ,T ,n2
Example: GE/RT = A x1 x2
We can get GE/RT as a function of
composition
E
G
 A21 x1  A12 x2
x1 x2 RT
From this equation we can calculate ln1 and ln2 as
functions of composition
and compare with the experimental values
  (nG E / RT ) 
ln  1  

n1

 P ,T ,n2
See derivation page 438,
Practice that derivation
Results
ln  1  x [ A12  2( A21  A12 ) x1 ]
2
2
ln  2  x [ A21  2( A12  A21 ) x2 ]
2
1
2-parameters Margules equations
What are the activity coefficients at infinite dilution for this model?
GE/RT = (0.198x1 + 0.372 x2)x1x2
This is what is called DATA REDUCTION