Non-Ideal Solutions (Chap. 9.7...
Transcript of Non-Ideal Solutions (Chap. 9.7...
Non-Ideal Solutions (Chap. 9.7 – 9.11)
Most solutions show some deviations from ideality
(Raoult’s Law)
In addition to Henry’s Law, we will examine:
– Regular Solutions
– Sub-Regular Solutions
Convenient to define activity coefficient (important):
– γi = 1 (ideal Raoultian Behaviour)
– γi > 1 (positive deviations)
– γi < 1 (negative deviations)
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Non-Ideal Solutions (Chap. 9.7)
Fe – Ni (negative deviation) Fe – Cu (positive deviation)
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Non-Ideal Solutions (Chap. 9.7)
Fe – Ni (negative deviation) Fe – Cu (positive deviation)
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Non-Ideal Solutions (Chap. 9.7)
The heat of mixing is not usually zero, and this is reflected
in the partial molar free energy of mixing and the heat of
mixing according to the following argument
We previously showed that:
This shows that if the activity coefficient is temperature
dependent, then the partial molar free energy of mixing is
not zero
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Non-Ideal Solutions (Chap. 9.7)
We can further manipulate this equation:
So plotting R ln γi versus 1/T will have a slope of the partial
molar free energy of mixing
Solutions become more ideal as T is raised
γi > 1 (positive deviations)
– Positive slope, positive heat of mixing
– γi decreases with T increase, becomes more ideal
γi < 1 (negative deviations)
– Negative slope, negative heat of mixing
– γi increases with T increase, becomes more ideal
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Example of Non-Ideal Solution (Fe – Si)
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Example of Non-Ideal Solution (Fe – Si)
Activity coefficient of
Si at infinite dilution:
Becomes closer to
unity as T increases
Note becomes more
ideal as T increases
Exothermic heat of
mixing, negative
deviations
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Example of Non-Ideal Solution (Fe – Si)
Activity coefficients for Fe
and Si
Note logarithmic scale
Both activity coefficients
increase with temperature,
tending towards ideality
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Demonstration of the Heat of Mixing
Silicon is added to steel as FeSi in various grades (50%,
70% Si) as lumps
FeSi is much cheaper to make than metallurgical grade Si
Problems in addition:
– Buoyant
– Steel shell forms
Problem investigated by Prof. S. Argyropoulos at University
of Toronto (videos kindly supplied by him)
– Cylinders of FeSi immersed into steel melts
– Steel cylinders immersed into FeSi melts
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Demonstration of the Heat of Mixing
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Demonstration of the Heat of Mixing
FeSi cylinder into steel melt
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Demonstration of the Heat of Mixing
Steel cylinder into FeSi melt
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Non-Ideal Solutions (Chap. 9.7)
Negative deviations
– γi < 1
– Negative heat of mixing, Exothermic
– A – B bonds very strong, tendency to ordering or compounds
Positive deviations
– γi > 1
– Positive heat of mixing, Endothermic
– A and B do not want to order from the bond energy
comparison
– However, the entropy contribution to the free energy is strong
enough to overcome the weak bonds between A – B and we
see the solution is made
– Free energy is always the combination of enthalpy and entropy
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Gibbs-Duhem Equation for Determination
of Activity (Chap. 9.8)
Often it is only possible to measure the activity of one
component in a solution, but we would like to know the
other one; the Gibbs-Duhem Equation makes this possible
For a binary solution
The partial molar free energy of mixing is:
Simple relationship for the changes in activities of A and B
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Gibbs-Duhem Equation for Determination
of Activity (Chap. 9.8)
If we know how aB changes
with composition we can
integrate that equation
Simple equation became
very nasty
– Experimental data (graphical)
– As Xb → 1, log ab → 0,
XB/XA → ∞
– As Xb → 0, ab → 0,
log ab → - ∞
The tail on the right is the
biggest problem15
Gibbs-Duhem Equation for Determination
of Activity (Chap. 9.8)
The tail on the right can be eliminated by using activity
coefficients for a binary solution
Subtract this equation from after converting to log
Gives another simple equation
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Gibbs-Duhem Equation for Determination
of Activity (Chap. 9.8)
Now do the integration
with activity coefficients
instead of activities
Now the curve does not go
to infinity because the
activity coefficient of B
must have a finite value at
XA = 1 (Henry’s Law)
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Worked Example of Gibbs-Duhem
Equation (Question 9.6)
Question solved with Equation (9.55), sufficient accuracy
Solution already presented as:
– Fig. 9.9 (p. 227)
– Fig. 9.11 (p. 228)
– Fig. 9.15 (p. 234)
Sufficient accuracy to use linear interpolation between data
points for the integration
Excel spreadsheet implementation, but you can use other
programs or even graph paper18
Worked Example of Gibbs-Duhem
Equation (Question 9.6)
19Last line because of Henry’s Law
Not sensitive to choice of γCu
Cannot go to infinity
Incremental area
Between points
Cumulative
AreaCu is "B" and Fe is "A"
Xcu aCu γCu "log γCu" Xcu/Xfe Δarea Σarea γFe
1 1 1 0 #DIV/0!
0.9 0.935 1.038889 0.016569 9 0.775958 5.969778
0.8 0.895 1.11875 0.048733 4 0.209066 0.566893 3.688863
0.7 0.865 1.235714 0.091918 2.333333 0.136753 0.43014 2.692402
0.6 0.85 1.416667 0.151268 1.5 0.113753 0.316387 2.071985
0.5 0.83 1.66 0.220108 1 0.086051 0.230336 1.699558
0.4 0.81 2.025 0.306425 0.666667 0.071931 0.158405 1.440142
0.3 0.78 2.6 0.414973 0.428571 0.059443 0.098962 1.25592
0.2 0.72 3.6 0.556303 0.25 0.047951 0.051011 1.124634
0.1 0.575 5.75 0.759668 0.111111 0.036719 0.014292 1.033457
0.05 0.4 8 0.90309 0.052632 0.011742 0.00255 1.005889
0 10 1 0 0.00255 0 1
Worked Example of Gibbs-Duhem
Equation (Question 9.6)
XCu/XFe vs. log γCu
Compare with Fig. 9.15
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Worked Example of Gibbs-Duhem
Equation (Question 9.6)
Comparison of calculated Fe activity coefficients with those
in text Fig 9.11 (p. 228)
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Regular Solutions (Chap. 9.9)
We have defined ideal (Raoultian) solutions:
The next step in developing models (combination of theory
and empirical evidence) is to recognize the heat of mixing is
not zero (we have already discussed clustering and bond
energies)
A Regular solution is defined as (important):
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Regular Solutions (Chap. 9.9)
The other aspect about regular solutions is:
Excess properties are those in excess of ideal solutions:
The free energy of mixing is has 2 components:
In general for a solution:
For an ideal solution there is no heat of mixing:
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Regular Solutions (Chap. 9.9)
So the excess free energy of a solution is:
A regular solution has the same entropy as an ideal one, so
the only difference in the free energy is the heat of mixing:
A regular solution has the same entropy of mixing as an
ideal solution, but the activity coefficients are not unity:
The ideal solution free energy of mixing is:
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Regular Solutions (Chap. 9.9)
The difference in free energy is the excess free energy:
Recall the activity coefficients for a regular solution:
Substituting and recognizing (XA + XB) =1
α is an inverse function of temperature:
Therefore the excess free energy (i.e. the heat term) is
independent of temperature:
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Regular Solutions (Chap. 9.9)
There are some useful relationships for regular solutions:
The Thallium – Tin binary is a regular solution:
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Regular Solutions (Chap. 9.9)
Tl – Sn system shows regular behaviour of activity
coefficients with composition
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Sub-Regular Solutions (Chap. 9.11)
Regular solutions have a parabolic dependency of the heat
of mixing on composition
Sub-regular solutions give more flexibility:
– Curve fitting
– Some justification of curve fitting from statistical
thermodynamics (Chap. 9.10)
The coefficients a and b are obtained by fitting to data
Examples of flexibility on the next slide
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Sub-Regular Solutions (Chap. 9.11)
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Worked Example (Chap. 9.13, Ex. 1)
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Note that
the text solution
has an error
Worked Example (Question 9.8)
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Worked Example (Question 9.10)
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