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)

1

Non-Ideal Solutions (Chap. 9.7)

Fe – Ni (negative deviation) Fe – Cu (positive deviation)

2

Non-Ideal Solutions (Chap. 9.7)

Fe – Ni (negative deviation) Fe – Cu (positive deviation)

3

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

4

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

5

Example of Non-Ideal Solution (Fe – Si)

6

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

7

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

8

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

9

Demonstration of the Heat of Mixing

10

Demonstration of the Heat of Mixing

FeSi cylinder into steel melt

11

Demonstration of the Heat of Mixing

Steel cylinder into FeSi melt

12

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

13

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

14

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

16

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)

17

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

20

Worked Example of Gibbs-Duhem

Equation (Question 9.6)

Comparison of calculated Fe activity coefficients with those

in text Fig 9.11 (p. 228)

21

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):

22

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:

23

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:

24

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:

25

Regular Solutions (Chap. 9.9)

There are some useful relationships for regular solutions:

The Thallium – Tin binary is a regular solution:

26

Regular Solutions (Chap. 9.9)

Tl – Sn system shows regular behaviour of activity

coefficients with composition

27

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

28

Sub-Regular Solutions (Chap. 9.11)

29

Worked Example (Chap. 9.13, Ex. 1)

30

Note that

the text solution

has an error

Worked Example (Question 9.8)

31

Worked Example (Question 9.10)

32