Lecture 9

Question 1

For an ideal solution and non-ideal solution what’s the chemical potential of a component>

Answer 1

ideal solutions:
μ = μ* + nRTln x = u* + kb*T ln x

non-ideal solutions:

μ = μ* + nRTln x = u* + kb*T ln a

where a is the activity, a = y*x

Question 2

What is a common characteristic of a regular solution?

Answer 2

the assumption that the non-ideal behavior of a mixture is a result of mixing enthalpies while mixing entropy is still considered to be ideal.

Question 3

What is the lattice model?

Answer 3

assume we have a binary mixture consisting of N1 and N2 molecules both with the same size but the interaction between N1 with N1 and N2 with N2 and N1 with N1 differ.

In this system the gibbs free energy can be written as

G = Gunmixed + mG, (mG = mixing )

where mG = mH - TmS

To find an expression for the enthalpy and entropy of mixing a lattice is considered. In this lattice, it is assumed that each of the N1 molecules and N2 molecules occupy a single lattice site with a size σ, where only the nearest neighbor interactions are taken into account.

in this lattice model, it is further assumed that all lattice sites are filled ( Hence, mH = mU and as a result mG = mA) and it is assumed that random mixing takes place which although strictly speaking incorrect means that N = N1 + N2 molecules are distributed randomly over the N lattice sites.

illustration of the lattice model is found in notes on page 1.

Question 4

What is the derivation of mixing entropy?

Answer 4

found in notes on pages 1-2

Question 5

What is the derivation of mixing enthalpy?

Answer 5

Found in notes on pages 2-3

Question 6

What is the derivation of chemical potential?

Answer 6

found in notes on pages 3-4

Question 7

What is the derivation for vapour pressure of regular solutions?

Answer 7

found in notes on pages 4 - 5

Question 8

What is the theory of phase separation in binary liquid mixtures based on the Kay constant?

Answer 8

First, it is important to note that two liquid phases that differ in composition can coexist at constant temp, and the pressure if the chemical potential of the liquids in each phase is equal. (This is phase separation condition)

To demonstrate the relation of the Kay constant and phase separation we must first plot the gibbs energy of mixing per molecule ‘g’ over the composition of one of the liquids ‘this is illustrated in note page 60’. From this graph, we view 3 scenarios:

1. the values of Kay below 2, the quantity of g has one minimum
2. As Kay approaches 2 the minima broadens as an inflection point (dg/dx = 0 = d^2g/dx^2) is reached at kay = 2 (also at) T = Tc
3. for T < Tc ( values of kay greater than 2) the function has two minima as a function of x, separated by a maximum

For scenario 3, if a common tangent can be drawn along the g-curve connecting both the x functions then the condition set in the first paragraph is achieved and the components phase separate.

Question 9

What are the binodal and spinodal curves? What are their conditions and equations?

Answer 9

Given G = Gunmixed + mG

G = Gunixed + (N1 + N2)g, where g literally is just mG/(N1+N2)

we can use the G to express the chemical potentials of the binary liquid mixture:

μ1 = μ*1 - x2(δ g/δ x2) + g, and

μ2 = μ*2 - (1-x2)(δ g/δ x2) + g

Since the common tangent line that connects the coexistence curves lies horizontally, the coexistence condition simplifies to:

(δ g/δ x2) = w(1-x2) + kbTln(x2/1-x2) = 0

The collection of these terms is denoted as the binodal. Any region not with the binodal curve is stable and doesn’t phase separate.

Furthermore, the spinodal is the region between the metastable and stable region characterized by:

(δ^2g/δ x2^2) = 0.

Where the binodal and spinodal concede is the inflection point of the critical temp where: (Kay = 2, and x = 0.5)

(δ g/δ x2) = 0 = (δ^2g/δ x2^2)

Note that the regular solution predicts spinodal and binodal around x = 1/2