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### Solutions and Mixtures

Chapter 15

# Components > 1

Lattice Model Thermody. Properties of Mixing (S,U,F,)

I. Entropy of Mixing

- Translational Entropy of Mixing
- Assume N lattice sites filled completely with NA A molecules and NB B molecules so that N = NA + NB.
- Then W = N!/( NA! NB!) See Ch 6
- Smix = k ℓn W = - k(NAℓn xA + NBℓn xB) = -Nk (xAℓn xA + xBℓn xB) Eqn 15.2, 3
- Ex 15.1

II. Energy of Mixing (1)

- Assume ideal soln, then, Umix= 0 and Fmix= -TSmix
- If the soln is not ideal, then Umix = sum of contact interactions of noncovalent bonds of nearest neighbor pairs 0.
- U = mAA wAA + mAB wAB + mBB wBB where mIJ = # I-J bonds and wIJ = I-J contact energies. All w terms < 0

Energy of Mixing (2)

- Define mAA and mAB = f(mAB, NA, NB)
- Each lattice site has z sides, so z NA = total number of contacts for all A
- Then zNA = 2mAA + mAB. And zNB = 2mBB + mAB . Solve for mAA and mBB
- Use to find U = (zwAA)NA/2 + (zwBB)NB/2 + ([wAB- ½ (wAA + wBB)] mAB) Eqn 15.8

Energy of Mixing (3)

- To simplify Eqn 15.8, use the Bragg-Williams or mean-field approximation to find average <mAB>. Note that this is a simplification and we assume that this average is a good approx to the actual situation. (i.e. one distribution dominates vs using a distribution of mAB values.

Energy of Mixing (4)Mean-Field Approximation

- Assume A and B are mixed randomly. Then the probability of finding a B next to an A ≈ z x(1-x) where pA = x = xA and (1-x) = pB.
- Then mAB = z Nx(1-x). Plug into Eqn 15.8 for U to get final eqn for U = Eqn 15.10.
- U = (zwAA)NA/2 + (zwBB)NB/2 + kT ABNANB/N

Energy of Mixing (5)Exchange Parameter

- U = (zwAA)NA/2 + (zwBB)NB/2 + kT ABNANB/N
- AB = energy cost of replacing A in pure A with B; similarly for B.
- AB = exchange parameter = - ℓn Kexch
- Umix = RT ABxA xB

Energy of Mixing (6)Exchange Parameter

- AB = exchange parameter = - ℓn Kexch
- A + B ↔ mixing eq. constant Kexch
- Umix = RT ABxA xB
- AB can be > 0 (AB interactions weaker than AA and BB); little mixing and Umix more positive, Kexch smaller
- AB can be < 0 (AB stronger than AA and BB), …

III. Free Energy of Mixing (1)

- F = U – TS = [Eqn 15.11] – T [Eqn 15.2] = Eqn 15.12
- Pure A + pure B mixed A + B has Fmix = F(NA + NB) – F(NA, 0) – F(0,NB)
- Note that F(NA, 0) = ½ x wAANA
- Fmix= [x ln x + (1-x) ln(1-x) +
AB x(1-x)]NkT Eqn 15.14

- This eqn describes a regular solution.

Free Energy of Mixing (2)

- If Fmix > 0, minimal mixing to form a soln.
- If Fmix < 0, then a soln forms
- If soln separates into 2 phases, Eqn 15.14 does not apply.
- Ex 15.2

IV. Chemical Potentials and Mixing

- A = (F/NA)NB,T = kT ℓn xA + zwAA/2 + kTAB (1-xA)2 = kT ℓn xA + corections due to AA interactions and exchange parameter. Eqn 15.15
- Also = 0 + kT ℓn x where = activity coefficient. x = effective mol fraction.

V. Free Energy of Creating Surface Area

- Consider interface or boundary between 2 condensed phases A and B.
- AB = interfacial tension = free energy cost of increasing the interfacial area between A and B.
- Calculate AB using the lattice model.

Surface Area (2)

- Assume (Fig 15.7)
- A and B are the same size
- NA = # A molecules and NB = B molecules
- interface consists of n A and n B molecules in contact with each other
- bulk molecules have z A nearest neighbors
- surface A molecules have (z-1) A nearest neighbors

Surface Area (3)

- U = Σ ni wij = term for A in bulk + term for A at surface + term for AB interactions + term for B in bulk + term for B at surface
- Then U = Eqn 15.19 = F since S = 0
- Let A = total area of interface = na
- Let a = area per molecule exposed to surface
- Then AB = (F/A)NB,NA,T = (F/n) (n/A) AB = [wAB – ½ (wAA + wBB)]/a

Surface Area (4)

- Then AB = (F/A)NB,NA,T = (F/n) (n/A) = [wAB – ½ (wAA + wBB)]/a
- AB = (kT/za) AB Eqn 15.22; see Eqn 15.11
- If there are no B molecules Eqn 15.22 reduces to Eqn 14.28 AB = - wAA /2a
- Ex 15.3 (mixing is not favorable, see p. 273)

Surface Area (5)

- Assumptions
- Mean field approximation for distribution
- Only translational contributions to S, U, F and μ are included.
- What about rot, vib, electronic? We assume that in mixing, only translational (location) and intermolecular interactions change.
- Then Fmix = F(NA + NB) – F(NA, 0) – F(0,NB) = NkT[x ln x + (1-x) ln (1-x) + AB x(1-x)]

Surface Area (6)

- However, if chemical rxns occur, rot, vib and elec must be included.

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