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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 N A A molecules and N B B molecules so that N = N A + N B .

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Solutions and mixtures

Solutions and Mixtures

Chapter 15

# Components > 1

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


I entropy of mixing
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
II. Energy of Mixing (1)

  • Assume ideal soln, then, Umix= 0 and Fmix= -TSmix

  • 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
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
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
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
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
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
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
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
IV. Chemical Potentials and Mixing

  • A = (F/NA)NB,T = kT ℓn xA + zwAA/2 + kTAB (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
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
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
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
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
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
Surface Area (6)

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


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