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(NA ℓn xA + NB ℓn xB) = -Nk (xA ℓn xA + xB ℓn 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 AB NANB/N

Energy of Mixing (5)Exchange Parameter

• U = (zwAA)NA/2 + (zwBB)NB/2 + kT AB NANB/N

AB = energy cost of replacing A in pure A with B; similarly for B.

AB = exchange parameter = - ℓn Kexch

Umix = RT AB xA xB

Energy of Mixing (6)Exchange Parameter

AB = exchange parameter = - ℓn Kexch

• A + B ↔ mixing eq. constant Kexch

Umix = RT AB xA 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.