The Behavior of Solutions

(1)

Chapter 9

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9.7 Nonideal Solutions

(9.46)

(9.47)

(3)

(4)

(5)

9.8 Application of the Gibbs-Duhem Relation to the Determination of Activity

(A) Method 1

• Binary A-B solution

• as

(9.48)

(9.50) (9.49)

(6)

(9.52)

(9.51)

9.8 Application of the Gibbs-Duhem Relation to the

Determination of Activity

(7)

(9.53)

(9.54)

(9.55) (B) Method 2

Determination of Activity

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(C) Method 3

(9.56)

(9.57) (9.58)

(9.59)

(9)

(9.61)

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Determination of Activity

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• Henry’s law

• Gibbs-Duhem eqatuion, Eq. (9.49), gives

• Integration gives

• By definition, a_i=1 when X_i=1, and thus the integration constant equals unity

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9.8 Application of the Gibbs-Duhem Relation to the

(14)

(9.62)

(9.63)

(D) Method 4

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(9.64) (9.65)

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9.8 Application of the Gibbs-Duhem Relation to the

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9.9 Regular Solutions

(9.66)

(9.54) (9.67)

(9.68)

(9.69)

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1.ΔH^M of R.S = G^XS

For any solution

For any ideal solution

For a regular solution, ΔS^M=ΔS^M,id and thus

(9.70) (9.71)

(9.73) (9.72)

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2. G

^XS

=ΔH

^M

≠ f(T)

For a regular solution, ln , and ln

(9.74a) (9.74b)

(9.75) (9.76)

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2. G

^XS

=ΔH

^M

≠ f(T)

It is thus seen that G

^XS

for a regular solution is independent of T

S

^XS

for a regular solution is zero,

then G

^XS

, and hence ΔH

^M

, are independent of T

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3. Application of RS

For a regular solution

Eq. (9.77) is of considerable practical use in

converting activity data for a regular solution at one temperature to activity data at another temperature

(9.77)

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9.9 Regular Solutions

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4. What can we use R.S?

where b and b’ are unequal

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9.10 A Statistical Model of Solutions

• Application of the statistical mixing model to two components which have equal molar volumes

• In solution the interatomic forces exist only between neighboring atoms

• The energy of the solution is the sum of the interatomic bond energies

• Consider 1 mole of a mixed crystal containing N_A atoms of A and N_B atoms of B

1. A-A bonds the energy of each of which is E_AA 2. B-B bonds the energy of each of which is E_BB 3. A-B bonds the energy of each of which is E_AB

N_o is Avogadro’s number

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9.10 A Statistical Model of Solutions

• The relative zero of energy to be that when the atoms are infinitely far apart, the bond energies E_AA, E_BB, and E_AB are negative quatities

• Let the coordination number of an atom in the crystal be z

• Similarly, for B, N_Bz = P_AB + 2P_BB or

(9.78)

(9.79)

(9.80)

(9.81)

<Quasi-Chemical Model>

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9.10 A Statistical Model of Solutions

• N

_A

atoms in pure A

• Thus

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• For the mixing process

• ΔV^M=0, then

• For given values of EAA, EBB and EAB, ΔH^M depends on PAB For the solution to be ideal, i.e., for ΔH^M = 0

• If |E_AB|>|1/2(E_AA+E_BB|), then, ΔH^M is a negative quantity negative deviations from Raoultian ideal behavior.

• If |E_AB|<|1/2(E_AA+E_BB|), then, ΔH^M is a positive quantity positive deviations from Raoultian ideal behavior

(9.82)

(9.83)

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9.10 A Statistical Model of Solutions

• If ΔH^M = 0, then the mixing of the N_A atoms with the N_B atoms of B is random  random mixing

• |ΔH^M|≤RT

The mixing of the atoms is also approximately random

• Consider two neighboring lattice sites in the crystal The probability that site 1 is occupied by an A atom is

• The probability that site 2 is occupied by a B atom is X_B

• Thus the probability of A-B pair is 2X_AX_B,A-A pair is X_A², B-B pair is X_B²

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9.10 A Statistical Model of Solutions

or

(9.84)

(9.85)

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• ΔH^M is a parabolic function of composition. As random mixing is assumed, the statistical model corresponds to the regular solution model, i.e.,

• Thus

(9.86) (9.87)

(9.88a) (9.88b)

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A Statistical Model of Solutions

• As the mixing is random, then

• Hence

• But

• The value of γ thus depends on the value of Ω, which, in turn, is determined by the values of the bond energies E_AA, E_BB, and E_AB If Ω is negative, then γA < 1, and if Ω is positive, then γA > 1

(9.89)

(9.90) (9.91)

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9.10 A Statistical Model of Solutions

• Henry’s law requires that γ

_A

, and hence lnγ

_A

, approach a constant value as X

_B

approaches unity

Thus,

• The applicability of the statistical model to real

solutions decreases as the magnitude of Ω increases,

i.e., if the magnitude of EAB is significantly greater or

less than the average of EAA and EBB then random

mixing of the A and B atoms cannot be assumed

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Figure 9.25 Illustration of the origins of deviation from regular solution behavior

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9.11 Subregular Solutions

(9.92) (9.93) (9.94a) (9.94b)

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9.10 Subregular Solutions

(9.95)

(9.96)

(9.97)