42C.1

Non-Ideal SolutionsThis development is patterned after that found in Molecular Themodynamics by D. A. McQuarrie and John D. Simon. Consider a molecular model of a regular binary solution in which similarly sized solute and solvent molecules are randomly distributed throughout the solution. Let the potential energy of this solution be given by the expression:

E = N11 11 + N22 22 + N12 12

Here N12 is the number of neighboring pairs of solvent and solute molecules and 12 is the energy of interaction between a solute and solvent molecule and is negative (why?).

What do N11 and e11 represent?

Since the molecules are randomly distributed , the probability that two solvent molecules are neighbors is given by:

(N1 - 1) / (N1 + N2 - 1) ~ N1 / (N1 + N2) = X1

or the mole fraction of solvent molecules in the solution. If z is the coordination number, i.e., the number of nearest neighbors for a given molecule, then on average a given solvent molecule will have:

X1 z

solvent molecules as nearest neighbors and, since there are N1 solvent molecules in the solution, the number of solvent - solvent neighboring pairs will be:

N11 = N1 X1 z / 2

Dividing by two prevents us from counting each solvent - solvent interaction twice.

42C.2

The total energy of interaction between all the solute and solvent molecules is thus:

E = (N1 z X1 / 2 ) 11 + (N2 z X2 / 2 ) 22 + (N2 z X1) 12

Could you justify the terms in this expression?

Here the coordination number has been assumed to be the same for both solute and solvent molecules, since we have assumed the solvent and solute molecules to be of similar size. Using the definitions of the mole fractions we can write:

E = [ (N12 z / 2 ) 11 + (N2

2 z / 2 ) 22 + (N1 N2 z) 12 ] / (N1 + N2)

We now define as a measure of non-ideality as:

= 2 12 - 11 - 22

What is for an ideal solution?

Using to eliminate 12 from the expression for E gives:

E = (N1 z / 2 ) 11 + (N2 z / 2 ) 22 + N1 N2 z / [2 (N1 + N2)]

Could you derive this result?

Since = 0 for an ideal solution, the 1st two terms represent the contribution to the total energy of interaction for an ideal solution and the last term represents the contribution from non-ideality:

E = Eideal + ( N1 N2 z / [2 (N1 + N2)] )

Sustituting into the definition of the Gibb’s free energy we have:

G = H - T S = E + P V - T S

= Eideal + P V - T S + ( N1 N2 z / [2 (N1 + N2)] )

42C.3

Since the solute and solvent molecules were assumed to be randomly distributed and of similar size, V and S are the same as for an ideal solution and we can write:

G = Gideal + ( N1 N2 z / [2 (N1 + N2)] )

We can convert the numbers of molecules in this expression to moles by dividing by avogadros number No:

G = Gideal + ( [n1 n2 z No / [2 (n1 + n2)] )

The chemical potential of the solvent is defined as:

u1 = ( G / n1 )T, P, n2 = ( Gideal / n1 )T, P, n2

+ [ n2 / (n1 + n2) - n1 n2 / (n1 + n2)2 ] z No / 2

= u1, ideal + ( X2 - X1 X2 ) z No / 2

= u1, ideal + X22 z No / 2

Can you justify these last few steps?

Using the definition of the chemical potential for an ideal solution:

u1, ideal = u1pure + R T ln X1

we can write for the chemical potential for our non-ideal solution:

u1 = u1pure + R T ln X1 + X2

2 z No / 2

= u1pure + R T ln ( X1 e + X2

2 z No / ( 2 R T ) )

Can you justify these steps?

42C.4

Comparing this expression with the expression for the chemical potential of the solvent in a non-ideal solution derived in the notes on Partial Molar Variables, Chemical Potential, Fugacities, Activities, and Standard States:

u1 = u1pure + R T ln a1

identifies the activity of the solvent in this non-ideal solution as:

a1 = f1 / f1o = f1 / f1

pure

= X1 e + X22 z No / ( 2 R T ) = X1 e + X2

2 / ( R T )

where = z No / 2.

Could you reproduce these derivations for the solute or for a multi-component solution?

This equation will predict how the fugacity of the solvent above a non-ideal solution will vary with mole fraction.

Under what conditions will this equation reproduce Raoult’s law?

Fugacity of Solute in a Non-Ideal Solution

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

mole fraction solute, X2

f2 /

f2 (p

ure)

postive deviation from Raoult's law, w > 0

Raoult's law, w = 0

42C.5

What does a positive deviation from Raoult’s law imply about the relative interaction energies, 11, 22, and 12?

How would a curve representing a negative deviation from Raoult’s law appear on this plot?

The molar Gibb’s free energy of mixing to form a non-ideal solution is given by:

Gmix = RT [ X1 ln a1 + X2 ln a2 ]

= R T [ X1 ln 1 X1 + X2 ln 2 X2 ]

= R T [ X1 ln X1 + X2 ln X2 + X1 ln 1 + X2 ln 2 ]

i is the activity coefficient of species i, which in an ideal solution is equal to one, hence:

Gmix, ideal = R T [ X1 ln X1 + X2 ln X2 ]

The excess Gibb’s free energy of mixing per mole of solution is defined as the difference between the Gibb’s free energy of mixing of a non-ideal solution and that of an ideal solution:

Gmix, excess = Gmix - G mix, ideal = R T [ X1 ln 1 + X2 ln 2 ]

Using the result we just derived

a1 = 1 X1 = X1 e + X22 / ( R T )

we can write for our regular solution:

Gmix, excess = [ X1 X22 + X2 X1

2 ] = X1 X2

Can you justify this derivation?

42C.6

In our regular solution in which the solute and solvent molecules are randomly distributed we have assumed that the entropy of mixing to form and ideal solution is the same as the entropy of mixing to form a non-ideal solution. Derive an expression for the excess enthalpy of mixing, Hmix, excess. What is Hmix, excess if the solution is ideal?

It is instructive to plot:

Gmix / = G mix, ideal + G mix, excess ) /

= ( R T / ) [ X1 ln X1 + X2 ln X2 ] + X1 X2

versus the mole fraction of one of the components, say X2 for different values of (R T / ):

Note that for either a temperature low enough or a deviation of interaction energies from idealilty high enough, Gmix shows two minima indicating that the solution has separated into two phases.

Gmix / vesus Mole Fraction of the Solute in a Regular Solution

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.000.00 0.20 0.40 0.60 0.80 1.00

mole fraction solute, X2

Gm

ix /

R T / = 0.40

R T / = 0.50

R T / = 0.60X2 = 0.14 X2 = 0.83

42C.7

Plot R T / versus the mole fractions of the solute corresponding to the minima in the plots of Gmix / versus mole fraction of solute to give the liquid - liquid phase diagram shown below for a regular solution:

Liquid-Liquid Phase Diagram for a Regular Solutiion

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

mole fraction solute, X2

R T

/

2 phase region

associated with an upper consulate temperature, above which thermal motion prevents phase separation

1 phase region

X2 = 0.14 X2 = 0.83

Note that for a given solute and solvent in a non-ideal regular solution there will be an upper consulate temperature above which thermal motion will prevent phase separation.