Chapter 4 Solutions • What is a solution?

A solution in thermodynamics refers to a system with more than one chemical component that is mixed homogeneously at molecular level.

1

– Solutions in thermodynamics include mixtures in liquid, solid and gas states.

– Homogeneous (single phase), multicomponent

4.1 Definitions of important composition variables in solution

Molar Fraction with

Atomic Percentage at% = 100% Xi

Weight Fraction with

Concentration or sometimes

2

ii

tot

nXn

= tot ii

n n=∑

ii

tot

WwW

= tot ii

W W=∑

ii

nCV

= ii

WCV

=

3

• Extensive and intensive properties

A property of a system is intensive it may be defined to have a value at a point in the system.

For example, temperature, pressure, density

A property of a system is extensive if it is the system as whole.

For example, volume, H, G

Extensive properties can be expressed as integrals of intensive properties over the extent of system.

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Two questions about solutions 1. Can the components be mixed and form a homogeneous phase?

Oil/water Ethanol/water Salt (NaCl)/water

2. How about the properties of solutions

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4.2 Definition of Partial Molal Properties

The state function V’ is a function not only of T and P but also of the number of moles of each component in the system,

Thus,

1 2' '( , , , ,..., )cV V T P n n n=

2

1, , 1 , , ,...,

' ' ''k k k

P n T n T P n n

V V VdV dT dP dnT P n

∂ ∂ ∂ = + + ∂ ∂ ∂

1 3 1 2 1

22 , , , ,..., , , , ,...,

' '...k c

ccT P n n n T P n n n

V Vdn dnn n

−

∂ ∂+ + + ∂ ∂

or 1, , , ,

' ' ''k k j k

c

kkP n T n k T P n n

V V VdV dT dP dnT P n= ≠

∂ ∂ ∂ = + + ∂ ∂ ∂ ∑

The coefficient of each of the changes in number of moles can be written:

unit is volume/mole.

_

, ,

'

j k

k

k T P n n

VVn

≠

∂= ∂

k = 1, 2,…,c Partial molal volume

4.2.1 Definition

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• An analogous definition can be devised for any of the extensive properties of the system.

Using B’ for any of the properties U’, S’, V’, H’, G’. Then , for a change in temperature, pressure and chemical content, the change in the properties, B’, is

The partial molal B for component k is the corresponding coefficient of dnk

1'

c

k kk

dB MdT NdP B dn−

=

= + +∑

_

, ,

'

j k

k

k T P n n

BBn

≠

∂= ∂

k = 1, 2,…,c

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4.2.2 Chemical Potential

μk is referred as chemical potential.

It is the coefficient of each compositional variable.

• Putting energy together, the potential differentials

1 , ,

'' ' 'j

c

kk k T P n

GdG S dT V dP dnn=

∂= − + + ∂

∑_

, ,

'

j

k kk T P n

G Gn

µ ∂

= = ∂

_' ' ' i idV V dT V dP V dnα β= − +∑

_' ' ' ' 'i i i idG S dT V dP G dn S dT V dP dnµ= − + + = − + +∑ ∑

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4.2.3 Consequences of definition of partial molal properties Considering a process in which the temperature and pressure are held constant, the system is formed by adding n1 moles of component n1, n2 moles of component n2,…, until a final state consisting of a homogeneous mixture of all the components at the initial temperature and pressure is achieved. At any step during the process, we have

_

,1

'c

T P k kk

dV V dn=

=∑

This is the first consequence.

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For the process under consideration, visualize the addition of all c components simultaneously in the proportions found in the final mixture. Thus, during the process the intensive properties (T, P and the set of Xk values) remain fixed and each of the terms is constant. In this case, integration is straightforward.

_ _

0 01 1

' k kc cn n

k k k kk k

V V dn V dn= =

= =∑ ∑∫ ∫_

1'

c

k kk

V V n=

=∑

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• This conclusion, that the total volume for the system is the weighted sum of the partial molal volumes, can be extended without complication to any extensive property:

• Accordingly, the second consequence of the definition of partial molal properties is the most rudimentary requirement of any strategy for assigning a part of a total property to each of the components and that is that the sum of the contributions must add up to the whole.

_

1'

c

k kk

B B n=

=∑

1'

c

k kk

G n µ=

=∑• For the Gibbs free energy, we have

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Beginning compute the differential of B’,

Since the differential of the sum of the differentials, differentiating the product yields

_

1' ( )

c

k kk

dB d B n=

=∑

_ _

1'

c

k k k kk

dB B dn n d B=

= + ∑

_ _

1 1'

c c

k k k kk k

dB B dn n d B= =

= +∑ ∑

• The third consequence is the Gibbs-Duhem equation.

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• The first summation is equal to the left side of the equation.

_ _

1 1'

c c

k k k kk k

dB B dn n d B= =

= +∑ ∑

Gibbs – Duhem equation

_

1'

c

k kk

dB B dn=

=∑• Accordingly, the second summation must be zero.

_

10

c

k kk

n d B=

=∑This equation demonstrates that the partial molal properties are not all independent.

For example, for a two-component solution, _ _

1 1 2 2 0n d V n d V+ =

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4.3 Molar quantities

• A molar quantity refers to an extensive thermodynamic variable that has been divided by the total number of moles in the system.

For example, molar volume

• It is frequently useful to normalize the description of properties of mixtures and express them on the basis of one mole of the solution formed,

_ '

tot

BBn

=

_ '

tot

VVn

=

_

_ _1

1

'

c

k k ck

k kktot tot

B dndBd B B dXn n

=

=

= = =∑

∑_ _

1

c

k kk

B X B=

=∑Thus,

4.3.1 Definition

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4.3.2 Relation between partial properties and total properties

as

then

We have

_ _

1

c

k kk

B X B=

=∑Two-component system

_ _ _

A A B BB B X B X= +

_ _

1

c

k kk

d B B dX=

=∑

_ _ _

A A B Bd B B dX B dX= +

1A BX X+ =

A BdX dX= −

__ _

B AB

d B B BdX

= −

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__ _

B AB

d B B BdX

= −

_ _ _

A A B BB B X B X= +

_ __ _ _

B A AB A

d B d BB B BdX dX

= + = −

__ _ _

(1 )A A A AA

d BB B X B XdX

= + − −

_ __ _ _

A A A A AA A

d B d BB X B X BdX dX

= + − − −

_ __

( 1 )A A A AA A

d B d BB X X XdX dX

= + − − +

__

(1 )A AA

d BB XdX

= − −

_ __ _ _

(1 )A A BA B

d B d BB B X B XdX dX

= + − = −

__ _

(1 )B BB

d BB B XdX

= + −Hence

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Related to real systems, we have _ _

_ _ _

(1 )A A BA B

d V d VV V X V XdX dX

= + − = −

_ __ _

(1 )A A BA B

d G d GG X G XdX dX

µ = + − = −

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4.3.3 Graphic interpretation

_ __ _

(1 )A A BA B

d G d GG X G XdX dX

µ = + − = −_

_

A BB

d GG XdX

µ = −

__

(1 )B BB

d GG XdX

µ = + −

__

(1 )B B B B BB

d GX G X X XdX

µ = + −

__

A A A A BB

d GX G X X XdX

µ = −

__

B A BB

d GG X X XdX

= +

_

A A B BG X Xµ µ= +Thus,

a. Free Energy

_

( )BG X

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• For example, the partial molar volumes can be obtained from the molar volume

This figure illustrates the intercept rule as applied to the molar volume of a binary A-B solution. It also graphically illustrates the different terms that appear in the expressions for the molar volumes in equations presented above.

__ _

A BB

d VV V XdX

= −

__ _

(1 )B BB

d VV V XdX

= + −

b. Volume

_

V

_

( )BV X_

( )BV X

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The importance to notice the different way of chemical potentials are obtained.

, ,

'

j i

ii T P n n

Gn

µ≠

∂= ∂

__

A BB

d GG XdX

µ = −

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4.4 Mixing Process

nA moles of pure A is mixed with nB moles of pure B. Before mixing, the combined volume of the two component is simply

where and are the molar volumes of pure A and B, respectively.

_ _0 0

A A B Bn V n V+

_

V

4.4.1 Volume change

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Dividing this premixing volume by ntot = nA + nB, gives the molar volume before mixing,

This represents the dashed line.

_ _0 0 _ _

0 0A A B BA A B B

A B

n V n V X V X Vn n+

= ++

_

V

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• When A and B are mixed, forming a solution, the change in volume upon mixing can be written as

Where is the molar volume after mixing.

Similarly we can have:

_ __ _0 0( ) ( )mix B A A B BV V X X V X V∆ = − +

_ __ _0 0( ) ( )mix B A A B BH H X X H X H∆ = − +

_ __ _0 0( ) ( )mix B A A B BG G X X G X G∆ = − +

4.4.2 Change in other functions

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At concentration of XB, • when ∆Hmix is negative, heat is released, the mixing is exothermic. •when ∆Hmix is positive, heat is absorbed, the mixing is endothermic.

exothermic endothermic XB

is equal to the heat exchange with environment upon mixing a total one mole of pure component A and B.

_

mixH∆

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The value of is an important quantity as its sign determines whether mixing will occur or not.

• A negative Gibbs free energy of mixing means that there is a thermodynamic driving force for mixing and the pure components when brought in contact will spontaneously form a solution. • A positive Gibbs free energy of mixing means that the components are immiscible and will not form a solution when brought together, but rather a two phase dispersion of a pure A phase mixed with a pure B phase.

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It is possible to apply the intercept rule to a plot of the molar Gibbs free energy of mixing.

__

A BB

d GG XdX

µ = −

_ _ __0 0( )mixA A B B

B B

d X G X G GdGdX dX

+ + ∆=

Start from

_ _ _ _0 0 0( )mixA B A B B

B

d G X G X G GdX

− + + ∆=

__ _

0 0 mixA B

B

d GG GdX∆

= − + +

__ _ _ __

0 0 0 0( )mixmixA A A B B B A B

B

d GX G X G G X G GdX

µ ∆= + + ∆ − − + +

__ _

0 mixmixA A B

B

d GG G XdX

µ ∆= + ∆ −Thus,

__ _

0 mixmixA A B

B

d GG G XdX

µ ∆− = ∆ −or

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4.4.3 Activity of component k

• It is defined as

Where µko is the chemical potential of k in its reference state.

Where ak is the activity of k in its reference state.

• Another convenient measure of solution behavior, is the activity coefficient of component k, γk

0 lnk k k kRT aµ µ µ− = ∆ =

k k ka Xγ=

Note: when µk = µk0, ak =1

Note: when ak = Xk, γk = 1

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4.5 Examples of solutions 4.5.1 Ideal solution

k ka X=_

ln lnk k k kG RT a RT Xµ∆ = ∆ = =

_ _

1 1ln

c c

mix kk k kk k

G X G RT X X= =

∆ = ∆ =∑ ∑

1 , ,

'' ' 'j

c

kk k T P n

GdG S dT V dP dnn=

∂= − + + ∂

∑,

''kP n

GST

∂ − = ∂

__

, ,, , , ,, , , ,

' ' '

k kj jj k k

k kk

P n P nk k kT P n T P nT P n P n P n

S G G GSn n T T n T T

µ ∂∂ ∂ ∂ ∂ ∂ ∂ = = − = − = − = − ∂ ∂ ∂ ∂ ∂ ∂ ∂

( ), ,

lnln

k k

kkk k

P n P n

RT XS R X

T Tµ ∂ ∂∆ ∆ = − = − = − ∂ ∂

_ _

1 1ln

c c

mix kk k kk k

S X S R X X= =

∆ = ∆ = −∑ ∑

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Mixtures obeying these relations, regardless of solids, liquid, or gas, are in general called ideal solutions.

29

30

4.5.2 Regular solution

For a regular solution,

1. The entropy of mixing is the same as that for an ideal solution:

2. The enthalpy of solution is not zero, but is a function of composition

_

lnk kS R X∆ = −

_

1 2( , ,...)k kH H X X∆ = ∆

The simplest regular solution model contains a single adjustable parameter in its description of heat of mixing,

Thus 1 2 1 1 2 2( ln ln )mixG aX X RT X X X X∆ = + +

31

32

4.5.3 A two parameter regular solution model

1 2 1 1 2 2 1 1 2 2( ) ( ln ln )mixG X X a X a X RT X X X X∆ = + + +