thermodynamics for solution
DESCRIPTION
Lecture note from NUS for thermodynamics of solutionsTRANSCRIPT
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.
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– 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.
_
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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
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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∆ = + +
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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∆ = + + +