introduction to the thermodynamics of materials-2014-chapter 1

8/21/2019 Introduction to the Thermodynamics of Materials-2014-Chapter 1

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IntroductionIntroduction ToTo TheTheThermodynamicsThermodynamics of Materialsof Materials

Dr.Dr. MingxiaMingxia GaoGao （高明霞（高明霞）

Department of Materials Science & EngineeringDepartment of Materials Science & Engineering

Zhejiang UniversityZhejiang University

EE--mail:mail: [email protected]@zju.edu.cn

Tel.: 0571Tel.: 0571--8795261587952615

Address Address：曹光彪楼：曹光彪楼419419--22



PrefacePrefaceCompositionComposition

StructureStructure PropertiesProperties

The central importance is to predominate the relations among them.

Substances Materials Apparatus

Characteristics Properties Functions

Any material is synthesized by “mixing” substances.

Thermodynamics of materials usually deals with the equilibriumprocess, and gives a right answer of what phases (solutions orchemical compounds) will form in the final state, i.e. theequilibrium state. It describes the basic principle between thecomposition and the structure.

j ji j j i

P C = =∑ ∑∑(j = α, β, γ …); (i= 1, 2, 3…)

synthesis



Thermodynamics of materials :

a. It evolves systematically a method of strictly describing the

behavior of matter in a manner which is devoid of temporal

theories. Use simple tool to solve real problem. (simplicity)

b. It is initial conceptual difficult and there are large numbers of

equations. (complicacy )

c. It is a subject between physical principal and practical

application.



TThe main contents of this course include five parts:he main contents of this course include five parts:

1.The behavior of solutions1.The behavior of solutions

((RaoultRaoult’ ’ ss law and Henrylaw and Henry’ ’ ss law; Thermodynamic Activity; Partial; Thermodynamic Activity; Partial

Molar QuantitiesMolar Quantities；；Relative Partial Molar QuantitiesRelative Partial Molar Quantities；；PropertiesProperties

of Ideal Solution; Gibbsof Ideal Solution; Gibbs--DuhemDuhem Equation and its application;Equation and its application;

Gibbs Free Energy ofGibbs Free Energy of formation;; NonidealNonideal Solutions; RegularSolutions; Regular

solutions; A Statistical Model of Solutions;solutions; A Statistical Model of Solutions; Subregular Subregular Solutions)Solutions)



2. Gibbs free energy-composition relations and phase

diagrams of binary system

(Gibbs Free Energy and Activity; the Gibbs Free Energy of

Formation of Regular Solutions; Criterion of Phase Stability of

Regular Solutions; Spinodal Separation; Liquid and SolidStandard States; the Curve of the Molar Gibbs Free Energy of

Formation vs. the Composition; Immiscible Binary System －

Phase Diagram; Binary Eutectic Systems ))



3. Reactions involving gases3. Reactions involving gases

(Reaction Equilibrium; The Effect of Temperature and Pressure;(Reaction Equilibrium; The Effect of Temperature and Pressure;

Reaction Equilibrium as a Compromise between Enthalpy andReaction Equilibrium as a Compromise between Enthalpy and

Entropy; TheEntropy; The Application in Scientific Research))

4. Reactions involving pure condensed phases and

gaseous phase(Reaction Equilibrium in a System Containing Pure Condensed

Phases and a Gas Phase; The Application in Scientific Research)



This course is the prelude of the study andThis course is the prelude of the study and

utilization of theutilization of the Thermodynamics of MaterialsThermodynamics of Materials..

The main purpose of the course is to introduce and

demonstrate some basic but important principles and theirapplicability, thus to evolve us a capacity of fully utilizing the

standard treatises and to a continuity of the development and the

application of the principles.

5. Reaction equilibrium in systems containing

components in condensed solution

Reaction Equilibrium Criteria; Alternative Standard States;

Application in Scientific Research.



Chapter 1Chapter 1

The behavior of solutionsThe behavior of solutions

The concept of solution in terms of the

thermodynamics of materials :

All kinds of mixtures composed of two or more than two

components that are homogenous in structure and

composition throughout, including gas, liquid and the solid

solution of single phase .



1.11.1 RaoultRaoult ’’ss law and Henrylaw and Henry’’s laws law

a quantity of pure species A (or B )condensed stateevaporatesaturated vapor pressureevaporation ratecondensation rate

A small quantity of liquid A adds to liquid B .

Condition 1:

The atomic diameters of A and B are comparable and assuming

the surface composition of the liquid to be the same as the bulk

liquid composition, and the A-A , A-B and B-B bond energies in

the solution are identical.

0( ) ( )c A A er A K p r A= ⋅ =



Raoult’s Law:0

0

A A A

B B B

p p X

p p X

= ⋅

= ⋅

Raoult’s law states that the vapor pressure exerted by a

component i in a solution is equal to the product of the molefraction of i in the solution and the vapor pressure of pure i at the

same temperature of the solution.

A p

( )e A A A Ar X K p⋅ = ⋅

0

AP

0

A p

What’s the vapor pressureexerted by A or B ?

r c= re0( ) ( )c A A er A K p r A= ⋅ =



The above discussion is based on that the intrinsic evaporation

rate of A and B , r e(A) and r e(B), are independent of the

composition of the solution. This requires that the magnitudes

of the A-A , B-B and A-B bond energies (interactions) are

identical, such that the depth of the potential energy well of an

atom at the liquid surface is independent of the types of atoms

which it has as nearest neighbors.



A A--BB bond energybond energy is considerably much negativeis considerably much negative thanthan A A-- A A oror

BB--BB bond energybond energy , (or to say that the, (or to say that the A A--BB interaction is stronger),interaction is stronger),

and assuming that the solution ofand assuming that the solution of A A inin BB is sufficientlyis sufficiently dilutedilute, that, that

everyevery A A atom at the surface of the liquid is surrounded only byatom at the surface of the liquid is surrounded only by BB

atoms.atoms.

( ) '( )e e

r A r A⇒

Henry’s Law:'

A A A p K X = ⋅

'

( ) 0

( )

e A

A A A

e A

r p p X r

=

r e (A) > r ’e (A)

negative deviation from

Raoultian behavior

Condition 2:Condition 2:

( )'e A A A Ar X K p⋅ = ⋅



With the increase of X A , the probability that all the A atoms at

the surface of the liquid are surrounded only by B decreases,and there are also some A-A atom pairs. So, the value of r ’e(A)

increases (not a constant). Beyond a critical value of X A , r ’e(A)

becomes composition- dependent. Then, is no

longer obeyed by A in the solution.

is obeyed only over an initial concentration

range of A in B . The extent is dependent on the temperature of

the solution and on the relative magnitudes of A-A , B-B and A-B

interactions.

'

A A A p K X = ⋅

' A A A p K X = ⋅



A A--BB attractionattraction is significantlyis significantly weaker thanweaker than A A-- A A oror BB--BB attractionattraction

re(A) < r’e (A)

positive deviation from Raoultian

behavior .

B

B

B

The components of a solution that Raoult’s law is obeyed are

said to exhibit Raoultian behavior.In the composition ranges where Henry’s law is obeyed, the

solutions are said to exhibit Henrian behavior .



1.21.2 The thermodynamic activity of a component inThe thermodynamic activity of a component in

a solutiona solution

(1) concept(1) concept

Three terms (2) the apparent reasonThree terms (2) the apparent reason

(3) application(3) application

For an ideal gas:For an ideal gas:0 lnG G RT p= +

For every component i of a mixture of gases: 0 ln

i i iu u RT p= +

For a solution obeys Raoult’s law: iii X p p ⋅= 0

Then:0 0 *ln ln ln

i i i i i iu u RT p RT X u RT X = + + = +



But the practical solution normally deviate from theBut the practical solution normally deviate from the RauotRauot’’ss Law ,Law ,

thus, thermodynamic activity is introduced instead of the molethus, thermodynamic activity is introduced instead of the mole

fractionfraction X X ii ::

iii

X a γ =iiiiii

X RT ua RT uu γ lnln 00 +=+=

With respect to a non-ideal gas : f RT C G ln+=

For a practical mixture of gases:iii f RT k u ln+=



The thermodynamic activity ( ) of a component in any state

at T is formally defined as the ratio of the fugacity of the

substance in that state to its fugacity in its standard state, i.e.,

for the species or substance i.

ia

0

i

ii

f

f a =

If the vapor above the solution is ideal:

ii p f =

0

i

ii

p pa =



Activities in Fe-Cr at1600℃, Raoultian

behavior

Activity of Ni in Fe-Ni at 1600 ℃

over the composition range in

which Henry’s law is obeyed bythe solution i :

i i ia k X =

ii X a =



1.3 Partial molar quantities

(1)The reason of the appearance of partial molar quantity

In order to measure and express the variations of the

thermodynamic properties of a solution when the concentration

of component i of the solution varies.

(2) The physical signification of partial molar quantity :

Normally, partial molar quantity is defined as the variation of

the thermodynamic property of a large solution produced by the

addition of 1 mole of component i at constant P, T and constant

amounts of all the other components in the system.



At constant T and P, the variation of with the variation of

the solution composition:

To suppose that is anTo suppose that is an extensive thermodynamic propertyextensive thermodynamic property of aof a

solution, a capacity property (solution, a capacity property (VV,,HH,,SS,,G,G,etcetc.). The value.). The value ofof ::

Q

Q

'Q

,......),,,,('' k ji nnnPT QQ =

K+⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂

+⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂

+⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂

= k

nnPT k

j

nnPT j

i

nnPT i

dnn

Qdn

n

Qdn

n

QdQ

jik ik j ,...,,,,...,,,,...,,,

''''

'Q

The partial molar quantity of of component i is defined as :

, , , ,...

'

j k

i

i T P n n

QQ

n

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

is the increase of the value of for

a mixture or solution when 1 mole of iis added to the solution of large

quantity at constant T and P .

iQ 'Q

Q

(Intensive property:(Intensive property: T,PT,P))



(3) Stipulation :

The stipulation is that the quantity is large enough so that theaddition of 1 mole of to the solution should not cause a

measurable change in the composition of the solution.

The partial molar quantity can be defined in another case, that is,

the variation of the property of a definite solution caused by adding

mole of i, which is taken as . Because of the small

quantity of , the composition of the solution is also considered

to be no change. Thus,

idn 'idQ

idn

'i

i

i

dQQ

dn= is also taken as the partial molar quantity of Q

of component i.



(4)(4) The extensive thermodynamic propertyThe extensive thermodynamic property can becan be ::

V V (the molar volume of a solution),(the molar volume of a solution), HH (enthalpy),(enthalpy), SS (entropy),(entropy), GG

(Gibbs free energy)(Gibbs free energy) , etc.

For example, the partial molar enthalpy of component i can beexpressed as:

, , j i

ii

i T P n n

H H

n≠

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

The partial molar Gibbs free energy of i is equal to its chemical

potential : i iG u=

So,for an ideal solution:

For a practical solution:

iii X RT uu ln0 += 0 lni i iG G RT X = +

iii a RT uu ln

0 += 0 lni i iG G RT a= +



There are some intrinsic relations between/among the different

partial molar quantities.

The general thermodynamic relationships between/among the

state properties of a system are applicable to the partial molar

properties of the components of a system. Such as:

ii iG H T S = −

ii

i P

GS

T

⎛ ⎞∂= −⎜ ⎟

∂⎝ ⎠i

i

T

GV

P

⎛ ⎞∂=⎜ ⎟

∂⎝ ⎠

( / )

(1/ )

ii

P

G T H

T

⎛ ⎞∂=⎜ ⎟

∂⎝ ⎠



1.41.4 GibbsGibbs--DuhemDuhem equation*equation*

K+⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

∂

∂+

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

∂

∂+

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

∂

∂=

k nnPT k

j

nnPT j

innPT i

dnn

Qdn

n

Qdn

n

QdQ

jik ik j ,...,,,,...,,,,...,,,

'''

'

,...,,,

'

k j nnPT i

in

QQ ⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂

=

If is the value of per mole of i in the solution, then the

value of for the solution (composed of several moles) itself is :

iQ Q

'i j k i j k

Q n Q n Q n Q= + + +L

'

i j k i j k

i j k i j k

dQ n dQ n dQ n dQ

Q dn Q dn Q dn

= + + +

+ + + +

L

L

d i f f e r e n t i a t i o n

' i j k i j k dQ Q dn Q dn Q dn= + + +L

Q



Thus, 0=+++ Lk k j jii Qd nQd nQd n

or 0=∑i

ii Qd n

The expressions of Gibbs-Duhem equation0=∑

i

ii Qd X

d i v

i d e d b y n

For a binary solution :

0=+ B B A A Qd X Qd X



Gibbs-Duhem equation describes the relationship among

the values of the thermodynamic properties of thecomponents and the composition of the solution.

Application:

If only one component (or some components) in a binary (or

multi-component) solution can be experimentally measured, thecorresponding property (properties) of the other (other

components) can be obtained by the Gibbs-Duhem equation.



1.5 Relative partial molar quantity

M oii iQ Q QΔ = −

0

iQ ：the partial molar quantity of pure i ，the molar

quantity.

Since correlates to the composition of a solution,

thus is also a function of composition.

he difference between and is defined as the

relative partial molar quantity, designated as .

iQ M

iQΔ

iQ

0

iQ M

iQΔ



For example:

M M M

ii iG H T S Δ = Δ − Δ

For a binary solution, the Gibbs-Duhem equation in the form of

can be expressed as : M

iQΔ0

M M

A B A B X d Q X d QΔ + Δ =

For enthalpy H : 0=Δ+Δ M B B

M A A H d X H d X

All of the thermodynamic relations that are applicable to the

partial molar quantities are also applicable to the relative partial

molar quantities.



1.6. The Gibbs free energy and other extensive

thermodynamic properties of a solution1.6.1 The molar Gibbs free energy of a solution and thepartial molar Gibbs free energies of the components of

the solution

For a binary solution A-B composed of n A moles of A and n B moles

of B , at fixed T and P , the value of the Gibbs free energy of thesolution, G’, is:

B B

A A

GnGnG +='



For 1 mole of solution, X A+ X B=1:

B B A A G X G X G +=

A B A B A B A BdG X dG X dG G dX G dX = + + +

D i f f e r e n t i a t i o n

= 0 (Gibbs-Duhem equation)

Since X A+ X B=1, dX A= - dX B,

B A

A

GGdX

dG−=Then,

B B A B

A

B G X G X dX

dG X −=

m u l t i p l y i n g b y X B

)( A B A

A

B X X G

dX

dG X G +=+

= 1



So,

A

B A

dX

dG X GG += Similarly,

B

A B

dX

dG X GG +=

1.6.2 The changes in Gibbs free energy andother thermodynamic properties due to theformation of a solution

(1) The difference in molar Gibbs free energy of a component i

between pure i and i in a solution at T :

It can be obtained from three steps:

These expressions relate the dependence on composition of thepartial molar Gibbs free energies of the components of a binary

solution and the molar Gibbs free energy of the solution.



(a)Evaporation of 1 mole of pure condensed i to vapor i at the

pressure at T . (equilibrium process, )

(b)Decrease in the pressure of 1 mole of vapor i from to at T .

(c) Condensation of 1 mole of vapor i from the pressure to the

condensed solution at T . (equilibrium process, )

0

i p

i p

)(aGΔ

)(cGΔ

0

i p

i p

Steps (a) and (c) are processes conducted at equilibrium, thus the

Gibbs free energies changed in the two processes are both zero.



The difference between and is the Gibbs free

energy change which accompanies the solution of 1 mole of i in

the solution.

)( solutioniniG − )( pureiG

i

i

ib pureisolutionini a RT

p

p RT GGG lnln

0)()()( =⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =Δ=−−

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ =Δ

0)( ln

i

ib

p

p RT G ln

RT dG Vdp dp RTd P

P= = =

The overall change in Gibbs free energy for the three-step

process equals that for only (b):

Here, = ;)( solutioniniG − iG )( pureiG 0

iG=



i

M

iii a RT GGG ln0 =Δ=−

This difference was designated as . This quantity is just the

relative partial molar Gibbs free energy of component i. It is also

called as the partial molar Gibbs free energy of mixing (solution)of i.

M

iGΔ

If, at constant T and P , n A moles of A and n B moles of B are

mixed， forming a binary solution, then

The Gibbs free energy before mixing= 00

B B A A GnGn +

The Gibbs free energy after mixing = B B A A GnGn +

The Gibbs free energy change due to mixing = M G'Δ



)()()()(' 0000

B B B A A A B B A A B B A A

M GGnGGnGnGnGnGnG −+−=+−+=Δ

M B B

M A A

M GnGnG Δ+Δ=Δ ' or )lnln(' B B A A M anan RT G +=Δ

In terms of 1 mole of binary A-B solution,

M

B B

M

A A

M G X G X G Δ+Δ=Δ )lnln( B B A A

M a X a X RT G +=Δor

M

GΔ : the molar Gibbs free energy change due to themixing of the components to form a solution.

The above two equations give the molar Gibbs free energy

change due to the mixing of the components forming a solution

with a given composition.

A typical form of the variation of with composition of a M

GΔ



A typical form of the variation of with composition of a

binary solution.

GΔ

The change of other molar properties due to the formation of a

solution, such as and , can also be obtained by the

similar method .

M H Δ

M S Δ

For any extensive thermodynamic property in a binary solution



For any extensive thermodynamic property in a binary solution,

M

B B

M

A A

M

Q X Q X Q Δ+Δ=Δ

M H Δ

M S Δ

M

ii

M

ii

M

B B

M

A A

M Q X Q X Q X Q X Q

∑ Δ=Δ++Δ+Δ=Δ LL

:the enthalpy change due to mixing X A moles of pure A and X B moles of pure B forming 1 mole of solution,

which is also called as the molar enthalpy (change) of

mixing of the solution.

In an A-B binary solution:

:the entropy change due to mixing X A moles of pure A and

X B moles of pure B forming 1 mole of solution, which is

also called as the molar entropy (change) of mixing of thesolution.

M



M GΔ : is the criterion to estimate whether the formation ofthe solution will spontaneously go along or not.

M GΔ

M GΔ M

GΔ

<0; the formation goes along

>0; solution cannot be formed

=0; equilibrium

M M M S T H G Δ−Δ=Δ M

P

M

S T

GΔ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂Δ∂

also fits the relations interrelated the extensivethermodynamic properties, , of the solution, such as:

M QΔ

Q

1.6.3 The method of tangential intercepts

A

B A

dX

dG X GG +=

B

A B

dX

dG X GG +=

The relation among the partial molar Gibbs free energy of a



g p gycomponent, the molar Gibbs free energy of the solution and the

composition is also applicable to the relative partial molar Gibbsfree energy and the molar Gibbs free energy of formation of thesolution. That is:

A

M

B

M M A

dX Gd X GG Δ+Δ=Δ

B

M

A M M

B

dX Gd X GG Δ+Δ=Δ

the slope of the tangent to thecurve at X A= X A = rs/rq

osrs pq

rq

rsrq pqG

M

A =+=+=Δ

OS =the tangential intercept at X A =1

=the tangential intercept at X B =1

M

BGΔ

r

sq

P

A B

The application of tangential intercepts:



The application of tangential intercepts:

It can be used to obtain the partial molar and relative partial molar

values of an extensive property of the components of a binary

solution from the variation of the integral values of the property

with the composition.

For any extensive thermodynamic properties :

A

M

B

M M

A dX

Qd

X QQ

Δ

+Δ=Δ , Q can be G, H, S, etc.

To obtain the relative partial molar quantities ( , , )，

one can be by the calculation from the above equation, and the

other can be by the graphical tangential intercept method.

M

iGΔ M

i H Δ M

iS Δ

,

1 7 The properties of Raoultian ideal solution



(2) Volume change of the formation of an ideal solution

For a species i occurring in a solution,

and for pure i :

ii

T

GV

p

⎛ ⎞∂=⎜ ⎟

∂⎝ ⎠0

0ii

T

GV

p

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

Thus, ( )0

0

,

i i

i i

T comp

G GV V

p

⎛ ⎞∂ −⎜ ⎟ = −⎜ ⎟∂⎝ ⎠

or,

M M ii

T comp

GV

p

⎛ ⎞∂Δ⎜ ⎟ = Δ⎜ ⎟∂⎝ ⎠

1.7 The properties of Raoultian ideal solution

(1) Activity

ii X a = X i is independent of T .

（partial derivation / derivative）

,

ln M id

i iG RT X Δ =For an ideal solution,



i

,

0

M id

iV Δ =

,

as X i is not a function of pressure, then:( ) ( ) ( ) ( )' 0 0 0 0

M M M

A B A B A B A B A A B B A A B B A BV n V n V n V n V n V V n V V n V n V Δ = + − + = − + − = Δ + ΔSince

Thus ,,

0

M id

V Δ =

,'

0

M id

V Δ =The volume of an ideal solution is equal to the sum of thevolumes of the starting pure components of the solution.

0

AV V = 0

B BV V =

The molar volume of an ideal

solution is a linear function of

the composition.

(3) Enthalp of fo mation of an ideal sol tion



(3) Enthalpy of formation of an ideal solution

From the Gibbs-Helmholtz equation:

) 2

,

ii

p comp

G T H

T T

⎡ ⎤∂

⎢ ⎥ = −∂⎢ ⎥⎣ ⎦

( )0 0

2

,

i i

p comp

G T H

T T

⎡ ⎤∂

⎢ ⎥ = −∂⎢ ⎥⎣ ⎦

for component i in solutionfor pure component i

（subtraction）

h l f h G bb l h l



This equation is applicable in a closed system at theprocess of constant P .

The complementarity for the Gibbs-Helmholtz equation:

p

GG H TS H T

T

∂⎛ ⎞= − = + ⎜ ⎟∂⎝ ⎠GdT HdT TdG= +

Division of the two sides of the equation by T 2, gives:

2 2

TdG GdT HdT

T T

−= −

2

( / )d G T H

dT T = − 2

( / )d G T H

dT T

Δ Δ= −

( )2( / ) ( ) /d x y ydx xdy y= −

2

1( )

dT d

T T

= −Since , thus ( )/

(1/ )

G T H

T

∂ Δ= Δ

∂

0

i iG G⎡ ⎤⎛ ⎞−

∂⎢ ⎥⎜ ⎟ ( ) M M

ii

G T H

⎡ ⎤∂ Δ Δ⎢ ⎥



0

2

,

i i

p comp

T H H

T T

∂⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥ = −

⎢ ⎥∂⎢ ⎥⎢ ⎥⎣ ⎦

or

( )2

,

i

p comp

H

T T

Δ⎢ ⎥ = −⎢ ⎥∂

⎢ ⎥⎣ ⎦

M

i H Δis the partial molar heat of solution of (the relative partial

molar heat of i ).i

For an ideal solution,,

ln M id

i iG RT X Δ =

( )2

ln M iid R X H

dT T

Δ= −

substituting into

Then

As X i is not a function of temperature, then:, 0 0

M id

i i i H H H Δ = − =

, 0 M id H Δ =

Th h t f f ti f l ti i bt i d th diff



The heat of formation of a solution is obtained as the difference

between the heats (enthalpies) of the components in the solutionand the heats (enthalpies) of the components before mixing.

( ) ( ) ( ) ( )' 0 0 0 0

M

A B A B A B A A B B A A B B

M M

A B A B

H n H n H n H n H n H H n H H

n H n H

Δ = + − + = − + −= Δ + Δ

,' 0 M id H Δ =

(4) Entropy of formation of an ideal solution



(4) Entropy of formation of an ideal solution

There are two contributions to the entropy of a solid solution:the thermal contribution, S th, and the configuration contribution,

S config.

In statistical thermodynamics, entropy is related to the

randomness by the Boltzmann equation, , where k is

Boltzmann’s constant, 1.38×10-23 J/ºC.

lnS k ω =

(a) With respect to the thermal entropy, is the total number of

ways in which vibration can be set up in the solution.

(b) In solutions, additional randomness exists due to the



different ways in which the atoms can be arranged. This is the

extra entropy of configuration, S config . In such condition, isthe number of distinguishable ways of the arrangement of the

atoms in the solution.

For the mixing of N A particles of A with N B particles of B forming

a substitutional solution, the change of the entropy of

configuration :

( )

( )!' ln ln ln

! !

ln ln

A B A Bconf A B

A B A B A B

A A B B

N N N N S k k N N

N N N N N N

k N X N X

⎡ ⎤⎛ ⎞ ⎛ ⎞+Δ = = − +⎢ ⎥⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎣ ⎦

= − +

N A =N 0 n A ; N B =N 0 n B (N 0 is the Avogadro’s number)

So ( )' ln ln ( ln ln )S kN n X n X R n X n XΔ = + = +



For an ideal solution, there is no volume change or heat change

during mixing. So, the only contribution to is the change in

the configuration entropy , thus:

So, ( )0 ln ln ( ln ln )config A A B B A A B BS kN n X n X R n X n X Δ = − + = − +

( ln ln )config A A B BS R X X X X Δ = − +For 1 mole of solution,

, M id S Δ

, ( ln ln ) M M

M id A B A A B B A BS R X X X X X S X S Δ = − + = Δ + Δ

Since X A <1, X B <1, thus , 0 M id S Δ >

The variation of with composition, M id SΔ



From the fundamental equation,

( )

,

,

,

ln ln

M id

M id

A A B B p comp

GS R X X X X

T

⎛ ⎞∂ΔΔ = − = − +

⎜ ⎟⎜ ⎟∂⎝ ⎠

identical with that from the entropy change in the configuration ofthe solution.

（

the increase of the number of spatial configuration)

The entropy increases during the formation of an ideal solution.

The increase in molar value of the entropy change of mixing is

dependent only on the mole fraction of the components of the

solution and is independent of T .

pin an ideal binary A-B solution

S Δ

For a solution contains n A moles of A and n B moles of B , theentropy change of formation is:

'

S

M M M

A B A BS n S nΔ = Δ + Δ

(5) The Gibbs free energy of formation of an ideal solution



For a binary A-B solution, the change of the extensive properties

when 1 mole of solution is formed can be presented as：

M M M

A B A BQ X Q X QΔ = Δ + Δ

To suppose that 1 mole of homogeneous solid solution is formedby mixing together X A moles of A and X B moles of B , X A +X B =1 .

before mixing: after mixing:

X A XB

0 0

1 A A B BG X G X G= +

0

AG and are the molar Gibbs free energy of pure A and pure

B, respectively.

0

BG

2 1

M G G G= + Δ

( ),

ln ln M id M M

A B A A B B A BG RT X X X X X G X GΔ = + = Δ + Δ

X A +XB

Variation of G1 (the Gibbs free energy per mola



Variation of G1 (the Gibbs free energy per mola

of A and B before mixing) with composition, 0 M id

S Δ >

, , , ,0 0 M id M id M id M id G H T S T S Δ = Δ − Δ = − Δ <

Since X A <1, X B <1, thus

So, the Gibbs free energy change of mixing for an idealsolution is only due to the change in entropy.

(negative values)

, M id GΔ is a function of T and composition.

T increasing , M id

GΔ more negative Δ

G M , i d

The actual Gibbs free energy of the solution, G 2, is certainly

related to the molar Gibbs free energy of pure A and pure B



related to the molar Gibbs free energy of pure A and pure B ( and ).

0

AG

0

B

G

( )0 0

2 ln ln A A B B A A B B

G X G X G RT X X X X = + + +

As T increases, and decrease,

and the Gibbs free energy curves

assume a greater curvature.

The decrease of and is due tothe decrease of the thermal entropy of

the both components.

0

AG

0

BG

0

AG 0

BG

p

GS

T

∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠

This means that G decreases with

increasing T at a rate of – S.

GG1(L) G2(L)

G2(H)

G1(H)

G0 A(L)

G0B(L)

G0 A(H)

G0B(H)

MM

The chemical potential of A and B for an ideal solution:



0 ln A A A

u G RT X = +

0 ln B B Bu G RT X = +

,

The relationship between the Gibbs free energy curve

and the chemical potentials for an ideal solution.

In practical, the solutions of Sn-Zn, Fe-Cr and Ag – Cu are closed

to ideal solutions.

1.8 Nonideal solution



i i X α ≠

i

ii

X

a=γ

The activity coefficient of a component i , , in a

solution is defined as the ratio of the activity of the

component to its molar fraction.

iγ

To know the variation of the values of with T and

composition is centrally important in solution thermodynamics.

iγ

Since,

to determine the equilibrium state of any chemical reaction

involving the components in the solution M

iGΔ is required

ia is required iγ is required



The variation of with X i

for a component i. Fe-Ni

system at 1600o

C.<1, negative deviation

from Raoult’s law.

ia

iγ


for a component i. Fe-Ni

system at 1600 oC.

iγ



iγ The variation of with X i

for a component i. Fe-Cu

system at 1550 oC.


for a component i. Fe-Cu

system at 1550 oC.

>1, positive deviation

from Raoult’s law.iγ

ia

,

ln ln ln M non id

i i i iG RT a RT RT X γ −

Δ = = +Since



iγ ( ) ( )

2

ln M

M i

iiG T R H

T T T

γ ∂ Δ ∂ ⋅ Δ= = −

∂ ∂ln

1

M i

i R H

T

γ ∂= Δ

⎛ ⎞∂⎜ ⎟⎝ ⎠

2

1( )

dT d

T T = −

,

0 M non id

i H −

Δ ≠

Whenvaries with T , thus:

(Gibbs-Helmholtz equation)

then for a nonideal solution

or ( )

Generally, an increase in the temperature of a nonideal solutiondecreases the extent of the deviation of its components from

ideal behavior.

iγ (a) When >1, an increase in temperature decreases the value of

toward unity;γ



From and decreases with increasing T ,

it is obtained that

1iγ > ( )1i iT γ γ ↑→ − ↓→ ↓ ( )1

iγ → AA

BB

E

AB E E <

toward unity;

,

,

,

ln1

M ii H R

T

γ ∂Δ = ⎛ ⎞∂ ⎜ ⎟⎝ ⎠

iγ

0. M

i H Δ > Thus, 0

M

H Δ >When >1, the molar heat of formation of the solution is a

positive quantity. The solution process is endothermic. is the

quantity of heat absorbed for the solution from the thermostating

heat reservoir surrounding the solution when 1 mole of the

solution is formed at T.

iγ

M

H Δ

iγ

iγ (b) When <1, an increase in temperature increases the value of

toward unity;γ



1iγ < ( )1i iT γ γ ↑→ − ↑→ ↑ ( )1

iγ → AA

BB

E

AB E E >

toward unity;

,

,

,

ln

1

M i

i

H R

T

γ ∂

Δ = ⎛ ⎞∂ ⎜ ⎟⎝ ⎠

From and increases with increasing T, it

is obtained that

i

γ

0, M

i H Δ <

Thus, 0 M H Δ <

When <1, the molar heat of formation of the solution is a

negative quantity. The solution process is exothermic. is the

quantity of heat absorbed by the thermostating heat reservoir

surrounding the solution when 1 mole of the solution is formed

at T.

iγ

M H Δ

iγ

For an A-B binary solution:



(b) For exothermic mixing, the A-B attraction is stronger than

either the A-A or B-B attractions ( <1, E AB <E AA ,E BB ). There is

a tendency toward compound formation between the two

components, i.e. toward “ordering” in the solution . A atoms

attempt to have only B atoms as nearest neighbors, and B

atoms attempt to have only A atoms as nearest neighbors.

(a) For endothermic mixing, the A-A and B-B attractions are

stronger than the A-B attraction ( >1, E AB >E AA ,E BB ). There is

a tendency toward phase separation or “clustering” in the

solution. A atoms attempt to have only A atoms as nearestneighbors, and B atoms attempt to have only B atoms as

nearest neighbors.

iγ

iγ



(c) In both exothermic mixing and endothermic mixing, the

equilibrium configuration of the solution is reached as a

compromise between the enthalpy factors and the entropy factor.

The enthalpy factors are determined by the relative magnitudes of

the atomic interaction, which attempt to either completely order

（）or completely unmix （）the solution.

And the entropy factor attempts to maximize the randomness of

mixing of the atoms in the solution.

(a) ordered (b) clustering (c) random

0 M H Δ < 0>Δ M H

1.9 The excess function of the extensivethermodynamic properties



thermodynamic properties

The excess function of an extensive thermodynamic property, ,

denotes the difference between the value of the property (Q )

such as H , S and G , etc., of a real solution and the value that thsolution would have if it was ideal.

E Q

0 0 , ,( ) ( ) E real id real ieal M real M id Q Q Q Q Q Q Q Q Q= − = − − − = Δ − Δ

For a binary system, A-B, the excess Gibbs free energy, E G

( ) ( )

( )

, ,

ln ln ln ln

ln ln

E M real M id

A A B B A A B B

A A B B

G G G

RT X a X a RT X X X X

RT X X γ γ

= Δ − Δ

= + −

= +

＋

again from the concept of the excess molar Gibbs free energy of a solution,

real id EG G G+ subtraction of from both sides gives:0 0

X G X G+



real id E G G G= + , subtraction of from both sides gives: A A B B X G X G+

, M M id E G G GΔ = Δ + As for any solution, , thus

The excess H and S can be expressed as :

( ), ,

ln ln) ln ln

A B

E E A B

p A A B B A B

p X p X

GS R X X RT X X

T T T

γ γ γ γ

⎡ ∂ ∂∂ ⎛ ⎞ ⎛ ⎞− = = − + +⎢ ⎥⎜ ⎟ ⎜ ⎟

∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣

－

2

, ,

ln ln

A B

E E E A B A B

p X p X

a aTS G H RT X X

T T

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞+ = = − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

( )ln ln E

A A B BG RT X X γ γ = +

The value of can be used to estimate whether the solution ispositive or negative from an ideal solution.

E G



If =0, the solution is an ideal solution.>0, positive deviation from the ideal solution.

<0, negative deviation from the ideal solution.

E

G E G

E G

The concept of excess thermodynamic function is also applicableto the partial molar quantity and relative partial molar quantity.

where Q can be G, H, S, etc.

( ) , ,

ln ln ln E

M M real M id

i i i i i i iG G G RT X RT X RT γ γ Δ = Δ − Δ = − =

( ) ( ) ( ), , 0 0

E M M real M id real id real id E

i ii i i i i i i iQ Q Q Q Q Q Q Q Q QΔ = Δ − Δ = − − − = − =

In terms of the excess relative partial molar Gibbs free energy:

In addition, in terms of excess partial molar Gibbs free energy,

0 0E real id real ideal M real M id



( ) ( ) ( )( )

, ,0 0

ln

E real id real ideal M real M id

i i i i i i ii i

E M

i i

G G G G G G G G G

G RT γ

= − = − − − = Δ − Δ

= Δ =

E E E A B A B

G X G X G= + E E E A B A BS X S X S = +

E E E

A B A B H X H X H = +

For a binary solution:

It should be noted, that is the criterion of the deviation type

from Raoult’s law. In the case of only one of the values of

and is known, normally, it cannot estimate the type of

deviation from Raoult’s behavior that the solution has .

E

iG E

i

H E

iS

1.10 Regular solution

(R i )



i

id

ia X = ,

0 M id

i H Δ = ,

0 M id

iV Δ = ,

ln M id

i iS R X Δ = −

, 0 M id H Δ = , 0 M id V Δ = , ln M id

i iS R X X Δ = − ∑

i ia X ≠ ,

0

M real

i H Δ ≠

,

0

M real

H Δ ≠

1) Raoultian ideal solution:

,

(Review)

2 ) nonideal solution:

Regular solution is the simplification of some nonideal solutions.

A regular solution is defined as the one which has a nonzero

heat of formation and an ideal entropy of formation.

;

,

ln M id

i iG RT X Δ =

, ln M id i iG RT X X Δ = ∑

, ln M real

i iG RT X aΔ = ∑

For component i :, 0 M reg

i H Δ ≠ , ,

ln M reg M id

i i iS S R X Δ = Δ = −

lM regS R X XΔ 0M reg

HΔ



, ln M reg

i iS R X X Δ = −

∑

, 0 M reg H Δ ≠

At any given temperature, it is suggested that the activitycoefficients, r A and r B ， of an A-B binary solution can be

represented by power series of the form:L+++= 3

3

2

213

1

2

1ln B B B A X X X α α α γ

L+++= 3

3

2

21 3

1

2

1

ln A A A B X X X β β β γ

1 2 3 1 2 3, , ......, , , ......α α α β β β are constants .

by Gibbs-Duhem equation: 0 ( ln ) M M M

A B i A B i X d G X d G G RT aΔ + Δ = Δ =

B B A A d X d X γ γ lnln −=Thus :

(a)If in the entire composition range, the solution obeys the above equation,

stipulations :



( ) p g , y q ,

then .1 1 0α β = =

(b) further, if the activity coefficient can be represented by the quadratic terms

only, then .2 2α β =

α 2ln ,

A B X γ α =

2

ln B A X γ α =

or where

A regular solution can also be assigned the one obeys the following Eqs:

2ln , A B RT X γ α ′=2

ln A B X RT α γ ′=where is independent of temperature.'α

is an inverse function of temperature,

RT

α

α

′

=

but independent of the composition of the

solution.

Hence, 2 2

1 2 B

1RT ln RT 'X

2 A B B X X γ α α α = +（）＝ Similar for RTln Bγ

The thermodynamic properties of a regular solution can be best

examined via the concept of excess function.



)()lnln( B A B A B B A A

E X X X X RT X X RT G +=+= α γ γ

, , M reg M id

i iS S Δ = Δ 0 0( ) ( )id id E

G G G G G G G− = − +Since, and

, , , ,

, , , ,

(or( ) )

( )

E reg M reg E M reg M id

M reg M reg M id M reg

G G G G

H T S S H

Δ = Δ − Δ

= Δ − Δ − Δ = Δ

2 B X α 2

A X α

, ,' E reg M reg

A B A BG RT X X X X H α α = = = ΔTherefore,

M GΔ id M G

,Δ

, E regG is independent of temperature, as



, ,

,

E reg E reg

P comp

G S T

⎛ ⎞∂ = −⎜ ⎟∂⎝ ⎠ 0, =reg E

S

Therefore, for a binary regular solution, at any given composition,

1 21 ( ) 2 ( )ln ln E

A A T A T G RT RT γ γ = =

2

1

( ) 1

( ) 2

ln

ln

A T

A T

T

T

γ

γ

=

This equation is practically used in converting activity data at

one temperature to those at other temperatures.

Additionally, for a regular solution, can also be obtained from E

iG



At given temperature, the plots of

vs. display a linear relationship, the

slop of which is . With the increase of

T , decreases.

2

2

' (1 ) ' ( )

' ln

E E

E A B A B A A A

A

B A

dGG G X X X X d X X

dX

X RT

α α

α γ

= + = + − −

= =

Tiγ log 2Sn X

α α

Example: the Ti-Sn system

vs.

Tiγ log

2

Sn X

RT

α α

′=( )

If is strictly adherence to the model,

it should be independent of T , ,

T α '

T R

α α =



But, b ≠ b ’ ， Thus, ,

but from experiment, decreases

slowly with T increasing.

R

It should be noted that a parabolic

relationship for or with composition

can not be taken as a demonstration that

the solution is regular.

M H Δ E G

B A

M X bX H =Δ B A

E X X bG ′=When or

M E H GΔ ≠, ,( ) E M M id M M M id

G G G H T S S = Δ − Δ = Δ − Δ − Δ M id M

S S Δ≠Δ ,So,

T α

Ti-Sn at 414 ℃

For examples:



1. Au( aurum / gold) －Cu, non-regular at 1550 K,

For examples:

AuCu

E X X G 060,24−= joules, but

M H Δ is asymmetric. Hence, 0 E S ≠

2. Au－ Ag, non-regular at 1350 K,

Au Ag

M X X H 590,29−=Δ joules, but E G is asymmetric. Hence, 0 E

S ≠

M id M S S Δ≠Δ ,

∫= A X

A

A

B

E dX RTX G

0 2

ln γ , while )

E

B

E E

AdX

dG X GG += A

E

A RT G γ ln=(from

Another form for GE:



For a regular solution, as ,

Thus, for a Raoultian solution, as ,

∫ A

B

B

X 0 2 AdX

1= Aγ 0 E G =

α

γ

=2

ln

B

A

X B A

E

X X RT G α =,

,

, , ,

, , ( )

( ln ln )

' ( ln ln )

M regular E reg M id

A A B B

M reg M reg id A B A A B B

G RT X X G G

X X RT X X X X H T S

α α

α

Δ = + = + Δ

= + + = Δ − Δ

' A B

X X α : the nonideality of the mixture.

If is positive, the heat of mixing is positive, endothermic process, 0 M H Δ >

0 M H Δ <If is negative, the heat of mixing is negative, exothermic process.

'α

'α

, , E M E M reg M id

For a regular solution,



, ,

or(( ) ) 0

g

S S S S Δ = Δ − Δ = , , ,( ) ( ) ( ) M E M E M E E M reg M id M regG H T S H H H H Δ = Δ − Δ = Δ = Δ − Δ = Δ

id M

B B A A

reg M S X X X X RS

,, )lnln( Δ=+−=Δ

)lnln(,,,

B B A A

reg M reg M reg M X X RT S T G H γ γ +=Δ−Δ=Δ

reg E reg M

i

id M

i

reg M

i

E reg M

i iG H H H H

,,,,,,

)( =Δ=Δ−Δ=Δ

i

reg M

i RT H γ ln,

=Δ, ,' E reg M reg

A B A BG RT X X X X H α α = = = Δ

1.11 A statistical model of solutions

(The quasi-chemical model)



In the statistical model, it is assumed that the heat of mixing,

,is only caused by the bond energies between the

adjacent atoms, and

M H Δ

(1) 0 0

A BV V =

(2) 0 M V Δ =0i iV V =

The statistical model of solutions is a theory about the

relationship between the macroscopical thermodynamic

properties and the microstructure of the solutions with

following features.

(3) For the two components, in both their pure state and in



Thus, the energy of the solution is the sum of the atom-atombond energies.

solution, the interatomic forces are only significant over shortdistance. Only nearest neighboring interactions need to be

considered, or, interactions only exist between neighboring

atoms.

(4) The factors of the atomic size and the structure of the

electron are ignored.

Consider 1 mole of A-B solution containing N A atoms of A and N B atoms of B ,

N N



AA E

This solution contains three types of atomic bonds:

1) A-A bonds, the energy of each bond is

2) B-B bonds, the energy of each bond is

.

BB E

3) A-B bonds, the energy of each bond is AB

E

0

A A A

A B

N N X

N N N = =+0

B B N X

N =

AB AB BB BB AA AA E P E P E P E ++=

AAP BBP ABPnumber of A-A bonds; number of B-B bonds; number of A-B bonds

(the energy of the solution)

2 AB AA

A

P P N

z z= +

2 2

A AB AA

N z PP = −

N z P

or



2 AB BB B P P N

z z= + 2 2 B AB BB

N z PP = −or

1 1 1( )

2 2 2 A AA B BB AB AB AA BB

E zN E zN E P E E E ⎡ ⎤= + + − +⎢ ⎥⎣ ⎦

1

2 AA A

P N z=For N A atoms of pure A : For N B

atoms of pure B : 1

2 BB BP N z=Thus, the change of the energy during mixing, M

E Δ M

E Δ

⎥⎦⎤

⎢⎣⎡ +− )(

2

1 BB AA AB AB E E E P

=(the energy of the solution) – (the energy of the unmixed components)

=

( Z : coordination number; FCC: 12; BCC:8) Each atom has z nearest neighbors

AB AB BB BB AA AA E P E P E P E ++=

BB B AA A E zN E zN 2

1

2

1+

M M M H E p V Δ = Δ − Δ

⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H 0 M V Δ =



AA E BB E AB E M H Δ ABPFor given values of , and , depends on

id M H ,Δ2

BB AAid

AB

E E E

+=For an ideal solution, =0, thus,

For ideal mixing, it is necessary that E AA =E BB =E AB , so, a sufficient

condition is that E AB should be the average of E AA and E BB .

( )ABE / 2 AA BB E E > +If 0, 1 M i H γ Δ < <

( )ABE / 2 AA BB

E E < + 0, 1 M

i H γ Δ > >

,

( ln ln )

M id

A A B BS R X X X X Δ = − +

For ideal solution, the mixing of N A atoms of A with N B atoms ofB is random, in this case, the measure of the randomness ofthe system, , is given as M S Δ

, exothermic mixing.

, endothermic mixing.

In solutions which do not depart too greatly from ideality, that is,

, it may be assumed that the randomness of the M

H RT Δ ≤



distribution of the atoms is approximately the same as in an ideal

solution. In such case, considering two neighboring lattice sites,

labeled 1 and 2, in the A-B crystal, 1 2

(a) The probability that site 1 is occupied by an A atom is X A :

0

the number of A atoms in the crystal

the number of lattice sites in the crystal

A

A

N X

n= =

(c) The probability that the site 1 is occupied by A and site 2 is

simultaneously by B is X A X B . The site 1 is occupied by B and

site 2 is simultaneously by A is also X A X B . Thus, the probability

that a neighboring pair of sites contains an A-B pair is 2 X A

X B

.

(b) The probability that site 2 is occupied by B atom is X B .

(d) The probability that a neighboring pair of sites contains an A-A

pair is X A 2, and that contains an B-B pair is X B

2.



As per mole of crystal contains pairs of lattice sites.021 zn

So,0 0

12

2

id

AB A B A BP zn X X zn X X = × = 2

0

1

2 AA AP zn X =2

0

1

2 BB BP zn X =

0

1( )

2

M

A B AB AA BB H zn X X E E E ⎡ ⎤Δ = − +⎢ ⎥⎣ ⎦ ⎥⎦

⎤⎢⎣

⎡ +−=Δ )(2

1 BB AA AB AB

M E E E P H ( )

0

1( )2

AB AA BB zn E E E ⎡ ⎤

Ω = − +⎢ ⎥⎣ ⎦ B A M X X H Ω=ΔTo take , then

M H Δ is a parabolic function of composition.

As random mixing is assumed, the statistical model correspondsto the regular solution, i.e.,

, M reg E

A B H G X X Δ = = Ω

Thus, for a regular solution,

, , , ( ln ln ) M reg M reg M reg

A B A A B BG H T S X X RT X X X X Δ = Δ − Δ = Ω + +



,10, , ( ( ), 0

2

M reg

AB AA BB AB AA BB E E E H ε ε ε Ω < < > + Δ <

A-B bond is preferred.

(1) when

0

1

( )2 AB AA BB zn E E E

⎡ ⎤

Ω = − +⎢ ⎥⎣ ⎦T M

GΔ

(2) when ,10, , ( ( ), 0

2

M reg

AB AA BB AB AA BB E E E H ε ε ε Ω > > < + Δ >



A-A , B-B bond is preferred.

Mixing at high T , 0 M GΔ < Mixing at low T , positive

value of appears. M GΔ

Since,

( ) M reg

B A

A

H X X

X

∂Δ= Ω −

∂, M reg

A B H X X Δ = Ω , then

, M reg



, , M reg M reg A B

A

H H H X X

∂ΔΔ = Δ + ∂

The value of relates to both the values of and T , in other

words, relates to the relative values of , , , z and T .

iγ Ω

AA E BB E AB E

Thus , 2( ) M reg

A A B B B A B H X X X X X X Δ = Ω + Ω − = Ω

and similarly, , 2 M reg

B A H X Δ = Ω

As the mixing is random, then A

M

A X RS ln−=Δ B

M

B X RS ln−=Δ

Hence, A B

M

A

M

A

M

A X RT X S T H G ln2 +Ω=Δ−Δ=Δ

Since ln ln ln M

A A A AG RT a RT RT X γ Δ = = +

So,22

ln B B A X X RT α γ =Ω

= 2 2

ln A B A X X RT γ α Ω

= = RT

Ω=α

Ωiγ Ω

iγ If <0, then <1; If >0, then >1.

A di h H ’ l h XB

0

l lA A RT

Ω

1



The applicability of the statistical model to actual solutionsdecreases as the magnitude of

According to the Henry’s law, when X B → ln ln A A RT γ γ → =1,

i X → Raoult’s law is approached asymptotically for thecomponent i.

As 1,

Ω Ω( ) increases. That is :

1( )

2 AB AA BB E E E +ffIf or ,

1( )

2 AB AA BB E E E +pp

the random mixing of the A and B atoms can not be assumed.

The equilibrium configuration of a solution at constant T and P is

which minimizes the Gibbs free energy G (G=H-TS ).Minimization of G occurs as a compromise between minimization

of H and maximization of S .

maximization of the number of A-B pairs (complete ordering of

(a) If ( ) / 2 AB AA BB E E E > + , minimization of H corresponds to the



the solution), and maximization of S corresponds to completely

random mixing. So, minimization of G occurs as a compromise

between maximization of P AB (the tendency increases withdecreasing , as there is a negative value) and random

mixing (the tendency increases with increasing T ) （）

Ω

If is appreciably negative, and T is not too high, then

P AB(actual) > P AB(random). It cannot assume that the mixing is random.

Ω

0 M H Δ <



⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H 0

1( )

2 AB AA BB

zn E E E ⎡ ⎤Ω = − +⎢ ⎥⎣ ⎦

Ω

-

(b) IF , minimization of H corresponds to

th i i i ti f th b f A B i ( l t l t i

( ) / 2 AB AA BB E E E < +



the minimization of the number of A-B pairs (complete clusteringin the solution). The minimization of G occurs as a compromise

between minimization of P AB (the tendency increases with

increasing , as here is a positive value) and random

mixing (maximization of S) .

Ω Ω

Then, if is appreciably positive, and T is not too high, thenP AB (actual) < P AB (random). It also cannot assume that the mixing is

random.

Ω

（）0 M H Δ >



⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H 0

1( )

2 AB AA BB

zn E E E ⎡ ⎤Ω = − +⎢ ⎥⎣ ⎦

+

For the statistical model to be applicable, the configuration of the

equilibrium solution should be not too far from random mixing.



then (1) For any value of , more-nearly random mixing occursas the temperature is increased.

(2) For any given T , more-nearly random mixing occurs with

smaller values of .

Ω

Ω



⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H 0

1( )

2 AB AA BB

zn E E E ⎡ ⎤Ω = − +⎢ ⎥⎣ ⎦

A-50at%B solution

0 M H Δ < 0 M

H Δ >1

0 M H Δ < 0 M

H Δ >1

1

3



The extreme configurations are complete ordering and complete clustering.

Random configuration occurs between the extremes when the probability of

an A-B pair is 0.5. In both extreme configurations (ordering or clustering),

=0. M

S Δ

Ω

Ω

1

2

3

4

1

2

3

4

**

*

3

2

4

Variations of the , and with the range of

spatial configuration available to a A-50 at%B solution.

M S Δ M H Δ M

GΔ

0 M H Δ < 0 M

H Δ >1

0 M H Δ < 0 M

H Δ >1

1

3



Curve 1 : the relationship between the entropy of mixing ( )

and the solution configuration. The maximum value is ,occurring in the random configuration.

M S Δ

id M S ,Δ

Curve 2 : M T S − Δ

Line 3: the variation of with configuration, linear to P AB. M

H Δcurve 4: The sum of curve 2 and line 3, ,the minimum in this curve occurs at the equilibriumconfiguration.

M M M S T H G Δ−Δ=Δ

(*/asterisk : the minimum of the curve). M

GΔ

⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H

Ω

Ω2 3

4

2

3

4

**

* 2

4

The equilibrium configuration of exothermic solution lies

between the ordered and random configuration; The equilibrium

configuration of endothermic solution lies between the clustered



configuration of endothermic solution lies between the clusteredand random configuration.

(1) That the random configuration to be the equilibrium

configuration only occurs in the case that =0 in all thecomposition range.

M

H Δ

0 M H Δ < 0 M H Δ >

Ω

Ω

1

2 3

4

0 M H Δ < 0 M H Δ >1

2

3

4

*

*

M

HΔ varies with Ω

0 M H Δ < 0 M

H Δ >1

0 M H Δ < 0 M

H Δ >1



(3) For any given system (of fixed ), as T and hence

increases, the position of the minimum on the curve

moves toward to the random configuration.

(4) Both extreme configurations are physically unrealizable.

(2) As for the A-B system increases, at a constant T , the

position of the minimum on the curve moves further away

from the random configuration. ( )

Ω M

GΔ

B A

M X X H Ω=Δ

Ω

M S T Δ

M GΔ

H Δ varies with Ω

Ω

Ω2 3

4Ω Ω2

3

4 T

T

⎥⎦

⎤⎢⎣

⎡ +−=Δ=Δ )(2

1 BB AA AB AB

M M E E E P E H

0

1( )

2 AB AA BB

zn E E E ⎡ ⎤Ω = − +

⎢ ⎥⎣ ⎦

In order to have the minimum on the curve coincide with

either extreme at definite T infinite values of would be

M GΔ

M HΔ



either extreme, at definite T , infinite values of would berequired (negative for complete ordering and positive for

complete clustering) .

In addition, with a nonzero , only at infinite T, the randomconfiguration could become the equilibrium configuration.

H Δ

M H Δ

0 M H Δ < 0 M

H Δ >

Ω

Ω

1

2 3

4

0 M H Δ < 0 M

H Δ >

Ω Ω

1

23

4 T

T*

*

The above two are

all unrealizable.

Ω

In the regular solution model:

is a constant;



Ω M

H Δis a constant;

shows a parabolic variation with composition;

E G

M GΔand are symmetrical about the composition X A =0.5.

00 B A V V ≠But if :

The lattice parameters of the crystal will vary

with the composition;

The interatomic distance varies with composition;

The bond energies will be composition-dependent.

M E

A B H X X GΔ = Ω = , M M id S S Δ = Δ

The regular solution model can be made more flexible byThe regular solution model can be made more flexible by

1.121.12 Subregular Subregular solutionssolutions



The regular solution model can be made more flexible byThe regular solution model can be made more flexible byarbitrarilyarbitrarily allowingallowing ΩΩ to vary with composition, such asto vary with composition, such as

LL++++=Ω 32

B B B dX cX bX a

The so called subregular solution model is that BbX a +=ΩE ( ) B A BG a bX X X = +

E 2 2 ( ) A B B B AG aX bX X X = + − E 2 22 B A A BG aX bX X = +

A

B A

dX

dG X GG +=( )

Empirical equation

a and b have no physical significance.

and a, b, c, d are constants.

Excess molar Gibbs free

energy curves generated by

the subregular solution



the subregular solutionmodel for several

combinations of a and b.

The maximum and minimum

in the curves occur atEG

E

0 B

dG

dX =

a

aabbab X B

6

2)(2 22 ++±−=

E ( ) B A BG a bX X X = +

a=13456 Jb 5412 8 J



, E

( ln ln )

( )

M M id

A A B B

B A B

G G G

RT X X X X

a bX X X

Δ = Δ +

= +

+ +

Fit the data of a and b to the

subregular solution model.

The variation ofThe variation of ΩΩ withwith

composition in the system of Agcomposition in the system of Ag--

Au, obtained from the Au, obtained from the

experimental measurementsexperimental measurements

1350K

b=5412.8 J

The influence of temperature on the behavior of subregular solution can be

accommodated by introducing a third constant, τ,

T



E

0 0( ) (1 ) B A B

T G a b X X X

τ = + −

EE 0 0( ) A Ba b X X GS

T τ +∂= =

∂

E E

0 0( ) (2 ) M

B A B

T H G TS a b X X X

τ

Δ = + = + −

About the example in the text book:

Au-Cu solid solution, between 410~889o

C, if Au Cu28,280 E

G X X = −



( ) ( ) ( )( )

, ,0 0

ln

E real id real ideal M real M id

i i i i i i ii i

E M

i i

G G G G G G G G G

G RT γ

= − = − − − = Δ − Δ

= Δ =

A

E

B

E E

i

dX

dG X GG +=

Calculate the activity coefficient of Au and Cu.

22

Cu Au28, 280 0.4ln 0.6248.3144 873

X RT

γ Ω ×= = − = −×

(at least not strict)

1.13 Application of the Gibbs-Duhem relation to

the determination of activity

1 13 1 Graphical integration



1.13.1 Graphical integration

To obtain ia ， one experimental method: electromotive force

0 lni

M i i i

G n F u RT aε μ Δ = − = − =

F is Faraday’s constant, 23,060 calories/volt·mole.

Normally for low temperature

At high temperature,

(1) 0

i

ii

p

pa =

(2) In some binary solutions that the variation with compositionof the activity of only one component can be experimentally

determined, and the activity of the other component can be

determined by means of the Gibbs-Duhem equation.

ii

i

a

X γ =

0=∑ ii Qd X i

M

i a RT G ln=Δ 0lnln =+ B B A A ad X ad X

B B

A ad X

X ad loglog −=log

log 1log ( ) log

B A A

A AB A

a atX X

B A X X B

a atX X a d aX

=

= == −∫integration



A X log 1g ( ) g

B Aa atX A X = ∫

Schematic representation of thevariation of with XB /X A ina binary solution

The shaded area under the curve

is the value of at X A =X A .log Aa

Blog a-

Two points should to be noticed:

1) As X B →

1, B

a →1, B

alog →0,

and XB / X A →∞. Then the curve

exhibits a tail to infinity as XB→1.

2) As X B →0, Ba →0, Balog →-∞,

and- →∞. Then the curveexhibits another tail to infinity

as XB→0.

log B

a

X=X A=X A2

0

？

The second point introduces an uncertainty into the calculation at any

composition. However, if to use activity coefficients instead of

activities in the Gibbs-Duhem equation:



q

In a binary solution,

0loglog =+ B B A A d X d X γ γ Then B

A

B

A d X X d γ γ loglog −=

1 A B

X X + = 0=+ B A

dX dX 0 A B A B

A B

dX dX X X

X X + =

ln ln 0 A A B B X d X X d X + = 0lnln =+ B B A A ad X ad X subtraction from

(denominator, numerator)

log

log 1log ( ) log

B A A

A A B A

atX X B

A X X BatX

A

X d

X

γ

γ γ γ

=

= == −∫

As X A 1 (X B 0), the solutionThe shaded area islogA

γ at XA = XA

log

log 1log ( ) log

B A A

A A B A

atX X B

A X X BatX

A

X d

X

γ

γ γ γ

=

= == −∫



A ( B ),

is taken as dilute solution:

0

0,

log log

B B

B B

γ γ γ γ =

=

g A

γ A A

The tail to minus infinity ( ) as XB 0 can be

avoided.

log Ba → −∞

Thus if the variation of ( , ) with composition is known,

the value of at the composition of X A can be obtained by

integration of the above equation .

Bγ Bγ log

Aγ log

1.13.2 Examples

Nia was experimentally measured.

Fea was obtained from Gibbs-Duhem equation (by means

(a) The Ni-Fe system at 1600 oC



of graphical integration)

Niγ Ni X Extrapolating to

the value (0.66) is the Henry’slaw constant for Ni in Fe.

= 0 at 0.66,

0.66 Nik =

Activities of Ni and Fe in thesystem Fe-Ni at 1600 oC

Ni Ni Nia k X =

In the composition range in which

Henry’s law is obeyed,

a

0

0.66 Ni Nik γ = =

γ

(1) In the case of is known

To get in the Fe-Ni binary system in the case that the

variation of with composition is known :

Niγ

Feγ

Niγ



( ) Niγ

Feγ can be obtained by means ofgraphical integration,

log

log 1log log

Ni Fe Fe

Fe Fe Ni Fe

at X X Ni

Fe X X Niat X

Fe

X d

X

γ

γ γ γ

=

= ==−∫

Niγ log As increases with increasing

XNi /XFe, The integrated area under the

curve between X Ni=X Ni and X Ni=0 is a

positive quantity. Thus, is

everywhere a negative quantity.

logFe

γ

positive

log Niγ at XFe=1

log Niγ at XFe=XFe

Variation of Niγ log with X Ni /X Fe

at XFe=1log Niγ log 0.66 0.18= = −

negative positive

(b) Fe-Cu system at 1550 oC

10.1



Cuγ Cu X

010.1Cu Cuk γ = =

Extrapolating to = 0 at 10.1

the value (10.1) is the Henry’slaw constant for Cu in Fe.

10.1Cu

K =

Cu Cu Cua k X =

In the composition range in

which Henry’s law is obeyed,

a

γ

XFe=XFe



decreases with increasing XCu /XFe , the integrated area

in is a negative quantity. is positive, .

Cuγ log

logFe

γ 1Fe

γ >

log

log 1log logCu Fe Fe

Fe FeCu Fe

at X X Cu

Fe X X Cuat X

Fe

X d X

γ

γ γ γ

=

= == −∫

negativenegative

positive

XFe=1

α 1.13.3 The - Function

-function is a further aid to the integration of the Gibbs-

Duhem equation.

α

l α



2

ln

(1 )

ii

i X

γ α =

− i X iγ -function is always finite due to the fact that

→1, →1.

α

as

For a binary A-B solution,

2

ln

B

A

A

X

γ α =

2

ln

A

B B

X

γ α =

2ln A A B X γ α =or

2ln B B A

X γ α =

Differentiation

B A A A B B d X dX X d α α γ 2

2ln +=ln ln B A B

A

X d d

X γ γ = −

Substitute into

B A B A B B B A

A

B

A A B

A

B

A d X X dX X d X X

X dX X

X

X d α α α α γ −−=−−= 22ln

2

ln 2 A B B A B A B

d X dX X X d γ α α = − −

Integration



( )

1 1ln 2

A A B A A

A B A

X X a at X X

A B B A B A B X a at X

X dX X X d γ α α = =

= ==− −∫ ∫

( ) ( ) B A B A B B B B A

X X d d X X d X X α α α = −∫ ∫ ∫( )d xy ydx xdy= +∫ ∫ ∫

1

ln 2

2 A A

A

A B B A B A B B B A B A B

B B A B A B B B A B A A

X X

B A B B A X

X dX X X X dX X dX

X dX X X X dX X dX

X X dX

γ α α α α

α α α α

α α =

=

= − − + +

= − − + −= − −

∫ ∫ ∫

∫ ∫ ∫∫

1 A B X X + = 0=+ B A dX dX ( , )

Bα

1ln

A A

A

X X

A B A B B A X

X X dX γ α α =

== − −∫



As is everywhere finite, thus the integration does not involve

a tail to infinity.

B B A X X α −

Bα

at X A =X A is the value of minus the area

versus X A from X A =X A to X A =1.

ln γ

under the plot of

Bα

(2) Combingα-function, ∫ =

=−−=

FeFe

Fe

X X

X Fe Ni Ni NiFeFe dX X X

1ln α α γ

As is everywhere negative theNiα

+ +

XFe=XFe



the variation of Niα with composition

As is everywhere negative, the

integrated area from X Fe=X Fe to

X Fe=1 is a positive quantity. But theresult of the above two terms cause

a negative value of , so .

Negative deviation from Raoult’s law.

Niα

lnFe

γ 1Feγ <

2 2

Fe

ln ln(1 )

Ni Ni Ni

Ni X X γ γ α = =

−

Aspositive

XFe=1

XFe=XFe XFe=1



2

ln CuCu

Fe X

γ α =

∫ =

=−−= FeFe

Fe

X X

X FeCuCuCuFeFe dX X X 1ln α α γ

As is everywhere positive, the integrated area fromX Fe=X Fe to X Fe=1 is negative. But the sum of which is ,

is positive. Positive deviation from Raoult’s law.

Cuα ln Feγ

- -

1.13.4 Direct calculation of the integral Gibbs freeenergy of mixing

M

B

M M

AdX

Gd X GG Δ+Δ=Δ rearranging and dividing by2

B X



AdX

The integral Gibbs free energy of mixing , , can be obtained

directly from the variation of with composition.

M GΔ

Aa

2 20 0

ln A A

M X X

M A A B A B A

B B

G aG X dX RTX dX

X X

ΔΔ = =∫ ∫

2 2

M M M M

A A B B

B B B

G dX X d G G dX G

d X X X

⎛ ⎞Δ Δ −Δ Δ

= = ⎜ ⎟⎝ ⎠

A A X X =

0 A X =integration A

B

M

A

B

M

dX

X

G

X

G∫

Δ=

Δ2

Example:

Ni and Cu in Fe, respectively

( ) 20

ln Ni X M Ni

Ni Fe Fe Ni

Fe

aG RTX dX

X −Δ = ∫

( ) 20

lnCu X M Cu

Cu Fe Fe CuFe

aG RTX dX

X −

Δ =

∫2l /( ) (



) vs.

2log /Cu Fe

a X Cu X curve (a) : ( ) versus

2

log / Ni Fea X Ni X curve (c) : ( ) vs.

2log /(1 )i i

X X −i X curve (b): (

2ln /(1 )i ia X − → −∞

The uncertainty is:

0i X →as

The shaded area ( the value of the integral of

between and )multiplied by the factor 2.303×8.3144×1823×0.5

is the in the system at .

0.5Cu X = 0Cu X =

M GΔ 0.5Fe

X =

2

log /Cu Fea X 0 1.0

XNi / XCu

With respect to the Raoultian solution (line b):

[ ]

20

ln ln(1 ) (1 ) ln(1 )

(1 ) 1ln (1 ) ln(1 )

i X M i i i

i i i i

i i

i i i i

X X X G RT X dX RT X X

X X RT X X X X

⎡ ⎤Δ = − = − + −⎢ ⎥

− −⎣ ⎦= + − −

∫



[ ]( ) ( )i i i i

The uncertainty due to the infinite tail as

→0 can be eliminated if the

equation is used to calculate the integral

excess Gibbs free energy.

i X

, , E M reg M id G G G= Δ − Δ

20

lnCu X E CuCu Fe Fe Cu

Fe

G RTX dX X

γ − = ∫

0

( 0 log log )Cu Cu Cu X γ γ = ⇒ =



The integral molar Gibbs free energies of mixing in the

systems Fe-Cu at 1550 ºC and Fe-Ni at 1600 ºC

20

A

M X

M A B A

B

QQ X dX

X

ΔΔ = ∫

A general equation which relates

the integral and partial molar

values of any extensive

thermodynamic function.



∫ Δ=Δ A X

A

B

M

A

B

M dX X H X H

0 2

∫ Δ=Δ A X

A

B

M

A B

M dX X S X S

0 2

2 20 0ln

A A

M X X

M A A B A B A

B B

G aG X dX RTX dX X X ΔΔ = =∫ ∫

1.13.5 The relationship between Henry’s law andRaoult’s law

For the solute B in a binary A-B solution, if B obeys Henry’s law,

then Xka = ln ln lna k X= +



then B B B X k a = ln ln ln B B B

a k X = +

ln ln B B

d a d X =Differentiation

ln ln B

A B A

X d a d a

X = − Inserting into Gibbs-

Duhem equation

A

A

A

A

B

B

B

A

B

B

A

B

A X d X

dX

X

dX

X

dX

X

X X d

X

X ad lnlnln ==−=−=−=

ln ln ln( ) A A

a X C = +

0 A Aa C X =

ln ln A A

d a d X =



when ,

Over the composition range of an A-B binary solution in which

the solute B obeys Henry’s law, the solvent A obeys Raoult’s law.

1=ia1=i X , hence, the integration constant ,C0=1.

A Aa X =

材料热力学数据库

1. 物理化学手册（包括英文电子版的，具有搜索功能，浙大

图书馆）http://www.hbcpnetbase.com/ CRC Handbook of Chemistry

and Physics (校图书馆“外文资源”－“ CRC Handbook”)



y ( )

2. JANAF Thermodynamic Tables (3rd edn). American

Chemical Society and American Institute of Physics for the

National Bureau of Standards, New York, 1985.

3. MSI EUREKA 相图数据库 (浙大图书馆，外文资源）

4. SGTE（Scientific Group Thermodate Europe）数据库

国际材料科学相图研究中心（ Materials Science International, MSI ），CEO

Gunter Effenberg 博士曾任 APDIC（国际合金相图委员会）主席。该数据库提供

了经评估过的相图和相关的构成数据，是全球最大的经评估的相图资源。包括的混合物超过45000种，评估过的材料系统约4000个.

introduction to the thermodynamics of materials-2014-chapter 1

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