09 - 11. termodinamika larutan dan aktivitas ion

09 - 11. TermodinamikaLarutan dan Aktivitas Ion

Zulfiadi Zulhan

Teknik MetalurgiFakultas Teknik Pertambangan dan PerminyakanInstitut Teknologi BandungINDONESIA

Termodinamika Metalurgi (MG-2112)

201 Zulfiadi Zulhan MG2112 Termodinamika Metalurgi 2021

Jangan Mengunggah

Materi Kuliah ini di

INTERNET!


1. Pendahuluan, istilah-istilah dan notasi

2. Hukum I Termodinamika

3. Hukum II Termodinamika

4. Hubungan Besaran-Besaran Termodinamika

5. Kesetimbangan

6. Kesetimbangan Kimia dan Diagram Ellingham

7. Proses Elektrokimia dan Diagram Potensial - pH (Pourbaix)

8. Ujian Tengah Semester

9. Termodinamika Larutan

10. Penggunaan Persamaan Gibbs – Duhem

11. Aktivitas Ion

12. Penggunaan Metoda Elektrokimia untuk menentukan Sifat-Sifat / Besaran-Besaran

Termodinamika

13. Keadaan Standar Alternatif

14. Koefisien Aktivitas dalam Larutan Encer Multi-Komponen

15. Diagram Fasa

16. Ujian Akhir Semester

Materi Perkuliahan


dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dP

dG = − S dT + V dP

At consntant temperature: dT = 0

dG = V dP

P ln d RT P

dP RT Gd ==For ideal gas: V = RT/P

Thermodynamic Activity (Module 04)

The change of molar Gibbs free energy of a single gas

with pressure at constant temperature



Fugacity of a condensed phase (solid or liquid) is equal to the fugacity of the vapor in equilibrium

with condensed phase.

At equilibrium, partial molar Gibbs free energies of vapor and condensed phase are equal.

For the difference in partial molar Gibbs free energy between two states at constant

temperature (in term of fugacity):

1

2

f

f

12

f

f ln RT f ln d RT G - G G

2

1

===

Sometimes, it is written as:

1

212

f

f ln RT - =

,...n,nP,T,a

a

cb

n

VV

=

i

nnS,V,innS,P,innV,T,innP,T,iijijijij

n

U

n

H

n

F

n

G=

=

=

=

if ln d RT Gd =



Thermodynamic activity of a material is defined as the ratio of fugacity

of material to the fugacity in its standard state:

i

ii

f

fa

1

2

f

f

12

f

f ln RT f ln d RT G - G G

2

1

=== 1

212

f

f ln RT - =

Thermodynamic Activity is evaluated relative to a standard state at the same temperature.

Activity is set equal to one at the standard state.

In terms of a quantity, Activity is called “fugacity”

Fugacity is property of a gas ~ pressure of non ideal gas.

Activity is the ratio of the fugacity of a material to its fugacity in the standard state.



Although the choice of standard state is arbitrary, in liquid solutions and solid

solutions, the pure materials at one atmosphere pressure and specified crystal

structure are usually taken as the standard state.

iio

ipurei,i a ln RT G-G G-G ==

Example:

Solutions of acetone and water, pure water as standard state for water and pure

acetone as standard state for acetone.

Solid solutions of iron and nickel, pure iron (with specified crystal structure, bcc or

fcc) as standard state for iron, and pure nickel as standard state for nickel.


Thermodynamic Activity

a ln RT f

f ln RT G - G dG i

i

iii

G

G

i ===

if ln d RT Gd =

i

ii

f

fa

P ln d RT P

dP RT Gd ==

Gi = Gi° + RT ln ai

dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dP

dG = − S dT + V dP

At consntant temperature: dT = 0

dG = V dP


Thermodynamic Activity

Δ ሜGA,pure→solution

A A

A

A

A

A

A

A

A

A

Pure A

A

A

A

A

Solution A in B

Gas

fA < fA

fA ~ xA – mol fraction

B B

B

B

B A

AAAA A

Gas

PA = P0A = fA

P0A = saturation

vapor pressure

A A

GA0

GA

ΔG1=0 ΔG3=0

GA,Solution = GA0+ RT ln(

fA

fA0)

GA,Pure = GA,Pure = GA0

Fugacity of a condensed phase (solid or liquid) is equal to the fugacity of the vapor

in equilibrium with condensed phase.

At equilibrium,

partial molar Gibbs

free energies of

vapor and

condensed phase

are equal.

න

𝐺𝐴°

𝐺𝐴

𝑑𝐺𝐴 = GA − GA° = RT ln

fA

fA°ΔG2=

GA − GA° = RT ln aAΔG4=

S: E. Jak

ΔG4= ΔG1 + ΔG2 + ΔG3


Aktivitas(Module 04)

https://www.pikiran-rakyat.com/

https://lokadata.id/

Murni

Nelayan

Murni

Petani

https://dialeksis.com/

https://idxchannel.okezone.com/

Aktivitas N

ela

yan

Aktivitas P

eta

ni

Jumlah Petani

Pure A Pure B

Activity o

f A

, a

A

Activity o

f B

, a

B

0

1

XB

aB =fB

fB° =

fBpXB

fB°

aA =fA

fA° =

fApXA

fA°

ideal solution:

fAp = fa°, then aA = XA

ideal solution:

fBp = fB°, then aB = XB


IDEAL Solution

ai =fi

fi° =

Xi fip

fi° =

Xi fio

fi° = Xi

Gi (solution) = Goi (pure)

+ RT ln Xi

1.0

Activity a

A0

Mol Fraction, XA

0

1.0

Ideal: aA = XA

Murni

NelayanMurni

Petani

Aktivitas N

ela

yan

Aktivitas P

eta

ni

Jumlah Petani

en.wikipedia.org

A

A

AB B

B

B

B A


NON IDEAL Solution (Positive Deviation)


+ RT ln i + RT ln Xi

1.0

Activity a

A

0

Mol Fraction, XA0

1.0aA = A XA

Murni

NelayanMurni

Petani

Aktivitas N

ela

yan

Aktivitas P

eta

ni

Jumlah Petani

A

A

AB B

B

B

B A


NON IDEAL Solution (Negative Deviation)


+ RT ln i + RT ln Xi

1.0

Activity a

A

0

Mol Fraction, XA0

1.0aA = A XA

Murni

NelayanMurni

Petani

Aktivitas N

ela

yan

Aktivitas P

eta

ni

Jumlah Petani

A

A

AB B

B

B

B A


Graphical Representation


Mixing of Ethanol and Water

commons.wikimedia.org

0

10

20

30

40

50

60

70

80

90

100

110

1

Water

50 mL

Ethanol

50 mL

50 mL Ethanol +

50 mL water


Change of Solution Volume by Addition of Moles B

Partial molar volume of a component in a

solution is volume change of solution when one

mole of particular component is added to it

if 50 cm3 water is added to 50 cm3 ethanol, total

volume will be ~ 97 cm3 (not 100 cm3).

The same principles applies to other

quantities: Enthalpy, Entropy, Gibbs free

energynB

V𝐒𝐥𝐨𝐩𝐞 =

𝛛𝐕

𝛛𝐧𝐁 𝐓,𝐏,𝐧𝐀

= ഥ𝐕

Volume of solution A-B as a a function of

moles of B added to the solution


Partial Molar Quantities (Module 03)

,...n,nP,T,a

a

cb

n

VVSlope

==

na

V

Constant T and P

na

V

Volume of a solution (a + b + c + …) as a function of

moles a (na) addedVolume of a as a function of na in

pure a (slopes is V/na = Va)

Constant T and P

If a is added to pure a, partial molar volume would be equal to the molar volume aa VV =


Partial Molar Quantities

Consider volume of solution consisting of materials A and B. Volume of this

solution, at constant pressure and temperature, is a function of the amount of A

(nA) and the amount of B (nB), n represents the number of moles.

P,TBA )n,n(VV =

B

n,P,TB

A

n,P,TA

dn n

V dn

n

VdV

AB

+

=

Bn,P,TA

A

n

VV

=

BBAA dn V dn V dV +=

For dV at constant T and P:

Partial molar volume of A:

Then:


Partial Molar Quantities

BBAA dn V dn V dV +=

Integration equation above :

BBAA n V n V V +=

Dividing by total number of moles (nA + nB)

BA

BB

BA

AA

BA n n

n V

n n

n V

n n

V

++

+=

+

BBAA X V X V V +=

BBAA dX V dX V Vd +=

XA and XB are the mole

fractions of A and B in the

solution.

V is molar volume.


Graphical RepresentationdV = VA dXA + VB dXB

Because XA + XB = 1,

then: dXA + dXB = 0,

dXA/dXB = -1

dV

dXB= VB − VA

At composition XB, tangential

line intersects A axis at , and

B axis atAV

Pure BPure A

XB

V

XB

V

B

AdX

VdX

BV

AV

XA

BV

1: XA𝑑𝑉

dXB T,P= XA𝑉𝐵 − XA𝑉𝐴

2: XB𝑑𝑉

dXB T,P= XB𝑉𝐵 − XB𝑉𝐴

3: 𝑉 = 𝑉𝐴 XA + 𝑉𝐵 XB

B

PT,B

A V dX

VdXV :31 =

++ A

PT,B

B V dX

VdXV :23 =

−−

: dXB


Relative Partial Molar Quantities

By mixing of A and B to form a solution, volume changes:

V- V V V initialfinalMmixing ==

Vfinal is the volume of the solution

Vinitial is the volume of components

before mixing.

Volume change upon mixing (Vm):

V n V n - V n V n V BBAABBAAM −+=

( ) ( ) V - V n V-V n V BBBAAAM +=

A.pure of volume molar to relative solution in A volume molar Partial

solution. enters it as A pure of volume in change The

V A,of molar partial relative called is V-V ermTrel

AAA

rel

BB

rel

AAM V n V n V +=


Relative Partial Molar Quantities

It can be written as:

rel

BB

rel

AAM V n V n V +=

For one mole of solution, this becomes:

In the above equations, volume is used as a representative of thermodynamic

property. The same forms of equations apply to Gibbs free energy, Enthalpy,

Entropy and similar quantities, example:

rel

BB

rel

AAM V X V X V +=

A

o

AA

rel

A a ln RT G G G =−=

For an ideal solution, ai = Xi

i

rel

i X ln RT G =

V n V n - V n V n V BBAABBAAM −+=

( ) ( ) V - V n V-V n V BBBAAAM +=

𝑇erm VA − VA is called relative partial molar of A, V𝐴𝑟𝑒𝑙


Entropy of Mixing: IDEAL SOLUTION

dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dPRelative partial entropy of component A:

( )B

B

n,PAn,P

rel

A

rel

A X ln RT T

-G T

- S

=

=

A

rel

A X ln R- S =

Molar entropy mixing for a solution A + B is:

rel

BB

rel

AAM S X S X S +=

BBAAM X ln X X ln X R- S += =i

iiM X ln X R- S

:componentsmany for general, in

dP V dT S dG +−=

dT S- Gd :ncompositio and pressure constant atrelrel

=


rel

BB

rel

AAM V X V X V +=


Entropy of Mixing, for two gases (Module 03)

Va Vb

Entropy changes for two gases

Consider two gases on either side of a partition, a on the left

and b on the right, each at the same temperature and

pressure.

b12ba12a12 )SS( n)SS( nSS −+−=−

VT

Upon removal of the partition, the two gases will mix.

Ideal gases do not interact, total entropy change is the sum of two entropy changes of the

individual gases:

S2 − S1 = n R ln𝑉𝑡𝑉𝑎

Untuk 1 gas

S2 − S1 = na R ln𝑉𝑎+𝑉𝑏

𝑉𝑎+nb R ln

𝑉𝑎+𝑉𝑏

𝑉𝑏


Entropy of Mixing, for two gases (Module 03)

S2−S1

𝑛𝑎+𝑛𝑏=

na

𝑛𝑎+𝑛𝑏R ln

𝑉𝑎+𝑉𝑏

𝑉𝑎+

nb

𝑛𝑎+𝑛𝑏R ln

𝑉𝑎+𝑉𝑏

𝑉𝑏

∆𝑆𝑚𝑖𝑥 = 𝑋𝑎 R ln𝑉𝑎+𝑉𝑏

𝑉𝑎+ 𝑋𝑏 R ln

𝑉𝑎+𝑉𝑏

𝑉𝑏

Va+Vb

Vb=

na+nb

nb=

1

Xb

∆𝑆𝑚𝑖𝑥 = −R 𝑋𝑎 ln 𝑋𝑎 + 𝑋𝑏 ln 𝑋𝑏

∆𝑆𝑚𝑖𝑥 = −R

𝑖=1

𝑖=𝑛

𝑋𝑖 ln 𝑋𝑖


Enthalpy of Mixing: IDEAL SOLUTION

Gibbs free energy change for a reaction as a function of temperature:

( ) dT S Gd −=

−= S T H G

T

G H S

−=

( ) dTT

H dT

T

G-Gd

−=

Multiply by 1/T

( ) dT

T

H dT

T

G-

T

Gd22

−=

T

1dH

T

Gd

=

Therefore:

rel

A

rel

A

H

T

1d

T

Gd

=

AA

rel

A X ln RT a ln RT G ==

( ) rel

AA H

T

1d

X ln Rd=

AA HH 0 −==

R ln XA is not a function of temperature

dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dP

Conclusion: Enthalpy mixing of

an ideal solution is zero

solution) (ideal 0H XH X Hrel

BB

rel

AAM =+=

rel

BB

rel

AAM V X V X V +=

GM = XA RT ln XA + XBRT ln XB


IDEAL SOLUTION

Volume change upon mixing for ideal solutions:

V P

G

T

=

rel

A

T

rel

AV

P

G=

( )0V

dP

X ln RTd rel

AA ==

solution) (ideal 0V XV X Vrel

BB

rel

AAM =+=

=i

iiM X ln X R- S

=i

iiM X ln X T R G

0 HM =

0 VM =

dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dP


IDEAL SOLUTION

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

VM

XB

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

HM

XB

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S M

XB

-12,000

-10,000

-8,000

-6,000

-4,000

-2,000

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

GM

XB

1800 K

1500 K

1200 K

900 K

600 K

300 K

Materials A and B can spontaneously form a solution

because the change in Gibbs free energy is negative

upon mixing.


Non Ideal Solutions: REGULAR SOLUTIONS

In 1895, Margules suggested the activity coefficients, A and

B, of the components of a binary solution can be

represented by power series:

ln γA = α1XB + 12 α2XB

2 + 13 α3XB

3 + …

ln γB = β1XA + 12β2XA

2 + 13β3XA

3 + …

ln γA =Ω

RTXB2 =

Ω

RT1−XA

2

In 1929, Hildebrand suggested the following equation to

predict activity coefficients, A and B, in binary solution A-B:

ln γA =α′

RTXB2 =

α′

RT1−XA

2

Ω = temperature dependent interaction parameter.

https://en.wikipedia.org/



Regular Solution: a non ideal solution whose thermodynamic properties can be described

using a simple agebraic function due to similarity between atoms A and B (no strong

interaction & both atoms randomly mixed

rel

BB

rel

AAM G X G X G +=

( )AAA

o

AA

rel

A X ln RTa ln RT G G G ==−=

( ) ( )BBBAAAM X ln RT XX ln RT X G +=

( ) ( ) ln X ln XRTX ln XX ln X RT G BBAABBAAM +++=

xs

M

ideal

MM G G G +=


GM = GMideal + GM

xs



A Regular Solution of two components is defined as one in which the activity coefficient of A

has the form:

ln 𝛾A =Ω

RTXB2 =

Ω

RT1−XA

2

GM = RT XA ln XA + XB ln XB + 𝑅𝑇 XA ln 𝛾A + XB ln 𝛾B

GM = RT XA ln XA + XB ln XB + XAΩXB2 + XBΩXA

2

GM = RT XA ln XA + XB ln XB + XA + XB ΩXA XB

GM = RT XA ln XA + XB ln XB + ΩXA XB



ln 𝛾A =Ω

RTXB2 =

Ω

RT1−XA

2

GM = RT XA ln XA + XB ln XB + ΩXA XB

GM = HM−T SM

HM = Ω XA XB

SM = −R XA ln XA + XB ln XB

Entropy mixing in regular solution is equivalent to entropy mixing in ideal solution.

xs

M

ideal

MM G G G +=

M

xs

M

xs

M HH G ==

0 Sxs

M =

SMideal = −R

i

Xi ln Xi

GMideal = R T

i

Xi ln Xi

HMideal = 0

HM = HMideal + HM

xs

SM = SMideal + SM

xs

GMxs = HM

xs − T SMxs


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Act

ivit

y (a

)

XB

Non Ideal Solutions: REGULAR SOLUTIONS (300)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Act

ivit

y co

effi

cien

t (

)

XB

-3000

-2500

-2000

-1500

-1000

-500

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

HM

XB

-4,500

-4,000

-3,500

-3,000

-2,500

-2,000

-1,500

-1,000

-500

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

GM

XB

ln 𝛾A =Ω

RTXB2 =

Ω

RT1−XA

2

HM = Ω XA XB

GM = GMideal + GM

xs

GMxs = HM

xs = HM

Ω = -10.000 JΩ = -7.000 J

Ω = -4.000 J

IdealIdeal


Non Ideal Solution: Negative deviation from Raoult’s law

D.R. Gaskell, 2003


0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

HM

XB

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Act

ivit

y (a

)

XB

Non Ideal Solutions: REGULAR SOLUTIONS (300 K)

ln 𝛾A =Ω

RTXB2 =

Ω

RT1−XA

2

HM = ΩXA XB

xs

M

ideal

MM G G G +=

M

xs

M

xs

M HH G ==

Ω = 10.000 JΩ = 7.000 J

Ω = 4.000 J

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Act

ivit

y co

effi

cien

t (

)

XB

Ideal

-2,000

-1,500

-1,000

-500

0

500

1,000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

GM

XB

Ideal


Non Ideal Solution: Positive deviation from Raoult’s law

D.R. Gaskell, 2003



M

xs

M

xs

M HH G ==

0 Sxs

M =

0S- T

G xs

M

xs

M ==

etemperatur of ntindenpende is Gxs

M

XS

BB

XS

AA

xs

M G X G X G +=

2

B)A(T2)A(T1

XS

A X ln RT ln RT G21

===

Practical use: converting activity data for regular solution at one temperature to activity data at

another temperature

2

1

1A

2A

T

T

T etemperatur at ln

T etemperatur at ln =

BBAA X V X V V +=

dU = T dS - P dV

dH = T dS + V dP

dF = -S dT - P dV

dG = -S dT + V dP



etemperatur of ntindenpende is Gxs

M

XS

BB

XS

AA

xs

M G X G X G +=

2

B)A(T2)A(T1

XS

A X ln RT ln RT G21

===

2

1

1A

2A

T

T

T etemperatur at ln

T etemperatur at ln =

D.R. Gaskell, 2003


Case 1

Copper and gold form complete ranges of solid solution at temperature between 410 and

889oC. At 600oC, the excess molar Gibbs free energy of formation of solid solution is given

by:

GMxs = - 28,280 XAu XCu Joule

Calculate the activities of Cu and Au in the solid solution of XCu =0.4 at 600oC. Calculate

partial pressures of Cu and Au!

ln 𝑝𝐶𝑢𝑜 (atm) = -

40,920

𝑇− 0.86 ln 𝑇 + 21.67

ln 𝑝𝑍𝑛𝑜 (atm) = -

45,650

𝑇− 0.306 ln 𝑇 + 10.81


Case 2

At 473 oC the Pb-Sn system exhibits regular solution behavior, and the activity coefficient of

Pb is given by:

log (Pb) = -0.32 (1-XPb)2

Write the corresponding equation for the variation of Sn with composition at 573 oC!


Case 3&4

3. In a liquid Cu-Sn alloy at 1400K and at XSn =0.4, the values of activitiy of Sn and Cu

are 0.333 and 0.362, respectively. Calculate Sn, Cu, , !

4. The variation with composition of Gxs for liquid Fe-Mn alloys at 1863 K is listed

below:

a. Does the system exhibit regular solution behavior?

b. Calculate GFexs

and GMn

xsat XMn = 0.6

c. Calculate GM at XMn = 0.4

d. Calculate the partial pressure of Mn and Fe exerted by the alloy of XMn = 0.2

XMn 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Gxs, J 395 703 925 1054 1100 1054 925 703 395

xs

MM G ,G MM S ,H


Non Ideal Solutions

In non ideal solutions, the curve of specific Gibbs free

energy of the solution as function of composition may

have many forms.

In the region between XB1 and XB2, solution

minimizes its free energy by decomposing

into two solutions. A single solution in this

region is unstable relative to mixture of

solutions of XB1 and XB2.


Non Ideal Solutions

Non ideal behavior: two immiscible materials, materials A and B form no solutions.

Molar Gibbs free energy against the

composition is simply a straight line

between the two points on the A and B

axes.


Non Ideal Solutions

Materials A and B form a compound AB, which does

not dissolve in either A or B


Thermodynamic Models of Solutions

Empirical models - polynomials• Regular

• Margules (2 and 3 suffix)

• Interaction parameter model for dilute solutions

Structurally-based models

• Masson polymer model

• Quasi-chemical

• Cell

• Central atom

• Sub-lattice (for solids)

• Associate Solution Model


Regular / Subregular Model

A

A

AB B

B

B

B A

Eo = ωAA EAA + ωBB EBB +ωAB EAB

𝜔𝐴𝐴 : number of bonds associated with the formation of AA

𝜔𝐵𝐵 : number of bonds associated with the formation of BB

𝜔𝐴𝐵 : number of bonds associated with the formation of AB

𝐸𝐴𝐴 : Energy associated with the formation of AA

𝐸𝐵𝐵 : Energy associated with the formation of BB

𝐸𝐴𝐵 : Energy associated with the formation of AB

If N atoms in solution and z coordination number,

number of bonds:

ωAA =1

2N z XA

2

ωBB =1

2N z XB

2

ωAB =12 N z XAXB

Eo =N z

2XA EAA + XB EBB + XA XB 2EAB − EAA − EBB

HM =N z

2XA XB 2EAB − EAA − EBB

HM = GMxs = ΩXA XB (REGULAR SOLUTION)

If the reference states are taken as pure A and B:

GMxs = XA XB Ω𝐴𝐵

𝐴 XA +Ω𝐴𝐵𝐵 XB

Kaufman & Bernstein, 1970, Subregular Model:GM = RT XA ln XA + XB ln XB + 𝑅𝑇 XA ln 𝛾A + XB ln 𝛾B

GM = GMideal + GM

xs


Extrapolation of Gibbs Excess Energy

A

A

AB B

B

B

B A

GM = RT XA ln XA + XB ln XB + 𝑅𝑇 XA ln 𝛾A + XB ln 𝛾B

GM = GMideal + GM

xs

Muggianu equation, 1975:

GMxs = XA XB 𝐿𝐴𝐵

𝑜 + 𝐿𝐴𝐵1 XA − XB + XB XC 𝐿𝐵𝐶

𝑜 + 𝐿𝐵𝐶1 XB − XC + XA XC 𝐿𝐴𝐶

𝑜 + 𝐿𝐴𝐶1 XA − XC

Kohler equation, 1960:

GMxs = XA + XB

2XA

XA + XB

XB

XA + XB𝐿𝐴𝐵𝑜 + 𝐿𝐴𝐵

1 XA − XB

XA + XB

+ XB + XC2 XB

XB + XC

XC

XB + XC𝐿𝐵𝐶𝑜 + 𝐿𝐵𝐶

1 X𝐵 − XC

XB + XC

+ XA + XC2 XA

XA + XC

XC

XA + XC𝐿𝐴𝐶𝑜 + 𝐿𝐴𝐶

1 XA − XC

XA + XC

L : excess interaction parameter

Toop equation, 1975:

GMxs = XA XB 𝐿𝐴𝐵

𝑜 + 𝐿𝐴𝐵1 XA − XB− XC + XA XC 𝐿𝐴𝐶

𝑜 + 𝐿𝐴𝐶1 XA − XC − XB

+ XB XC 𝐿𝐵𝐶𝑜 + 𝐿𝐵𝐶

1 XA − XC +XB − XC

XB + XCXA


Quasichemical Model


NON-IDEAL SOLUTIONS

A lot of substances formordered solutions with complex functionsof properties in terms of compositions

Appropriate models have to be used to adequately describethermodynamic properties

e.g.:molten salts slag – molten oxidesquasi-chemical solution model is used

Solution with

strong interatomic

interactions

resulting in very

negative MH

(-84) as well as in

strong ordering

(so that that MS

is close to 0)

Enthalpy MH

of mixing

Entropy MS

of mixing


Gibbs-Duhem Relation

Molar Gibbs free energy of an A-B solution can be written as:

BBAA X G X G G +=

BBAABBAA Gd X GdX dX G dX G Gd +++=

Differentiation:

P,TBA )n,n(GG =

B

n,P,TB

A

n,P,TA

dn n

G dn

n

GdG

AB

+

=

Bn,P,TA

A

n

GG

=

Compare:

BBAA dX G dX G Gd +=

0Gd X GdX BBAA =+

BBAA X V X V V +=

XAdഥHA + XBdഥHB = 0

XAdതSA + XBdതSB = 0

en.wikipedia.org


Gibbs-Duhem Relation: Activity

This equation is known as Gibbs-Duhem equation.

0Gd X GdX BBAA =+

( ) ( ) 0a ln RTd X a ln RTdX BBAA =+

( ) ( )B

A

BA a lnd

X

X- a lnd =

If the activity of one component of a binary solution is known as a

function of composition, then the activity of the other can be

determined.

aNi experimental

aFe ??

Gibbs Duhem Reation is used to find activity of a

component based on experimental activity of another

component in binary system if the solution is not a

regular solution


As XB → 1, aB → 1, log aB → 0

and XB / XA → ∞.

The curve exhibits a tail to infinity

as XB → 1

As XB → 0, aB → 0, log aB→

-∞ and XB / XA → 0.

The curve exhibits a tail to

minus infinity as XB → 0

Gibbs-Duhem Relation: Activity

( ) ( )B

A

BA a lnd

X

X- a lnd =

If the variation of aB with

composition is known, then

integration of above equation from

XA = 1 to XA gives the value of log

aA at XA:

( ) ( )B

A

BA a logd

X

X- a logd =

( )=

==

AB

AB

X at a log

1 X at a logB

A

BAA a logd

X

X - X at a log


Case 1:

A and B are in a solution, material B behaves ideally.

Activity of B is equal to mole fraction of B (pure B as standard state)

( ) ( )B

A

BA a lnd

X

X- a lnd =

( ) ( )B

A

BA X lnd

X

X- a lnd =

( )B

B

A

BA

X

dX

X

X- a lnd =

1X X BA =+

BA dX- dX =

( )A

AA

X

dX a lnd =

AA X a =

If material B in a solution of A and B behaves ideally, then material A

behaves also ideally.


Gibbs-Duhem Relation: Activity Coefficient

( ) ( ) 0X ln d X X ln d X BBBAAA =+

( ) ( ) ( ) ( ) 0X ln d XX ln d X ln d X ln d X BBAABBAA =+++

( ) AAA dXX ln d X =

( ) BBB dXX ln d X =

0dXdX BA =+

( ) ( ) 0 ln d X ln d X BBAA =+

( ) ( )B

A

BA lnd

X

X- lnd =


Activity and Activity Coefficient

At XB/XA=0 → aB = 0

ln aB= ∞ (infinite) At XB/XA=0 → γB = γBo

ln γB = ln γBo

(finite number)

lnγA = නXA=1

XA=XA

−XBXA

dlnγB = නXA=XA

XA=1 XBXA

dlnγBln𝑎A = නXA=1

XA=XA

−XBXA

dln𝑎B = නXA=XA

XA=1 XBXA

dln𝑎B


Case 2:

Copper zinc alloy. Activity of zinc is easy to measure because zinc is vollatile; it is possible to

measure zinc pressure in equilibrium with zinc containing liquids or solids. Activity of zinc is zinc

vapor pressure relative to vapor pressure of pure zinc. In temperature range 1400-1500K, an

expression that fits the data for activity coefficient of zinc is:

2

CuZn X -38,300 ln RT =

Applying Gibbs Duhem equation:

( ) ( )Zn

Cu

ZnCu lnd

X

X- lnd =

( )

−=

RT

X 300,38d

X

X- lnd

2

Cu

Cu

ZnCu

( ) CuZnCu dX RT

300,38 x 2X lnd =

ZnCu dX dX −=

( ) ZnZnCu dX XRT

300,38 x 2 lnd −=

2

ZnCu XRT

300,38 ln −=


Case 3:

lnγCd|XCd=0.1 = නXCd=0.1

XCd=1 XZnXCd

dlnγZn

The activity of zinc in liquid cadmium-zinc alloys at 708K is related to the alloy composition

by the following equation:

ln γ𝑍𝑛 = 0.87 𝑋𝐶𝑑2 − 0.3 𝑋𝐶𝑑

3

Calculate the activity of cadmium at XCd = 0.1!


Case 3:

lnγCd|XCd=0.1 = නXCd=0.1

XCd=1 XZnXCd

dlnγZn

XCd XZn ln γZn XZn/XCd

0.10 0.90 0.01 9.00

0.20 0.80 0.03 4.00

0.30 0.70 0.07 2.33

0.40 0.60 0.12 1.50

0.50 0.50 0.18 1.00

0.60 0.40 0.25 0.67

0.70 0.30 0.32 0.43

0.80 0.20 0.40 0.25

0.90 0.10 0.49 0.11

1.00 0.00 0.57 0.00


0.10 0.90 0.01 9.00

0.20 0.80 0.03 4.00

0.30 0.70 0.07 2.33

0.40 0.60 0.12 1.50

0.50 0.50 0.18 1.00

0.60 0.40 0.25 0.67

0.70 0.30 0.32 0.43

0.80 0.20 0.40 0.25

0.90 0.10 0.49 0.11

1.00 0.00 0.57 0.00


0.10 0.90 0.01 9.00

0.20 0.80 0.03 4.00

0.30 0.70 0.07 2.33

0.40 0.60 0.12 1.50

0.50 0.50 0.18 1.00

0.60 0.40 0.25 0.67

0.70 0.30 0.32 0.43

0.80 0.20 0.40 0.25

0.90 0.10 0.49 0.11

1.00 0.00 0.57 0.00


0.10 0.90 0.01 9.00

0.20 0.80 0.03 4.00

0.30 0.70 0.07 2.33

0.40 0.60 0.12 1.50

0.50 0.50 0.18 1.00

0.60 0.40 0.25 0.67

0.70 0.30 0.32 0.43

0.80 0.20 0.40 0.25

0.90 0.10 0.49 0.11

1.00 0.00 0.57 0.000

1

2

3

4

5

6

7

8

9

10

0.00 0.10 0.20 0.30 0.40 0.50 0.60

XZnXCd

lnγZn

lnγCd = 0.59 → γCd = 1.81 → 𝑎Cd = 0.18

Area of trapezoid 1 =9 + 4

2(0.03 − 0.01)


Case 4:

From vapour pressure measurement, the following values have been determined for the

activity of mercury in liquid mercury bismuth alloys at 320oC. Calculate activity of bismuth in a

50 atom-percent Bi alloy at this temperature!

XHg 0.949 0.893 0.851 0.753 0.653 0.537 0.437 0.330 0.207 0.063

aHg 0.961 0.929 0.908 0.840 0.765 0.650 0.542 0.432 0.278 0.092

( ) ( )Hg

Bi

Hg

Bi a logd X

X- a logd =

( )=

==

BiHg

BiHg

X at a log

1 X at a logHg

Bi

Hg

BiBi a logd X

X - X at a log


Aqueous Solution

Solutions of electrolytes are non‐ideal at relatively low concentrations.

The greater the charge on the ions the larger the deviations from ideality.

For example, for one mole of CaCl2 dissolved the deviation from ideal behavior is larger than

from one mole NaCl due to the 2+ charge of calcium ion.

Perhatikan disosiasi dari molekul berikut:

V+ = koefisien stoikiometrik

Z+ = muatan positif

Z- = muatan negatif

Contoh: 2

4

2

42 OS H 2 OS H−+−+ +=

i

o

ii a ln TR +=

μi = μio + RT ln γi mi

ai = i mi

ai = aktivitas i

i = koefisien aktivitas i

mi = konsentrasi molal i

Untuk larutan ideal, i = 1 : ai = mi

z zz

v

z

v B v A v B A −

−

+

+

−

−

+

+ +=


Aqueous Solution

i

o

ii mln T (ideal) R+=

ln T mln T (real) ii

o

ii RR ++=

ln TR (ideal) -(real) iI-iii ==

Menurut teori Debye-Hueckel, perubahan potensial kimia yang

timbul dari interaksi ion-ion adalah sebagai berikut:

( )1-

2

0iAI-i

2

e z N -

=

( )1-

2

0iAi

2

e z N - ln TR

=

NA= bilangan Avogadro (6,022 x 1023 / mol)

zi = bilangan muatan ion π = 3,14

eo = muatan elektron (1,602 x 10-19 C)

= kostanta dielektrik (78 for water)

-1 = Debye Hueckel reciprocal length (m)

k = konstanta boltzmann (1,380 x 10-23 J/K)

ci = Konsentrasi dalam molal


I = ionic strength

𝜅 =8 𝜋 𝑁𝐴 𝑒𝑜

2

1000 𝜀 𝑘 𝑇

I = 12σ ci zi

2


Debye-Huckel Equation


Limiting Law (for I < 10-2)

Individual ion: log 𝛾𝑖 = − 𝐴 𝑧𝑖2 𝐼

Mean activity coefficient: log 𝛾 = − 𝐴 𝑧+ 𝑧− 𝐼

A = 0.509 at 25 C

Extended DH (for I < 10-1)

Individual ion: log 𝛾𝑖 = −𝐴 𝑧𝑖

2 𝐼

1+ 𝑎𝑖 𝐵 𝐼

Mean activity coefficient: log 𝛾 =−−𝐴 𝑧+ 𝑧− 𝐼

1+𝑎 𝐵 𝐼

A = 0.509 at 25 C

B = 0.33


Davies Equation

Davies (for I < 0.5)

Individual ion: log 𝛾𝑖 = −𝐴 𝑧𝑖2 𝐼

1+ 𝐼− 0.2 𝐼

Mean activity coefficient:

log 𝛾 = −𝐴 𝑧+ 𝑧−𝐼

1 + 𝐼− 0.2 𝐼

A = 0.509 at 25 C


Case 1

Calculate the ionic strength and the mean activity coefficient of 2.0m mol kg-1 Ca(NO3)2 at 25 oC.

Solution:

I = ½(22*0.002 + (-1)2*(2*0.002)) = ½ *6*0.002 = 0.006

Debye-Huckel limiting equation:

log 𝛾 = − 𝐴 𝑧+ 𝑧− 𝐼

log 𝛾 = - |2*1|*A*(0.006)1/2

= - 2*0.509*0.0775= -0.0789

𝛾 = 0.834

I = 12σ ci zi

2


Terima kasih!Zulfiadi Zulhan

Program Studi Teknik Metalurgi

Fakultas Teknik Pertambangan dan Perminyakan

Institut Teknologi Bandung

Jl. Ganesa No. 10

Bandung 40132

INDONESIA

www.metallurgy.itb.ac.id

[email protected]

09 - 11. termodinamika larutan dan aktivitas ion

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