Thermodynamic properties of compound-forming liquid alloys

Journal of Non-Crystalline Solids 117/118 (1990) 505-512 North-Holland

505

Section 4: Thermodynamics

THERMODYNAMIC PROPERTIES OF COMPOUND-FORMING LIQUID ALLOYS Ferdinand SOMMER

Max-Planck-Institut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestr. 75, D-7000 Stuttgart 1, F.R.G. The experimentally determined properties of compound-forming liquid alloys show specific concentration and temperature dependences,which are caused by the existence of chemical short range order in these alloys,The concentration and temperature dependences of thermodynamic mixing functions are explained on the basis of homogeneous equilibria reactions within an association model for a simple description of chemical short range order,The ability of this model to predict thermodynamic properties of a binary liquid alloy from experimental results and to extrapolate it to ternary liquid alloys is demonstrated. I. INTRODUCTION

mation is small with a maximum value of the enthalpy

The experimental determination of thermodynamic pro-

of mixing ( 4kJmo1-1 > A H > - 4 kJmol-1).The in-

perties of alloys is the basis to understand the energetics

tegral values of the thermodynamic mixing functions for

of alloy formation.Recent progress in this area is mainly

these alloys exhibit a parabolic shaped concentration de-

achieved in application of well known principles resulting

pendence and are implicitly temperature independent

in improved accuracy of the measurements and its extended scope. 1,2 The integral and partial enthalpies of for-

as described by the regular solution model.Deviations from this concentration dependent behaviour of the ther-

mation as a function of concentration and temperature

modynamic mixing functions are very sensitive indica-

can be obtained directly by calorimetric methods.They

tors for the occurrence of chemical short range order

can also be determined from the temperature dependence of the partial Gibbs free energy of formation obtained by

centration the A H - z - c u r v e changes from parabolic

partial vapour pressure measurements or e.m.f, measu-

to a triangular shaped curve with increasing CSRO and

rements on suitable galvanic cells,however,generally with a lower precision.The entropy of mixing is then given by

also the maximum negative value of A H increases with

the Gibbs-Helmholtz equation.The thermodynamic mixing functions of liquid alloys with compound-formation

(CSRO).With maximum CSRO at the equiatomic con-

increasing CSRO.For example Li-Mg,A1-Li and Sn-Te systems show this trend (see Figs.l-3). The corresponding entropy of mixing/kS shows a minimum at the eqniato-

tendency show,in most cases, strong deviations from regular solution behaviour ( see paragraph 2,3 ), which

lues o f / k S with increasing C S R O . / k S exhibits negative

mic concentration which decreases even to negative va-

is caused by the existence of chemical short range or-

values if the positive configurational contributions are

der.This has been established separately by the results

overcompensated by contributions due to changes in eiec-

of structural investigations and other structure sensitive properties 3,4 The use of homogeneous equilibrium reac-

trouic,vibrational or magnetic behaviour.The position of

tions to describe these chemical short range order ten-

the maximum o f / k H and the minimum of/kS differs, for most of the liquid alloys with compound-forming ten-

dencies enables a simple description of the concentration

dency, from the eqniatomic concentration.The position

and temperature dependences of the thermodynamic mi-

of these extremas conforms , however, often to the stoichiometry of intermetallic phases or compounds. The

x-ing functions of liquid alloys with compound-forming tendency(see paragraph 4,5).

position coincides generally with the highest melting and therefore most stable phase , if compounds with dif-

2. SPECIFIC C O N C E N T R A T I O N DEPENDENCES OF T H E R M O D Y N A M I C MIXING FUNCTIONS OF LIQUID A L L O Y S WITH C O M P O U N D - F O R M I N G T E N D E N C Y

ferent stoichiometries exsist. The thermodynamic mixing functions for some liquid alloys show on the other

The atoms in a liquid alloy are nearly statisticallydis-

hand extrema at concentrations different from the stoi-

tributed if the change in pair interactions by alloy for-

chiometry ofintermetallic phases. The A H - and/kS-values of liquid Mg-Ca alloys,for example,have a maximum

0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

506

F. Sommer / Compound-forming liquid alloys and a minimum at equiatomic concentration , respecti-

"7

-g

vely , whereas the intermetailic phase has the composition Mg2Ca s . ~"urthermore in some simple eutectic systems

TAS

the extremas are found at concentrations where meta-

0

stable crystalline phases can be obtained by rapid solidification (e.g.Au-Ge 4). The partial eltropies ASi of a binary alloy with compound-forming tendency often show an S-shaped beha.2

.4

L i

.6

.B

xMe

1.0 M8

Figure 1 : Enthalpy, entropy and Gibbs free energy of mixing of liquid Li-Mg alloys at 940 K calculated using the association model (ooo ~ experimental). 5

"7

viour. The position of the inflection point corresponds to the maximum in CSRO (e.g.Zn-Sbg). If the change in concentration by mixing the liquid components during a calorimetric measurement is smM1 for each successive measurement (i.e. < 1 at.%) then the derivative of A H can be directly determined in a good approximation by

0

aA/~(x) I

E

~

~

• ,+T

0

=

all(x1 + Zx~)-/,H(~I) A~

(1)

The directly measured derivatives of the enthalpy of mixing are more sensitive to deviations from regular solution behaviour compared to integral values ( e.g. Fig.4,5). The second derivative of the Gibbs free energy with re-

-5

-

-i0

spect to concentration or the first derivative of the acti0

.2

.4

hl Figure 2 :

.6

.8

xu~

.0

Li

Enthaipy, entropy and Gibbs free energy of mixing of liquid and undercooled A1-Li alloys at 973 K calculated using the association model (ooo 6 experimental).

vity with respect to concentration,the so-cailed stability function of Darken,is a further good indicator of deviations from statistical distribution of the atoms.The stability function of liquid alloys with compound-forming tendency is positive and its maxima coincides with the maxima of CSRO .The stability function is furthermore directly connected with the concentration fluctuations Sxx(q --~ 0) in an alloy for a wave number q in the long

"7 0

E

wavelength limit,otained from diffraction experiments tl S,x(q -~ o) = RT/(O2aa/Ox~)T,p

-10

(2)

The concentraton fluctuations are small at concentrati-20

ons having large stability functions and show there strong deviation from S~d for the ideal solution ( S ~ = XAXB)

lllllllalllltlllllhllt!llllll'lllllllllillllm

.2 Sn

.4

.6 xTe

.8

1.0 Te

Figure 3 : Enthalpy,entropy and Gibbs free energy of mixing of liquid Sn-Te alloys at 1140 K calculated using the association model (ooo 7 experimental).

(see Fig.6). Phase separation in liquid alloys can only occure if A H is positive according to the regular solution model.However some alloys with strong compound-forming tendency and high negative values of A H (e.g. Ag-Te ,Ga-Te ,T1-Te ) show phase separation in the liquid state on the Ag-,Ga-,

F. Sommer / Compound-forming liquid alloys

507

and Tl-rich sides, respectively. In these systems exists i

L

0

j

zTAq

f

a high CSRO with maximum value at the concentrati-

E

0

ons of the following intermetallic phases Ag2Te, Ga2Tls

-

-10

gap A H exhibits a positive deviation instead of a tri-

"

-20

and TI2Te. In the concentration range of the miscibility angular shaped concentration dependence 4 , which may be expected in the case of complete ordering. The po-

f~" -30

sitive deviation of the &H-values from linear concentral~lhl

Hh

0

IHI

llllhll

.2

I]I HlillHhl

.4

llilHll

.6

.8

xpb

Li

I HI

1.0

Pb

tion dependence can be explained assuming positive interactions between the ordered atoms and the excess metal atoms which are present in this concentration range.

Figure j : F,nthalpy,entropy and Gibbs free energy of

The positive interactions between the ordered atoms and

mixing of liquid Li-Pb alloys at 1000 K calculated using the association model (ooo 10 experimental).

the excess metal atoms in these systems,which exhibit understandable if one takes into account that the partial ionic bond in ordered atoms has an entirely different

5O

I

altogether strong compound-forming tendency, becomes

character from the metallic bond of excess atoms.In liO

E

quid metal salt mixtures e.g. Cs-CsCl,one finds, positive

0

..M

A H values and herewith positive interactions between

-50

the metal atoms and the iouically bonded salt.

x--o \ -I00

3. SPECIFIC T E M P E R A T U R E DEPENDENCES O F T H E R M O D Y N A M I C MIXING F U N C T I O N S O F LIQUID ALLOYS WITH COMPOUND-FORMING TENDENCY

-150 t l l ~ l l

0 Li

it ill

till

.2

I It lilt

lit

.4

11 I I l l l l l

I IIIllllll,

.6

.8

x~

1.0 Pb

Figure 5 : Derivative of the enthalpy of mixing of liquid Li- Pb alloys at 1000 K calculated using the association model (ooo 10 experimental).

The excess molar heat capacity for a binary alloy is given by

ACp(x, T) = C,(z, T) - zACA(T) -- xsC~(T)

(3)

Cp(x, T), C~(T) and C~ are the molar heat capacities of the alloy and the pure components respectively. Experimental values of ACp(x, T) give,regarding the thermo-

.25

dynamic properties , the most direct evidence of CSRO. •

20

p i

f

•

i

•

ACp(x, T) of compound- forming liquid alloys should be •

positive because the energy which is necessary to heat

c~ .15

an alloy melt with CSRO is bigger compared to an alloy X .10 ~O

where the atoms are statistically distributed.The posi-

05

tion of the maximum positive value of ACp(x, T) thus

•

yields a direct indication of maximum CSRO.AC~(x, T)

O. 0 Li

.2

.4

.6 Xpb

.8

1.0 Pb

Figure 6 : Concentration fluctuations of liquid Li- Pb alloys at 930 K calculated using the association model ( - - - S ~ ,ooo 12 experimental).

should additionally decrease with increasing temperature because the CSRO decreases with increasing temperature.For some liquid alloys the measured ACp values are constant in the temperature range of measurement.These systems exhibit a very strong compound- forming tendency and the CSRO is within the experimental uncertainty constant in the temperature interval of measu-

F. Sommer / Compound-forming liquid alloys

508

rement 13 .This temperature interval is generally small

and

because of experimental problems. For a small number of binary alloy melts the ACp de-

/kS = --R(nA~InxA~ + nntInxst

+ nA, SjAS~,s~ (5)

pendence on concentration exhibits two maxima and the temperature dependences of these maxima are often different e.g.(In-Sb 14, Na-Pb xS).Thermodynamically there are no strong experimental indications of the existence of more than two different kinds of CSRO in binary liquid alloys . With positive and temperature dependent ACp values the negative values of A H increase with decreasing temperature and the positive values of /kS decrease with decreasing temperature.

+ nAiBjlnxA,,~)

where hA1 and ns~ moles free A and B atoms are in equilibrium with nA,Bj moles of associates having the composition AiBj (i~j=l,2...,n) .XA~,Xs, and XA,Sj are the molar fractions of the assumed species for 1 tool of a binary alloy.The eqnilil~rium value for nAiB j is determined by a mass action law with an association constant defined by

gAiZj

=

exp[-(AH~t,,¢ - TAS~,Bj)/RT]

(6)

The partial enthalpies at infinite dilution A]~i can be determined calorimetrically in many binary liquid alloys

where /kH~B~ and AS~4~B~ represent the enthalpy and

in a large temperature range (e.g.Fig.15).For the same

entropy of formation of the associates.For the activity coef~cient ~ one obt~ns ~om eqns(4) to(6)

binary a l l o y / k ~ of one component is temperature dependent w h i l e / k ] ~ of the second component is tempetemperatures A ~

..t- f-yreg

assumes constant values. --

XA1)XAIBj -- CrB:g,A,BjXBIXA,Bj)/RT]

4. BASIC FORMULAE OF THE ASSOCIATIONMODEL The use of homogeneous equilibrium reactions for a simple description of CSRO in liquid alloys with compound-

,~,eg

--

(l

(7)

(1

X B , ) X A I B j -- CrA:g,A i B j X A I X A , B j ) / R T ]

(8)

forming tendency enables to describe the different kinds The expression of excess molar heat is given by 20

of concentration and temperature dependences of the integral and partial thermodynamic mixing functions, which have been discussed in paragraph 2 and 3.The

ACp(x,T) = nT[R(-ilnXAt - jlnxB1 + InZAiB~)

short range ordered volume parts are summarily descri-

[2(c~:~s, [ij + (i + j - 1){-jXA, - ixs~

bed as associates with a well-defined composition,while

+ ( i + j - 1)~A,s~)]

the rest of the atoms axe regarded to be randomly dis-

+ c ~ : ~ A ,s~ [ - i + (i + j - 1){~A, -

tributed.The associates are in a steady dynamic equili-

+ ( i + j - 1)~AI~A,B~}]

brium with the non associated atoms,which is governed

+C~:g,A,sj [-j + (i + j - 1){ZSl - jzA,sj

by the mass action law.The lifetime and the spatial ar-

+ ( i + j - 1)xs,~A,s~}])

rangement of the short range ordered regions does not

RT[

enter into the model.Particulars about the basis of this

5

+ j= + ~ XB1

-

o 2 ASAIBj]

i~A,sj

- (i + j -1)21] -1

(9)

ZAIBj

model , the development and application by other authors may be taken from the following references 1L16,1T

with n = nAt 4- nB 1 -4- nAiBj .The temperature depen-

The expressions for A H and /kS for binary alloys with

stoichiometry of the associates.For i , j > 1, A H ° exhi-

the formation of a single association type while neglecting volume effects are 4,t8

bits no temperature dependence 4,1s

AH

-]-

rt

+

reg nBtnA'BjCBt'AIBj n

n

~- n A , B j A H ~ I B j

(4)

dences of the partial enthalpies are influenced by the

~

= c~:~

(10)

A~

= CA1 "~g,BI

(11)

For i = j = l , o n e obtains for / X ~ using eqns. (4)-(6)

F. Sommer / Compound-forming liquid alloys

AM

=

~A~,S, + ~ g

RT2OInK1/OT

&,B1 +

-

RT2OInK2/OT reg

(12) (13)

reg

where K1 = I ( AIB j exp[( C A~.B, -- CB, ,AIBj ) / RT] and K2 = K,,~m,p[(cZ~,i = 1,j > 1

CZ~,,~)/RT).For

,eqn.(12) is valid for A M

509

where hAl, riB,, nc, are the number of moles of free A,B and C atoms in equilibrium with nAisj, nB~cl and nAhCl moles of associates of the composition A i B j ( i , j = 1, 2..., n), B , Co(u,v = 1 , 2 . . , n ) , A k C l ( k , l = 1,2 .... n) and the and x ahc~ are the mole fractions of the respective species in 1 mole of ternary alloy. X A1, X BI , XCt , X AiBj ~X BuC.

and eqn.(ll)is valid for A ~ .

For i > 1,j = 1, A ~ is temperature independent (eqn. (10)) and A ~ is temperature dependent (eqn.13)). It

5. APPLICATION TO LIQUID BINARY AND TERNARY ALLOYS

can be shown that the association model predicts an S-

It is necessary to define the stoichiometry of the asso-

shaped temperature dependence for A H-° with two limiting values at high and low temperatures 19,20

ciates to calculate the thermodynamic functions using

For alloys with high associate fractions and high values for the enthalpy and entropy of formation per atom of the associates, the A H ° and A S ° generally cannot be regarded as independent of temperature.The following relationship can be used for a temperature T2 if A H ° and A S ° are known at TI(T2 > TI) 21 ° T 2) = AH~,sj(T1) + A(T2 - T~) AHA,Bj(

(14)

the expressions given in paragraph 4.The results of all measured structure sensitive properties for a given liquid alloy should give indications for maximum CSRO at the same concentration.This information can be used to fix the composition of the associates.The model parameters are then obtained by fitting measured integral or partial values of A H mad A G by the method of least squares 20 .The model parameters of same exemplary systems,which are discussed in the following paragraph ,

AS°&B~(T2) = AS°A,&(T1) + A l n ~

(15)

are given in Table 1. The change of the concentration dependence of AH, A G

The association model enables the calculation of thermodynamic functions of ternary systems,if the model parameter of the respective basic binary systems are known and no additional association reaction occurs.Then enthalpy and entropy of the ternary alloy are given by the following expression 22 reg

AH

nAx n B t C A t , B l

reg

Li and Sn-Te (see Figs.l-3).The A H - x -

curve of liquid

Sn-Te alloys shows on the Sn-rich side positive deviation from a triangular shaped concentration dependence as discussed for other alloy systems in paragraph 2.Due to the positive interactions between Sn and SnTe associates

nAt nAIBjCAx,AIBj

-

there is no miscibility gap in the liquid state,but a flat

n

+

n reg reg n B l n A I B j C B I ' A I B j + nBxTtc1CB*,CI n n reg reg n B t nB~C~ CB,,BuC~ + no1 nBt, C~ Cct,BuC~

Further examples of AH, AG and T A S -

+ +

n n f~reg reg nA*nC*~'~AI,Ct .~_ nAtnA~c~CA*,A~C~

equiatomic composition are given in Figs. 4 and 7

n

+

,,-~reg nC~nA~'CIt'~CI'A~CI

liquidus curve exists in that concentration range. curves for li-

quid alloys with CSRO at concentrations different from

n

The comparison between experimental values of partial

{- nAiBjA H ~ B

Gibbs free energies or activities with calculated values are given in Figs. 8-10.The S ~ ( q --* 0) calculated from thermodynamic data 1on5 with the association model

o

AS

and T A S from nearly regular solution behaviour to complete CSRO can be followed along the systems Li-Mg,A1-

agrees with the values obtained from neutron and x-ray measurements 12

=

--R(nA, lnxA~ + nB~Inxsl + nc~Inxc,

+

nA~BjlnXA~Bj + n B ~ C . I n X B ~ C .

The model parameters of the association model of li-

+

nAkctlnxAkCl) + nAIBjAS~IB j

+

n

quid Ce-Cu alloys have been determined by fitting experimental A H values at 1095,1130 and 1473K 22 .The cal-

o o B~c~ASB.c. + nAhc~ASAkcl

(lr)

510

F. Sommer / Compound-forming liquid alloys

"7

o

o

7

-5

-20

-~

O

--40 6

E

-10

~

-15 illhlllhlllhll,hlll!i.llh,

qlllll*hlllh

.4

.2 S1

.6

-I00 ,,,h,,,h,,,h,,,h,,,h,..I,.,,h,,,h,,,I

.2

iii

Cu

Xc.

1.0

1.0

.8,

Zr

Partial Gibbs flee energy of mixing of liquid and undercooled Cu- Zr alloys at 1499 K calculated using the association model (ooo 2e experimental).

Figure 10 :

Enthalpy of mixing of liquid and undercooled Cu-Si alloys at 1600 K calculated using the association model (ooo 23 experimental).

Figure 7 :

.6

Xz,

Cu

1.0

.8

....

.4

T

v

.6

i

o

.6

d

E

.4

¢0

'k~] l

.2 O. 0

,2

.6

.4

St

.6

xc.

.............................. L,,,;:,,,,L,,J....

1.0

0

Cu

.2

.4

Cu Figure 8 : Activity of liquid and undercooled Cu-Si alloys at 1499 K calculated using the association model (ooo 24 experimental).

-i

o E

Figure I1

:

.6

1.0

.8

Xc~

Ce

Excess molar heat of liquid Cu- Ce alloys at 1850 K calculated using the association model (ooo 27 experimental).

o

AcP'Jm°l-lK-1 [ 10

-10

5 I~J -40

2000 ,,h,..I..,,I.,,,h,,.!..,.I....I,.,,h,HI

0

A1 Figure 9 :

.2

.4.

.6

XLt

5ooc

....

,6

T,K

1.0

LI

Partial Gibbs free energy of mixing of liquid A1- Li alloys at 970 K calculated using the association model (o,Z~ 2s experimental)

Figure 12 :

joo% 5UU

'U..6 0

2

" Cu

100

Xzr

Concentration and temperature dependence of excess molar heat of liquid and undercooled Cu-Zr alloys calculated using the association model 26

F. Sommer / Compound-forming liquid alloys

&cP, jmol-lK-1

, 15 t

:

/

511

the La-Sn system it is possible to fix the model parameter using only experimental values of A ~ a ( T )

and

and some additional values of AH(x, T) for

A~,(T)

lanthanum rich La-Sn alloys (Fig.16).This is especially important for alloy systems with intermetallic compounds having melting points above 1900K where it is difficult to measure AH(x) or AG(x) over the entire concentration range.

2000 "~ T,K

1000 O.

'/

Xc e

AI

7

~ -150

Figure 13 : Concentration and temperature dependence of excess molar heat of liquid and undercooled A1-La alloys calculated using the association model 27

o

-tO0

"

i i 0

-200 -250

~

-300

i

-350

............

s

, i , t , l . l , t , l , i ,

500 ~-~-150

,i,lll,J,i,

1000 1500 T , K

Figure 15 : Partial enthalpies of mixing at infinite dii

~ I

-200

...............

lution of liquid and undercooled La-Sn alloys as a function of temperature calculated using the association model (ooo 2s experimental).

~Jr . . . . .

'~Itltttlttti~Itlti~t~ti~t~

500

1000 T,

1500 K

Figure 1~ : Partial enthalpies of mixing at infinite dilution of liquid and undercooled A1-La alloys as a function of temperature calculated using the association model (ooo 10 experimental).

O E

-20

culated ACp at 1850K with these model parameters give a good agreement with experimental values Fig.ll.This shows the utility of the model for extrapolating the measured values for the thermodynamic functions over a

• Hlhlllhlllhjllh'"!

.2 La

',''hlllhlljhll'hlH

.4

.6 Xs.

.8

1.0 Sn

wide range of temperatures even for undercooled liquid

Figure 16 : Enthalpy, entropy and Gibbs free energy of

alloys.The last point is especially important to describe

mixing of liquid and undercooled La-Sn alloys at 1283 K calculated using the association model (ooo 2s experimental).

the solidification behaviour of rapidly solidified alloys.Figs. 12 and 13 show the calculated ACp(x, T ) - behaviour of liquid and undercooled Cu-Zr and A1-La alloys. Experimental results and calculated values of AH-Ti(T) of liquid La-Sn and A1-La alloys are shown in Figs. 14 and 15.The experimental values show the behaviour expected from the association model (see eqns.(10)to(13)).For

CMculated results of ternary alloys using only the model parameters of the respective basic binary systems are given in Figs. 17 and 18 .The agreement between measured values and calculated values show the good ability of this model for extrapolation to higher component systems.

F. Sommer / Compound-forming liquid alloys

512

Table 1: Association model parameters in [kJmo1-1] ] Li-Mg I A1-Li I S ~ T e Associate

AiBj

AH~4~Bj AS~4~Bj C~gB~ CrA:a,AIB~ CrB:g,AiBj 7

I Li-Pb I Cu-Si I Cu-Ce [ Cu-Zr [ AI-La I La-Sn I Ce-Mg I Cu-Mg I

LiMg -31.8 -0.0392

A1Li -30.6 -0.0166

SaTe -53.8 -0.0103

Li4Pb -143.7 -0.0238

CusSi -84.1 -0.0169

Cu3Ce -81.5 -0.0466

Cu2Zr -61.5 -0.0241

A12La -138.2 -0.0506

LaSn -218.1 -0.117

Mg3Ce -56.4 -0.026

Cu~Mg -40.5 -0.0115

-6.27 0

0 0

-28.2 11.6

-60.7 0

-22.8 0

-46.0 -60.5

-23.5 -11.4

-98.1 -50.9

-166.0 -92.6

-35.5 0

-26.1 0

0

0

-6.3

-64.3

0

-28.7

-30.3

-86.3

-58.9

-25.1

-12.2

"7 o E

I0

O

E

E-

10

0

0

-10

-10

~--20

-20 ,llhlllhHllllllhll,

.2

h,llhlllhlllhlllhlll~

.4

CuQ.= M~Q,=0

.6

.8

.0

i

i

i

.2

.4

.6

Ceo.33 Cuo.s~

Ce

I

.8

.0 H~

Figure 17 : E n t h a l p y , e n t r o p y a n d G i b b s free e n e r g y o f m i x i n g o f liquid (CusoMg2o)~Ce(l_x) al-

Figure 18 : E n t h a l p y , e n t r o p y a n d G i b b s free e n e r g y of m i x i n g of liquid (Ce3~Cu6v)~Mg(1-x) al-

loys at 1125 K c a l c u l a t e d using t h e assoc i a t i o n m o d e l (ooo 22 e x p e r i m e n t a l ) .

loys at 1131 K c a l c u l a t e d using t h e assoc i a t i o n m o d e l (ooo ~2 e x p e r i m e n t a l ) .

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87 (1983) 709. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14.

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24. 25.

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