On the Relationship between Interaction Coefficients M.Z. SUKIENNIK and R. W. OLESINSKI The widely used relationship between Wagner coefficients, E} j) = 6,~i), has been analyzed. Based on various representations of the independent composition variables, the validity of the relationship has strictly been limited to the cases when the ratio ln(yj/%), i,j = 2 . . . . . s is constant at the point x~ = 1 and in the neighborhood of this point, provided that the derivatives of the activity coefficients have finite values within the specified limit. The clarification of misconceptions about the Wagner coefficients, frequently encountered in the literature, has been attempted.

I.

INTRODUCTION

THERMODYNAMIC analysis of liquid dilute solutions, particularly the iron-based alloys, frequently utilizes the formalism introduced by Wagner and Chipman. ~,2,3The activity coefficient, Y. of the solute (its mole fraction x~ ~ 0, i -- 2 . . . . . s) is presented in this formalism as a Taylor series expansion about the point (xl -- 1, x~ = 0, i = 2, . . . . s). Assuming Henrian ideal solution as the reference state, the expansion takes the following form: In% =

~ (O- ln %) -

j=2\

C3Xj / x k ~ j

~,=l

X)

II. DIFFERENTIATION OF THERMODYNAMIC FUNCTIONS W I T H R E S P E C T TO MOLE FRACTIONS

The first terms of this series are called the first order interaction coefficients and are denoted by elJ~: e~j) =

(a In yi]

1

[2]

The second derivatives of series (1) are known as the interaction coefficients of the second order, el j,~>. The relationships among the interaction coefficients have been investigated quite extensively, a,5 A major controversy still seems to surround the popular relationship involving the first order coefficients, namely: E (J) i = ~.)i)

[31

Probably the most elegant demonstration of relation [3] is based on the following thermodynamic identity:

Zi = Z' + OZ____~'_~ XF(OZ'I OXi

r=2

[4]

\OXr/xk~x r

where Z' denotes a molar property of the phase and Z~ is the partial molar property of the component i:

njCni

Z: property of the whole phase n~: number of moles of component i Applying Eq. [4] to the partial excess Gibbs free energy of the components i and j, then differentiating with respect M.Z. SUKIENNIK is Associate Professor, Academy of Mining and Metallurgy, Cracow, Poland. R.W. OLEStNSKI is Research Associate, Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611. Manuscript submitted December 12, 1983. METALLURGICALTRANSACTIONS B

to xj and xe, respectively, and comparing (OZi/Oxl)x~,,
The starting set of independent variables, with respect to which the behavior of thermodynamic functions can be analyzed, is the representation of the number of moles of the phase components, (n~, n2. . . . . n,). Since the forthcoming analysis will utilize the directional derivatives, it is appropriate to define a basis of orthogonal unit vectors: a l = (1,0 . . . . . 0) or, = (0, 0 . . . . .

1)

[61

which permits any vector a to be expressed as a linear combination of the base vectors:

ot = ~ Aia, = ( A , , . . . , A , )

[7]

i=I

The derivative of the function Z in the direction of the vector a is defined as the scalar product of the two vectors, a and grad Z: DoZ = a . g r a d Z

=(A, ..... A,)" 'OZ = ~ AiZi

. . . . .

anZ) [81

i=1

Replacing the set of the independent variables, (n~,n2, . . . . n,), by the set of dependent variables, (x~, x2 . . . . . x,), it is possible to distinguish (s) different representations of the independent variables: (x2 . . . . .

x,)

............ (x, . . . . .

[9]

x,_,)

VOLUME 15B, DECEMBER 1984--677

A derivative of the function Z, expressed as:

,,0, x 2, . . . ,xj - i , x j +l . . . . ,Xs

describes the variation of the function Z in the direction of the vector a), ~: aj, l = a j -

01

[11]

which is identical with the directional derivative, D.),~Z:

x 2 . . . . . xj

It may be added that the literature abounds 6-9 in expressions of the derivatives with respect to the mole fractions at unspecified set of independent variables. A particularly deplorable error occurs when such a derivative is identified 9 with the partial molar function Z:.

I11.

ANALYSIS OF R E L A T I O N r ~ ~ = eJ'~

Relation [3] will be analyzed for two representations of independent variables:

A/(x2 . . . . . x,)

I,Xj +1 . . . . . x s

B/(xl . . . . . x~-i, xk+l . . . . . x,)

D,~j,~ Z = %, ~ 9 g r a d Z

[19]

/

~

(--130

O,l,O

. . . . .

..... ' "" "

-

OZ 8nj

3Z On|

---Z/-

On,~

Z1

The analytical method has been expounded in the preceding section. For the first set, A, of variables, the difference of the interaction coefficients:

[12]

Derivative [10] expressed within the set of independent variables (x~ . . . . . x,_~) describes the variation of the function Z in the direction of the vector a~,,: aj,, = r 1 6 2 a ,

[13]

which is identical with the directional derivative D,,j.Z: (3~)

A = el ;) - eJ`) can be written in the following form: A a = D%,(ln %) - D.,,l(ln y))

[211'

*Henceforth, all derivatives o f In 3' are at x, = 1.

which, after rearrangement, leads to the relation:

=- D , : X I , . . . ,Xj I,Xj+I . . . . . Xs_ 1

D,,:,Z

=

Aa = - [t41

Following the above rationale, the variation of Z in the direction of any vector a;,k: ot;~ = ~ : -

~

For the set B of variables, difference [20] can be expressed as follows: AB = D,j.~(ln %) - D,,,~(ln %)

[23]

AB -

[24]

[15]

can be expressed by the relation:

D,,jkZ = aj, k" g r a d Z = Zj - Zk

3nk

[16]

The above derivations are based on the principles explained previously and also on the fundamental relation:

Since Z is an extensive property, it follows that

Z = nZ',

n = ~ni

OZi_

[17]

i=1

Onj

which leads to the relationship between the directional derivatives of the functions Z and Z ' :

D,,:kZ = D,,:k(nZ') = nD,,jkZ' + Z'D.;~(n)

02Z On, 3nj

O2Z '

0Zi -

D.j, kZ = nD.:.~Z' = Zj - Zk

[18]

OXj

It may thus be concluded that the derivative:

OZj OX i

.....

.....

~ n i = n i=l

6 7 8 - - V O L U M E 15B, DECEMBER 1984

=

1

_ OZj 3ni

r= I ram

02Z '

~ ~

OXj O X i

02Z'

4,

O X i OXj

x.

can be identified as the derivative of the function Z in the direction of the vector aj, k, the derivative being expressed for a subset of the variables (n~ . . . . . n,) defined by the condition:

02Z 3n~ 3n~

2_,

-

oz' x.,,x.l

_

Let us now differentiate identity [4] with respect to xj and the same identity written for Zj with respect to x~, within the set of variables (xl . . . . . Xm-l,Xm+~ . . . . . X,), m 4= i, m 4= j:

or, because D~jk(n ) = O,

-ggx/Xl .....

[221

On1

grad Z =

aj,~"

= Zj - Z~

(

[201

x ~ - -

OX r OXj

02Z ' x,

r= 1 r@m

[251

OXr OXi

Subtracting Eqs. [25] it follows:

OZi 3xj

3Zj 3x~

: o2z ' --

~

,=1

Xr - -

kOx~Ox~

s dx~OxJ

[26]

r~m

Applying relation [26] to the excess Gibbs free energy, G E, one obtains the following expression for difference [20]:

METALLURGICAL TRANSACTIONS B

02(GE)']

I ~ X,(O2(GE)' A = -~r=l r#m RT

\OX'~r-~iXi Xr

variables, the ratio of the activity coefficients of any two solutes must be constant in the neighborhood of the point (Xl = 1,x~ = 0, i = 2 . . . . . s). This condition can hardly be fulfilled within the composition ranges that make it possible to determine thermodynamic properties of the solution with an acceptable accuracy.

OXrOXj /I

(G~ - GEm -- G E -~ GEm)

r=l r~-m

IV.

: ~ Xr~--~xSJ /

CLOSING

REMARKS

[27]

r=l rein

Equation [27] can be written for both sets of variables [19]:

Apart from the attempt to prove relation [3] using identity [4], there occur in the literature rather unacceptable demonstrations and essentially erroneous interpretations of relationship [3]. For instance, it is mistaken for a thermodynamic principle; some authors 7'8'9 try to derive it from the equation: 02(GE) t _ 02(GE) '

r=l rO.r,( n

' eor o :,

[29]

rT~m

OXi OXj

OXj OXi

and utilize an impossible relation:

which are identical with expressions [22] and [241:

a(Ge) '

0 ln( ) AA-" r=2XrO~

T-~2) :

--On~

cqxi

[30]

0 ln( ) A8 r=l XrD

tr

:

--Ol'lk

[3t1

r4-k As mentioned earlier, when proving relation [3] it is assumed that the derivatives have finite values at the point (x~ = 1, x~ = 0, i = 2 . . . . . s). This assumption makes the left-hand side of identity [30] equal to zero, so it follows:

-

G~

As shown in the preceding sections, relation [3] is by no means a general thermodynamic principle and may be valid only under strictly defined conditions: 1. that the ratio ln(yff7~), i , j = 2 . . . . . s is constant at the point xl = 1 and in the neighborhood of this point; 2. that there exist finite derivatives of the activity coefficients at the point xl = 1 and in the neighborhood of this point. These conditions, however, cannot be satisfied within any realistic range of composition because interactions among the solutes vary with composition and that mutual influence (affecting 7ff%) obviously increases with the departure from the point (xl = 1, x~ --- O, i = 2 . . . . . s). It follows then that an application of relation [3] to experimental results may lead to serious errors in the thermodynamic description of the solution.

Introducing the same assumption to Eq. [31] one obtains: NOTATION LIST As -

On----~

r=l

r4=k dln(~l = D~,.k(ln ~ ) =

_

c3:-~%/

din( Z] 0~---~d

0 ln( t

\ Yfl

[33]

Onk

The above implies that the difference of interaction coefficients expressed for the sets of variables other than that of A is not necessarily equal to zero:

01.(:) AB -

-

-

0nk

Excess Gibbs free energy Number of moles Gas constant Temperature Mole fraction Property of the whole phase Molar property Partial molar property of the component i Vector Activity coefficient Difference of the interaction coefficients Interaction coefficient Vector component

REFERENCES [34]

If relation [3] is to be valid for any representation of the METALLURGICAL TRANSACTIONS B

GE n R T x Z Z' Zi a 3/ A e A

1. C. Wagner: Thermodynamicsof Alloys, Addison-Wesley Press, Cambridge, MA, 1952. 2. J. Chipman: J. ISI, 1955, vol. 180, pp. 97-106. 3. J. Chipman: Trans.AIME, 1967, vol. 239, pp. 1332-36.

VOLUME 15B, DECEMBER 1984--679

4. C. H. P. Lupis and J. E Elliott: Acta Metall., 1966, vol. 14, pp. 529-38. 5. L. S. Darken: Trans. A1ME, 1967, vol. 239, pp. 90-96. 6. D. R. Gaskell: Introduction to Metallurgical Thermodynamics~ 2rtd ed., McGraw-Hill, New York, NY, 1981. 7. O. Kubaschewski and C.B. Alcock: Metallurgical Thermochemisto, , Pergamon Press, New York, NY, 1979.

680--VOLUME 15B, DECEMBER 1984

8. C.B. Alcock: Principles of Pyrometallurgy, Academic Press, New York, NY, 1976. 9. L. Coudurier, D.W. Hopkins, and I. Wilkomirsky: Fundamentals of Metallurgical Processes, International Series on Materials Science & Technology, D.W. Hopkins, ed., Pergamon Press, New York, NY, 1978, vol. 27.

METALLURGICALTRANSACTIONS B