An alternative gibbs-duhem method for the calculation of activities from the redox data for iron oxide in ternary oxide
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I.
then the mole fractions may be expressed by
INTRODUCTION
A M O N G the various methods tl-7] proposed for the integration of the Gibbs-Duhem equation, Schuhmann's approach t31 has found most applications in the calculation of activities in a ternary iron oxide-containing slag from the oxygen isobars. However, because of the way in which the key data, i.e., the "tangent intercepts" to the oxygen isobar, are obtained and because of the number of steps involved, the results are prone to error. The current article presents an alternative set of equations that are particularly suitable for applications to redox systems.
II.
DERIVATION OF THE E Q U A T I O N S
Let F be an extensive property of a system. If n~, n2, and n3 denote the number of moles of each of the three components in a ternary system, the Gibbs-Duhem equation, for constant temperature and pressure, may be written as
ntdfl + n2df2 + n3df3 = 0
[la]
x,df, + x2dA + x~dA = 0
[lb]
Or
wherefi = (3F/3ni),:o/i is the partial molar quantity and xi = nJY, nj is the mole fraction of component i. Define variables y and t as
xl = 1 - y x2 = (1 - t)y x3 ty
Substituting these in Eq. [lb] and regrouping, one gets (1 -
y)dfl + ydf2 + tydAf = 0
[21
X2 "~- X3 S. SUN, Senior Research Scientist, and S. JAHANSHAHI, Program Manager and Principle Research Scientist, are with G.K. Williams Cooperative Research Centre, CSIRO Division of Mineral and Process Engineering, Box 312 Rosebank MDC, Clayton, Vic. 3169, Australia. Manuscript received April 19, 1993.
METALLURGICAL AND MATERIALS TRANSACTIONS B
[4]
where
a f = f 3 -f2
[5]
The term Af is used as the third function in place of f3. When F is taken to represent the free energy of mixing, dAf becomes d In (a~o,,/a~o) which in turn is 1/2d In ao, where afro, afro, 5, and ao (defined as pU202 or pCOJpCO, etc.) are the activities of FeO, FeOls, and O, respectively. In redox-type measurements, it is usual that In a o is determined as a function of y and t (or rather t as a function of y and ao). The following derivation establishes the ways to obtain each f from the known Af. Dividing Eq. [4] by dr, assuming y constant, the following partial differential equation results: (1
-
Y)
(3f'l (~t) kat/[3Af~ \cgt/y
+ y
y
+ ty/-s----/y = 0
Similarly, by dividing Eq. [4] by following is obtained:
{ ~ = 1 -- X 1 X3
[3]
(1 - y ) \Oy/t
+ y
dy for constant t, the
+ t
[6]
\
Oy /t
= 0
[7]
Taking partial differentiation of Eq. [6] with respect to y, and of Eq. [7] with respect to t, then subtracting one of the resulting second-order equations from the other, one obtains the following:
(
Ofl~ - (3f2~ + y(OAfl _ t ( 3 A f ~ = 0 OtJy \ O t / y \G/, \ Ot /y
[8]
VOLUME 25B, APRIL 1994--277
Now the combined Eqs. [6] and [8] may be solved to give
Of21 =
Ot/y
_t(OAfl (OAf~ \ at /ly -t- y(1 -- y) \ - - ~ y / ,
the integral quantity as a function of the variables t and y is given by f=
[91
(1 - y)f~ + Yf2 + tyAf
[16]
Partial differentiation with respect to t leads simply to
and y2(0Af I
Oflt =
Ot/r
-
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