An algorithm and program in C-language for computation of standard free energy of formation of clay minerals

Abstract

A program has been developed for the computation of standard free energy of formation (ΔGf0) of end member as well as other layer silicate clays. This program utilizes information on the chemical composition of the mineral and values of ΔGf0 of other oxides, hydroxides and silicates, available in the literature to derive the required ΔGf0 value. Previous studies have shown that the procedure gives a fairly accurate estimate of ΔGf0 of clay minerals. This program will be useful for rapid computation of ΔGf0 values. With minor modifications, it can also be used for solving non-linear regression equations of any type. It has been developed on a IBM RS/6000 workstation for X-window in C-language under UNIX environment.

Full text

An algorithm and program in C-language for computation
of standard free energy of formation of clay minerals
Mohammad Kudrat
a,
*, K.P. Sharma
a
, Chandrika Varadachari
b
,
Kunal Ghosh
c
a
Regional Remote Sensing Service Centre, Indian Space Research Organisation, IIRS Campus, P.O. Box 135, 4 Kalidas Road,
Dehradun 248 001, India
b
Raman Centre for Applied and Interdisciplinary Sciences, 11 Gangapuri, Calcutta 700 093, India
c
Department of Agricultural Chemistry and Soil Science, University of Calcutta, 35 B.C. Road, Calcutta 700 019, India
Received 19 February 1998; received in revised form 25 November 1998
Abstract
A program has been developed for the computation of standard free energy of formation (DG
f
0
) of end member as
well as other layer silicate clays. This program utilizes information on the chemical composition of the mineral and
values of DG
f
0
of other oxides, hydroxides and silicates, available in the literature to derive the required DG
f
0
value.
Previous studies have shown that the procedure gives a fairly accurate estimate of DG
f
0
of clay minerals. This
program will be useful for rapid computation of DG
f
0
values. With minor modi®cations, it can also be used for
solving non-linear regression equations of any type. It has been developed on a IBM RS/6000 workstation for X-
window in C-language under UNIX environment. #1999 Elsevier Science Ltd. All rights reserved.
Code available at http://www.iamg.org/cGEditor/index.htm
Keywords: Computer program; Thermodynamics; Silicates; Non-linear regression
1. Introduction
Standard free energy of formation (DG
f
0
) values of
silicate minerals are invaluable for the thermodynamic
treatment of geochemical processes. Various theoreti-
cal, empirical and experimental methods have been
used for determining the DG
f
0
of silicates. For the layer
silicates, particularly the non-stoichiometric clay min-
erals, available experimental data are limited. Because
of the time-consuming and laborious nature of the ex-
perimental methods, it is virtually impossible to obtain
DG
f
0
values for the innumerable non-stoichiometric sili-
cate clays that exist in nature. Theoretical and empiri-
cal methods, however, provide a simple and rapid
means of obtaining such values with reasonably good
accuracy.
A theoretical method based on crystal energy par-
ameters was developed by Slaughter (1966a, 1966b,
1966c). The inherent diculty with this method is the
non-availability of accurate crystal energy parameters.
Tardy and Garrels (1974) adopted a simpli®ed empiri-
cal method for the determination of DG
f
0
for 2:1 sili-
cate clays which exhibit a large variation in chemical
composition resulting in variation in tetrahedral, octa-
hedral and inter-layer occupations and distribution of
layer charge. In this method (Tardy and Garrels,
Computers & Geosciences 25 (1999) 241±250
0098-3004/99/$ - see front matter #1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 3 0 0 4 ( 9 8 )0 0 1 7 0 - 8
* Corresponding author. Fax: +91-135-745-439.
E-mail address: rrsscd@nde.vsnl.net.in (M. Kudrat)

1974), DG
f
0
from elements of a given layer silicate is
obtained as equal to the sum of DG
f
0
from elements of
its constituent `silicated oxides' which are assumed to
have ®xed values for all silicates, but dier from those
of the pure oxide and hydroxide phases. The initial
principle of evaluation of DG
f
0
was later substantially
revised by Tardy and Garrels (1976, 1977) based on
the fact that within a family of compounds (i.e. metasi-
licates, orthosilicates, etc.) DG
f
0
of a compound from
their constituent oxides are linearly dependent on a
parameter called DO2ÿ
M2which is a function of electro-
negativity of the constituent cations (M
2+
). This new
method was later used by Tardy and Duplay (1992)
for evaluating DG
f
0
of clay minerals of various degrees
of hydration and crystallinities. Using a somewhat
similar approach, Nriagu (1975) applied a small correc-
tion term to the free energy change accompanying the
combination of silicon hydroxide with metal hydrox-
ides. However, the average dierence between the pre-
dicted and experimental data was about 9.5 kcal/mol.
This was further re®ned by Sposito (1986) with a con-
sequent increase in accuracy of the data to about 2.6
kcal/mol. In this technique, the estimation is based on
the combined contribution of hydroxide components
rather than both hydroxide and oxide polyhedra.
Karpov and Kashik (1968) used a multiple linear-re-
gression technique to estimate the DG
f
0
contribution of
36 dierent oxides to the overall DG
f
0
of silicate min-
erals; however they ignored the hydroxide contri-
butions. The precision of this method is said to be 1.6
to 2.4 kcal/mol, but their calculated value for grossular
dier from the experimental value (Robie and
Waldbaum, 1968) by 16 kcal/mol. Chermak and
Rimstidt (1989) utilised this concept for obtaining
DG
f
0
, wherein multiple linear-regression was involved.
Powell and Holland (1993) observed that the least
square technique is not only sound, but also the most
appropriate.
By far, the most versatile and promising method
having an acceptable error range is proposed by Chen
(1975). The procedure consists of ®rst representing the
mineral in question in the form of a combination of
various compounds from which the minerals are
assumed to be formed. For each combination the sum
of DG
f
0
of the components (i.e. aDG
f,i
0
) is computed
and arranged in descending order of aDG
f,i
0
. Brie¯y, an
equation of the form:
XDG0
f,iaebx c1
is obtained, where `a' and `b' are constants to be deter-
mined by regression analysis, `c' gives the value for the
DG
f
0
of the mineral of interest and `x' is an indepen-
dent variable representing the rank of a particular
aDG
f,i
0
in the whole series of aDG
f,i
0
values arranged in
descending order according to their magnitude. Earlier,
the value of rank `x' was chosen arbitrarily without
any mathematical base. Varadachari et al. (1994) later
modi®ed and improved the exponential curvilinear re-
gression method by improving the assignment of rank
`x'. Brie¯y, the value of aDG
f,i
0
is computed by ®rst
representing the mineral in the form of combinations
of various compounds and subsequently adding the
DG
f
0
of the component minerals. The values of aDG
f,i
0
are then arranged in descending order and an integral
value `x' is assigned which represents the rank of the
aDG
f,i
0
. The possible combinations of `x' are computed
utilising a `tree-diagram' procedure (Varadachari et al.,
1994). Curve-®tting is done to obtain that combination
which gives the best curve ®t. The corresponding value
of the constant `c' is the value of DG
f,i
0
of the mineral.
Eq. (1) is solved using an iterative least squares
method. Here, certain initial estimates of a,band care
assumed (i.e. a
0
,b
0
and c
0
) and the error aE
2
is evalu-
ated. The iteration is continued with revised values of
a
0
,b
0
and c
0
and at each stage of the iterative pro-
cedure, aE
2
is evaluated. The iteration is continued
until the solution converges, i.e. aE
2
reaches a mini-
mum.
Varadachari et al. (1994) applied this method to
obtain the free energy of formation of various silicate
clays. Comparison of the DG
f
0
values derived, with the
standard values available in the literature, revealed
good agreement between them. It was also possible to
apply this method to determine the DG
f
0
of non-stoi-
chiometric clay minerals.
The objective of this study was to develop a compu-
ter program for determining DG
f
0
of clay minerals uti-
lising the procedure developed by Varadachari et al.
(1994). Essentially, the program consists of (a) arran-
ging the set of aDG
f,i
0
values for a mineral in descend-
ing order, (b) computing the various possible
permissible combinations of `x' for the aDG
f,i
0
values,
(c) solving Eq. (1) with each set of `x' values and
obtaining that combination showing the best ®t and
(d) displaying the results.
2. Algorithm
Details of the procedure have been enumerated by
Varadachari et al. (1994). For convenience, the math-
ematical basis of the procedure is reviewed brie¯y.
In the ®rst step, the mineral in question is rep-
resented as combinations of various compounds, e.g.
kaolinite, Al
2
Si
2
O
5
(OH)
4
, may be written as
Al
2
O
3
+2SiO
2
+2H
2
O, Al
2
SiO
5
+SiO
2
+2H
2
O,
2AlO(OH) + 2SiO
2
+H
2
O, etc. For each set of
equations, the DG
f
0
of the components (values obtained
from Helgeson et al., 1978) are added to obtain aDG
f,i
0
.
The values of aDG
f,i
0
are then arranged in descending
order and a rank `x' is assigned to each. The ranks are
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250242

assigned following a `tree-diagram' procedure (which is
explained below). Utilizing the values of aDG
f,i
0
and x,
nonlinear regression equations are derived. The
equations are of the form:
XDG0
f,iaebx c
where a,band care constants and the value of `c'is
the DG
f
0
of the mineral. For solving this equation, an
iterative least square technique (Draper and Smith,
1981; van Heeswijk and Fox, 1988) was adopted which
is as follows (Scarborough, 1976).
The generalized form of Eq. (1) may be written as
yfx,a,b,c2
where xand yare variables and a,band care con-
stants. To solve this equation some initial values of the
unknowns are assumed as a
0
,b
0
and c
0
. Then,
aa0a,
bb0b,
cc0g,3
where a,band gare the values by which the assumed
values dier from the actual value of a,band c. If the
experimentally obtained values of xand yare x
1
,x
2
,
...,x
p
and y
1
,y
2
,...,y
p
, then by substituting in Eq.
(2), we have
yn0fxn,a0,b0,c04
where nranges from 1 to p. The approximated value
y
n
'diers from the observed value y
n
by a value, say,
r
n
. Therefore,
yn0ynrn:5
Again, the observed value of y
n
diers from the value
obtained from the best ®tted curve by a value E
n
,
which is related as
Enfxn,a,b,cÿyn:6
On combining Eqs. (3) and (6), we get
Enynfxn,a0a,b0b,c0g:7
This, when expanded in Taylor series as a function of
a,band cgives
Enynfxn,a0,b0,
c0adf=da0bdf=db0gdf=dc0
8
when higher order terms are ignored. However, since
y
n
'=f(x
n
,a
0
,b
0
,c
0
) and r
n
=y
n
'ÿy
n
, Eq. (8) may be
rewritten as
Enadf=da0bdf=db0gdf=dc0rn:9
This equation is linear in a,band g. The best represen-
tative values of a,band care those for which a
n=1
p
E
n
2
is a minimum.
To solve Eq. (8) for a,band gby the least-squares
technique, an initial estimate viz. a
0
,b
0
and c
0
is taken
for the ®rst iteration. Subsequently, the initial values
a
0
,b
0
and c
0
are increased or decreased by small values
to obtain the revised values a
1
,b
1
and c
1
and iteration
is done again. This process is repeated with successive
values a
2
,b
2
and c
2
,a
3
,b
3
and c
3
and so on. Iteration
is continued until the solution converges, i.e. until g,a
and breach selected small values. The value of E
n
is
also evaluated at each stage to observe if a reduction
in its value is actually achieved. If convergence does
not occur or is very slow, the parameter increments are
halved or doubled (Draper and Smith, 1981).
In the particular situation of Eq. (1) which has the
form
fxaebx c10
the partial derivatives are
df=daebx 11
df=dbaxebx 12
df=dc113
On substituting Eqs. (11)±(13) into Eq. (9), we have
Enaeb0xnba0xneb0xngrn14
or
Enaeb0xnba0xneb0xnga0eb0xnc0ÿyn:15
Eq. (15) is a generalized form of the equation termed
the `residual equation'. By substituting the values of n,
plinear equations are obtained as shown below:
E1aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy116
E2aeb0x2ba0x2eb0x2ga0eb0x2c0ÿy217
Epaeb0xpba0xpeb0xpga0eb0xpc0ÿyp18
Eqs. (16)±(18) are linear (of ®rst order) in the correc-
tions a,b,gand, therefore, can be solved by the
method of least squares. For solving the unknowns a,
band gfrom the presidual equations, the principle
.
.
..
.
..
.
.
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250 243

elucidated by Scarborough (1976) and Draper and
Smith (1981) has been used.
According to the principle of least squares the best
values of the unknown parameters (a,b,g) are those
which make the sum of the squares of the residuals a
minimum. Thus,
X
p
n1
E2
nminimum E2
1E2
2...E2
p19
Hence,
X
p
n1
E2
naeb0x1ba0x1eb0x1ga0eb0x1
c0ÿy12aeb0x2ba0x2eb0x2
ga0eb0x2c0ÿy22
...aeb0xpba0xpeb0xp
ga0eb0xpc0ÿyp2fa,b,g20
The condition that f(a,b,g) be a minimum is that its
partial derivatives with respect to a,band gshall each
be zero. That is,
df=da df=db df=dg 0:
By partial dierentiation of Eq. (20) with respect to a,
band gand dividing by 2, we get the following normal
equations:
aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1eb0x1
aeb0x2ba0x2eb0x2ga0eb0x2
c0ÿy2eb0x2...aeb0xpba0xpeb0xp
ga0eb0xpc0ÿypeb0xp021
aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1a0x1eb0x1
aeb0x2ba0x2eb0x2ga0eb0x2
c0ÿy2a0x2eb0x2...aeb0xpba0xpeb0xp
ga0eb0xpc0ÿypa0xpeb0xp022
aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1
aeb0x2ba0x2eb0x2ga0eb0x2c0ÿy2
...aeb0xpba0xpeb0xpga0eb0xp
c0ÿyp0:23
These normal equations can be solved for the three
unknowns by simple algebraic methods. It needs to be
mentioned here that the number of equations is always
the same as the number of unknown constants to be
determined, whereas the number of residual equations
is equal to the number of observations. The number of
observations must always be greater than the number
of unknown constants if the method of least squares is
to be used in solving the equation.
The two sets of input data required to obtain the
values of a,band gand hence the values of the re-
gression constants a,band care the computed values
of DG
f,i
0
(i.e. the y
n
value) and the corresponding
assigned values of rankings `x
n
'(n=1, ...,p).
Assignment of the xvalues is done by ®rst comput-
ing the various possible combinations of `x' values for
a set of aDG
f,i
0
data; subsequently, that combination of
xvalues which gives the best ®tted curve, i.e. smallest
residual error, is chosen.
The procedure for computing the various possible
combinations of xhas been described in detail by
Varadachari et al. (1994). First, a `tree-diagram' is con-
structed (Varadachari et al., 1994). The ®rst branching
of the `tree' begins with 0; this has three branches 0, 1
and 2. Each of these ®gures 0, 1 and 2 again has 3
branches each, with the ®gures of the branches being
n,n+ 1 and n+ 2 where nis the parent ®gure from
which the branching originates. Once the tree is con-
structed, the combinations are obtained by following
the branches beginning from the top 0. However, com-
binations that contain less than three dierent values
of xare ignored.
Suppose, a set of 5 dierent equations for a mineral
is obtained, giving 5 dierent values of aDG
f,i
0
. Thus,
kaolinite, Al
2
Si
2
O
5
(OH)
4
may be represented as
(Varadachari et al., 1994):
(i) Al
2
O
3
+2SiO
2
+2H
2
OaDG
f,i
0
=ÿ897.496 kcal/mol
(ii) Al
2
SiO
5
+2SiO
2
+2H
2
OaDG
f,i
0
=ÿ898.613 kcal/mol
(iii) 2AlO(OH) + 2SiO
2
+2H
2
OaDG
f,i
0
=ÿ900.482 kcal/mol
(iv) 0.5Al
2
SiO
5
+Al(OH)
3
+1.5SiO
2
+0.5H
2
OaDG
f,i
0
=ÿ901.775 kcal/mol
(v) Al(OH)
3
+AlO(OH) + 2SiO
2
aDG
f,i
0
=ÿ902.710 kcal/mol
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250244

To obtain the various combinations of `x' values
that can be assigned to these 5 aDG
f,i
0
, values, a `tree
diagram' for a 5-set data is constructed. This is shown
in Fig. 1. By following the branches beginning from
the extreme left, the combinations obtained are: 00000,
00001,00002,00011, 00012, 00013, 00022, 00023,
00024, 00111, 00112, 00113, 00122, 00123, 00124,
00133, 00134, 00135, 00222, 00223, 00224, 00233,
00234, 00235, 00244, 00245, 00246, 01111, 01112,
01113, 01122, 01123, 01124, 01133, 01134, 01135,
01222, 01223, 01224, 01233, 01234, 01235, 01244,
01245, 01246, 01333, 01334, 01335, 01344, 01345,
01346, 01355, 01356, 01357, 02222, 02223, 02224,
02233, 02234, 02235, 02244, 02245, 02246, 02333,
02334, 02335, 02344, 02345, 02346, 02355, 02356,
02357, 02444, 02445, 02446, 02455, 02456, 02457,
02466, 02467, 02468.
The combinations which are shown in bold typeset
are discarded since they give less than 3 dierent
values of `x' which is necessary for regression. Thus, a
®gure such as 02244 implies that the values of x, for
the chemical combinations (i), (ii), (iii), (iv) and (v) are
0, 2, 2, 4 and 4, respectively. Using all the combi-
nations, iteration is done and aE
2
is evaluated for each
combination. The combination of `x' values giving the
smallest residual error, i.e. the best curve-®t is chosen
and the value of DG
f
0
is obtained from the value of `c'
in Eq. (1). In the example of kaolinite, the combi-
nation 01357 gives the best curve-®t. A regression
equation of the form, aDG
f,i
0
=7.6414e
ÿ0.165062x
ÿ905.1208, is obtained. The required value of DG
f
0
for
kaolinite is thus, ÿ905.1208 kcal/mol.
A ¯ow chart of the procedure used here for comput-
ing DG
f
0
is given in Fig. 2(A) and (B).
3. Description of the program
Code was developed in 'C'-language. Here, a pre-
vious Fortran subroutine CURVE-FIT, by van
Heeswijk and Fox (1988), has been incorporated with
necessary modi®cations and additional features. The
software structure is given in Fig. 2(A) and the ¯ow
chart in Fig. 2(B).
Essentially, the program consists of three sections.
First, the values of aDG
f,i
0
, (input data) are arranged in
descending order; subsequently, the various combi-
nations (sets) of the ranking `x' for the aDG
f,i
0
values
Fig. 1. Tree diagram for generating the various possible combination of ranks (x) for a set of ®ve data (aDG
f,i
0
).
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250 245

are computed by the method described earlier. Finally,
for each set of rankings x, Eq. (1) is solved and the re-
sidual error for the curve ®t is estimated; this process
is repeated for every set of rankings and that set show-
ing smallest residual error is selected.
The program MAIN, is a set of dierent functions
and two sub-programs. The sub-program FREE
ENERGY is for actual calculation, whereas the second
one PLOTCURVE draws curves on the video screen.
Both the sub-programs are linked together to work in
unison to give the desired results.
The sub-program of FREE ENERGY determines the
best ®t for each of the generated data sets (xranks).
The initial estimates of a,band care a
0
,b
0
and c
0
.If
the values of the initial estimates a
0
,b
0
and c
0
are clo-
ser to the true values of a,band c, convergence of the
solution will be easier.
The function Obes() generates the various sets of
ranks (x) from the given number of observations, i.e.
the number of aDG
f,i
0
, values. The Obes() function
stores the `x' sets in a ®le after ®ltering, using the logic
that at least three dierent values of `x' must be avail-
able to solve the three unknowns in Eq. (1). The func-
tion Obes() uses two other functions, FUN for
calculation of values xand FUN1 for storing the gen-
erated ®ltered values in a data ®le. At the time of ex-
ecution, the program prompts for the number of
observations, i.e. number of free-energy values derived
from the chemical equations.
The smoothness of the curve represents the best ®t
of the data set. Options for Cardinal Curve and Bezier
Curve ®ts can also be displayed. The sub-program
FREE ENERGY is supported currently for eight
unknowns; if necessary this can be modi®ed to take
care of an unlimited number of parameters. It contains
three subroutines: NONLIN, GJINV and GUASS.
Another subroutine TESTVAL will check for zero
divides and also checks for large exponents which may
occur in the subroutine NONLIN. Depending on the
function, the subroutine TESTVAL may not be needed
in all situations.
The partial derivatives of the equation are calculated
in functions f
1
(), f
2
() and f
3
(). The actual function, i.e.
Eq. (1), is contained in f
0
(). The subroutine NONLIN
gives the value of the parameters a,band cafter sub-
stituting the values of dierent parameters (a
0
,b
0
,c
0
).
If the function has more than three unknown par-
ameters the number of partial-derivative functions
need to be increased accordingly. The variable x,
which is a necessary input for NONLIN is generated
Fig. 2. (A) Software structure. (B) Flow chart for computing DG
f
0
by an iterative method.
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250246

Fig 2 (continued)
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250 247

automatically from the subroutine Obes(). The sub-
program FREE ENERGY can be used for solving
non-linear equations of any type with minor modi®-
cations of the program, such as de®nition of the
equations and its partial derivatives.
A graphic user interface is also part of the program
using the AIX graphic facilities on X-Windows. The
program also provides the facility to visualize the ®nal
results directly or display the results at each and every
step. After a ®nal output has been achieved, option is
provided to draw Bezier,Cardinal or B-spline curves
for the output results and the data set. The sub-pro-
grams B-CURVE, INITIALIZE and DRAWCURVE
have been used for graphics output in the program
PLOTCURVE.
This program has been developed on an IBM-RS/
6000 workstation for X-Window in `C'-language in
UNIX environment. However, the program can be exe-
cuted in any other system with the same environment.
The program is modular with a display option in the
X-Window environment which can be separated out
and implemented on other systems according to the
user's needs. The source code is available from the
authors or by anonymous FTP from IAMG.ORG.
4. Results
The values of standard free energy of formation
(DG
f
0
) for various layer silicates (clay minerals) were
derived by Varadachari et al. (1994) and Kudrat
(1995), using this program. On comparison of the esti-
mated DG
f
0
data of various standard clay minerals,
with the data available in the literature, a good agree-
ment between the values was observed. For example
the computed values by Varadachari et al. (1994) for
kaolinite, muscovite, pyrophyllite and phlogopite were,
respectively, ÿ905.121, ÿ1337.991, ÿ1257.550 and
ÿ1395.221 kcal/mol; these compare favourably with
the experimental values reported by Helgeson et al.
(1978) which are, respectively, ÿ905.614, ÿ1336.301,
ÿ1255.997 and ÿ1396.187. The program was also suc-
cessfully used for evaluating DG
f
0
values of other clays
with variable composition, such as illite, montmorillo-
nite, vermiculite, saponite, nontronite, etc.
Two examples of the use of this program for deriv-
ing the DG
f
0
value of an end-member layer silicate (pyr-
ophyllite) and a non-stoichiometric clay mineral
(greenish-yellow vermiculite, North Carolina) are pre-
sented here.
Table 1 lists the various chemical combination
equations and the values of aDG
f,i
0
obtained from the
summation of DG
f
0
of the individual components. The
ranks (x) of the aDG
f,i
0
shown in Table 1 represent
values which give the best ®tted equation. Fig. 3(A)
and (B) are the exponential curves which show the re-
lation between aDG
f,i
0
and xvalues. The values of the
constants a,band care obtained from the equation.
The value of cgives the required DG
f
0
of the mineral.
It may be observed from Table 1 that for pyrophyl-
lite, six equations were constructed; their aDG
f,i
0
had
ranks 0, 1, 3, 3, 5 and 6. The best-®t curve is shown in
Fig. 3(A). The derived value of DG
f
0
for this mineral is
ÿ1257.55 kcal/mol which compares well with the
Table 1
aDG
f,i
0
and the rankings of the combinations for some clay minerals
Clay minerals and their combinations aDG
f,i
0
(kcal/mol) Rank
Pryophyllite Al
2
Si
4
O
10
(OH)
2
(1) Al
2
O
3
+4SiO
2
+H
2
Oÿ1250.098 0
(2) Al
2
SiO
5
+3SiO
2
+H
2
Oÿ1251.215 1
(3) 0.5Al(OH)
3
+0.75Al
2
SiO
5
+3.25SiO
2
+0.25H
2
Oÿ1252.796 3
(4) 2AlO(OH)+ 4SiO
2
ÿ1253.084 3
(5) 0.5Al
2
Si
2
O
5
(OH)
4
+0.5Al
2
O
3
+3SiO
2
ÿ1254.154 5
(6) 0.5Al
2
Si
2
O
5
(OH)
4
+0.5Al
2
SiO
5
+2.5SiO
2
ÿ1254.716 6
Regression equation: aDG
f,i
0
=7.445103e
ÿ0.159289x
ÿ1257.551758
Greenish-yellow vermiculite: Si
2.905
Al
1.590
Fe
0.166
3+
Fe
0.215
2+
Mg
2.3410
O
10
(OH)
2
(1) 2.905SiO
2
+0.795Al
2
O
3
+0.0830Fe
2
O
3
+0.215FeO + 2.3410MgO + H
2
Oÿ1295.454 0
(2) 0.795Al
2
SiO
5
+1.1705Mg
2
SiO
4
+0.083Fe
2
O
3
+0.215FeSiO
3
+0.7245SiO
2
+H
2
Oÿ1314.313 2
(3) 0.795Al
2
Si
2
O
5
(OH)
4
+1.1705Mg
2
SiO
4
+0.1445SiO
2
+0.083Fe
2
O
3
+0.215FeO ÿ0.59H
2
Oÿ1319.173 3
(4) 0.72Mg
3
Si
2
O
5
(OH)
4
+0.795Al
2
SiO
5
+0.083Fe
2
O
3
+0.215FeSiO
3
+0.0905Mg
2
SiO
4
+0.3645SiO
2
ÿ0.44H
2
Oÿ1322.931 4
(5) 0.4682Mg
5
Al
2
Si
3
O
10
(OH)
8
+0.3268Al
2
O
3
+0.083Fe
2
O
3
+0.215FeSiO
3
+1.2854SiO
2
ÿ0.8728H
2
Oÿ1326.682 6
(6) 0.4682Mg
5
Al
2
Si
3
O
10
(OH)
8
+0.3268Al
2
SiO
5
+0.215FeSiO
3
+0.083Fe
2
O
3
+0.9586SiO
2
ÿ0.8728H
2
Oÿ1326.784 6
(7) 0.72Mg
3
Si
4
O
10
(OH)
2
+0.795Al
2
O
3
+0.083Fe
2
O
3
+0.215FeO + 0.025SiO
2
+0.181MgO + 0.3705H
2
Oÿ1326.978 6
(8) 0.4682Mg
5
Al
2
Si
3
O
10
(OH)
8
+0.3268Al
2
Si
2
O
5
(OH)
4
+0.083Fe
2
O
3
+0.215FeSiO
3
+0.6318SiO
2
ÿ1.5264H
2
Oÿ1329.418 8
Regression equation: aDG
f,i
0
=35.2433908e
ÿ0.373689x
ÿ1330.758301
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250248

values ÿ1255.997 and ÿ1259.383 kcal/mol reported by
Helgeson et al. (1978) and Robie et al. (1978), respect-
ively.
The chemical combination equations for greenish-
yellow vermiculite from North Carolina (analytical
data from Deer et al., 1967) is also shown in Table 1.
Here, eight equations with their aDG
f,i
0
were obtained
with ranks xas 0, 2, 3, 4, 6, 6, 6 and 8. Fig. 3(B)
shows the best-®t curve and the corresponding
equation gives the values of a,band c. The estimated
DG
f
0
of this mineral is ÿ1330.758 kcal/mol. No value
of DG
f
0
for a vermiculite of similar composition is
available in the literature. This further emphasizes the
advantage with this process whereby a hitherto
Fig. 3. (A) Regression of aDG
f,i
0
vaues for pyrophyllite. (B) Regression of aDG
f,i
0
vaues for green-yellowish vermiculite.
M. Kudrat et al. / Computers & Geosciences 25 (1999) 241±250 249

unknown value of DG
f
0
for a clay mineral can be
obtained from a knowledge of its chemical compo-
sition alone.
Acknowledgements
Thanks are due to Dr. M.L. Manchanda, Head,
Regional Remote Sensing Service Centre and Shri B.
Prabhakaran, Scientist, RRSSC, Dehradun and Mrs.
Shashi Kudrat, IIRS, Dehradun for their kind encour-
agement and support.
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