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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 diculty 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 dier 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

M2which 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 dierence 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 dierent 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

dier 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,iaebx c1

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,iaebx 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

yfx,a,b,c2

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,

aa0a,

bb0b,

cc0g,3

where a,band gare the values by which the assumed

values dier 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

yn0fxn,a0,b0,c04

where nranges from 1 to p. The approximated value

y

n

'diers from the observed value y

n

by a value, say,

r

n

. Therefore,

yn0ynrn:5

Again, the observed value of y

n

diers from the value

obtained from the best ®tted curve by a value E

n

,

which is related as

Enfxn,a,b,cÿyn:6

On combining Eqs. (3) and (6), we get

Enynfxn,a0a,b0b,c0g:7

This, when expanded in Taylor series as a function of

a,band cgives

Enynfxn,a0,b0,

c0adf=da0bdf=db0gdf=dc0

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

Enadf=da0bdf=db0gdf=dc0rn: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

fxaebx c10

the partial derivatives are

df=daebx 11

df=dbaxebx 12

df=dc113

On substituting Eqs. (11)±(13) into Eq. (9), we have

Enaeb0xnba0xneb0xngrn14

or

Enaeb0xnba0xneb0xnga0eb0xnc0ÿyn: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:

E1aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy116

E2aeb0x2ba0x2eb0x2ga0eb0x2c0ÿy217

Epaeb0xpba0xpeb0xpga0eb0xpc0ÿyp18

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

n1

E2

nminimum E2

1E2

2...E2

p19

Hence,

X

p

n1

E2

naeb0x1ba0x1eb0x1ga0eb0x1

c0ÿy12aeb0x2ba0x2eb0x2

ga0eb0x2c0ÿy22

...aeb0xpba0xpeb0xp

ga0eb0xpc0ÿyp2fa,b,g20

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 dierentiation of Eq. (20) with respect to a,

band gand dividing by 2, we get the following normal

equations:

aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1eb0x1

aeb0x2ba0x2eb0x2ga0eb0x2

c0ÿy2eb0x2...aeb0xpba0xpeb0xp

ga0eb0xpc0ÿypeb0xp021

aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1a0x1eb0x1

aeb0x2ba0x2eb0x2ga0eb0x2

c0ÿy2a0x2eb0x2...aeb0xpba0xpeb0xp

ga0eb0xpc0ÿypa0xpeb0xp022

aeb0x1ba0x1eb0x1ga0eb0x1c0ÿy1

aeb0x2ba0x2eb0x2ga0eb0x2c0ÿy2

...aeb0xpba0xpeb0xpga0eb0xp

c0ÿyp0:23

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 dierent values

of xare ignored.

Suppose, a set of 5 dierent equations for a mineral

is obtained, giving 5 dierent 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 dierent

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 dierent 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 dierent 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 dierent 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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