International Journal of Geosciences, 2011, 2, 227-230
doi:10.4236/ijg.2011.23024 Published Online August 2011 (http://www.SciRP.org/journal/ijg)
Calculation of Standard Thermodynamic Potentials for
Na-Zeolites with the Use of Linear Programming Problems
Oleg Viacheslavovich Yeriomin
Establishme nt of the Russ i a n Aca demy of Sciences, Institute of Natural Resources, Ecology, and Cryology,
The Siberian Branch of the Russian Academy of Sciences, Chita, Russian Federation
Received April 1, 2011; revised June 2, 2011; accepted July 6, 2011
Abstract
Linear programming problems for Na-Al-Si-O-H system have been formulated and solved for calculations of
standard enthalpies and Gibbs potentials of zeolites with unknown thermodynamic properties. The calcula-
tions are based on dual solutions of linear programming problems. Comparison of numerical results with
published data gives relative mistakes of estimations less than one percent. On the basis of calculated poten-
tials the standard entropies have been estimated. The standard thermodynamic potentials for eight natural
zeolites with unknown properties have been calculated. The presented method does not demand any infor-
mation about crystal structure of zeolites and can be applied to any of their stoichiometric presentation.
Keywords: Na-Zeolites, Standard Thermodynamic Potentials, Linear Programming
1. Introduction
Zeolites are very important aluminosilicate microporous
substances of multipurpose usage. Possessing specific
framework they actively participate in processes of sorp-
tion and ion exchange, catalysis, that has caused their
wide application in the industry, agriculture, medicine,
environment protection.
The knowledge of thermodynamic properties of zeo-
lites provides the investigations of their behaviour in
nature, technological and biochemical processes, in syn-
thesis of new materials.
Alongside with experimental methods of determina-
tion of physical-chemical properties of zeolites there are
some predictive methods now available. One group of
methods is based on the additivity of oxides (or hydrox-
ides) components of zeolites [1]. These models have a
good accuracy for anhydrous forms but in cases with
zeolitic water the divergence of estimations is more sig-
nificant. Another group of methods uses the framework
data of zeolites [2].
Whereas natural and synthesized zeolites are often
presented by stoichiometry variable compounds the role
of empirical methods of calculation of their thermody-
namic properties is very important.
Methods of thermodynamic modelling in geochemis-
try allow to analyze the physical and chemical properties
of systems and components on the basis of dual theorems
in convex programming [3]. With the use of linear pro-
gramming problems for complex copper sulfates the
relative mistakes of standard Gibbs poten tials calculation
have been obtained at about 1% [4]. We’ll consider the
application of linear programming methods for an esti-
mation of thermodynamic potentials for Na-zeolites.
2. Methodology
For five chemical elements Na-Al-Si-O-H we’ll con-
struct the set from six substances including zeolites
in such a way that stoichiometric matrix
S
A
has a full
rank. Then for it is possible to write down the unique
chemical reaction [5]. For example, for
S
S = {NaAlSi2O6·H2O (analcime); NaAlSi2O6 (dehy-
drated analcime); Na2Al2Si3O10·H2O (natrolite); NaAl-
SiO4 (nefeline); 33 102
N
aAl SiO(OH) (paragonite); }
reaction can be written: 2
O
262 23102
26 24
NaAlSiONaAl Si O2HO
2NaAlSi OH ONaAlSiO


(1)
Let’s define a following linear programming problem:
min,, 0
fHxAx bx


, (2)
where f
H
—values of standard enthalpy formation
from elements for substances from ,
S

O. V. YERIOMIN 228
x
- vector of their quantity,
b - vector of chemical elements mass balance.
If
x
 is a nonsingular so lution of (2), then:
f
H
xby



, (3)
where
y
—solution of equivalent to (2) dual problem
[6]:
max ,f byAyH
 , (4)
where - index of transposing.
'
If
x
 contains nonzero quantity of any zeolite
, then from (3) follows:
0
z
x


f
H
zAzy


, (5)
where

f
H
z

—the potential of zeolite at equi-
librium, z
A
z—vector-column of
A
matrix, corre-
sponding to stoichiometric formula of zeolite .
z
Equality (5), namely the
y

value can be used for
calculation of unknown

f
H
z for Na-zeolites by
means a formula:
 
f
H
zYzy


, (6)
where —stoichiometric vector of zeolite .

Yz z
Let’s consider a numerical example.
The f
H
 values for component according to
their written sequence are taken from [7]:
S
f
H
 = – (3291100, 2974800, 5718600, 2073800,
5943500, 0) J/mole.
Vector of mass balance is defined by reaction (1):
(,,,,)
NaAlSi OH
bbbbbb (3, 3, 5, 18, 4) mole.
The problem (2) has solution:
x
= (0, 1, 1, 0, 0, 0)
mole – mole of dehydrated analcime and mole of natro-
lite.
The problem (4) has solution:
(32157, –380213, –26705,
–428889, 51073) J/mole.
*****
(,,,,)
NaAlSi OH
yyyyyy

The equation (6) for dachiardite can be written:
442048 2
442048 2
(NaAl SiO13HO)
Y(NaAl SiO13HO)y
(4,4,20,61,26)
(32157, 380213, 26705, 428889, 51073)
=26760655 (J/mole).
fH







Similarly we’ll evaluate the values in (2). For
data [7]: fG
f = – (3291100, 2974800, 5718600, 2073800,
5943500, 0) J/mole,
G
solution (2):
x


= (0, 1, 1, 0, 0, 0) mole,
solution (4): (19309,
–358452, –29039, –401930, 67990) J/mole.
*****
(,,,,)
NaAlSi OH
yyyyyy
standard entropies . We’ll use the thermodynamic
at
Gibbs potentials and enthalpies are connected with
data consistency equion of “Selektor” software [3] for
calculation of S:
S


f
GTL
f
SH

 

, (7)
where -erature (298.15 K),
T temp


, LYzs
(,,,,)
NaAlSi O H
s
sssss

= (51.2, 28.3, 18.8,) 102.5, 65.0
J/moes of le/K. Valu
s
 - entropies of chemical ele-
ments are taken from [8
Results of calculations o
].n (6) and (7) and comparison
w
. Results and Discussion
he set S defihe forms of problems (2). The choice
lculation of un-
kn
hermodynamic poten-
tia
can be considered as the chemical ele-
m
calculations on (6) depends on vari-
ab
ith published data [7] are presented in Table 1.
3
Tnes t
y*
of
of components S is rather arbitrary. It is desirable that the
included zeolites have experimental or estimated ther-
modynamic data. The condition of full rank A matrix
allows to make the preliminary chemical interpretation of
system. The problem (2) for Gibbs energy can be con-
sidered as thermodynamic equilib rium calculation for the
heterogeneous mixture S under standard temperature and
pressure. The solutions of (2) define zeolites for which
the condition (5) will be satisfied. They can be named
“basic” zeolites - on the basis o f which the estimation on
(6) will be applied for other substances. Natrolite and
dehydrated analcime are the “basic” zeolites in our cal-
culations. We can see from Table 1 the implementation
of an optimality criterion (5) - exact equality of thermo-
dynamic potentials for “basic” zeolites.
For the existing solution y* the ca
own enthalpies (or Gibbs energies) of substances by
means (6) is simple and represents the scalar multiplica-
tion of two vectors. For any stoichiometric forms of zeo-
lites the use of (6) gives the molal dimension of poten-
tials. As a example, the estimations on (6) and compari-
son with calorimetry measurements [10] for some hy-
drous and anhydrous forms of sodalite family zeolites are
presented in Table 2.
The calculated on (6) and (7) t
ls for some natural Na-zeolites [11] with unknown
properties are presented in Table 3. These data may be
used in geochemical calculations of processes with their
participation.
The vector
ents contributions to potentials of zeolites. The en-
thalpy solution of (2)—y* are presented as a bar diagram
on Figure 1. We can see from figure that oxygen and
aluminium atoms provide the most contributions in en-
thalpy potentials. Hydrogen and sodium have a positive
energy values.
The stability
ility of y* components in connection with data uncer-
tainty for “basic” zeolites. This problem demands the

O. V. YERIOMIN 229
-zeolites calculated on (6) and (7). In brackets - a relative
[7] Calculated on (6) and (7)
Table 1. Values of standard thermodynamic potentials for some Na
mistakes of calculations (%) with published data [7,9].
Chemical formula (mineral)
(kJ/mol
fG
e)
f
H


( kJ/mol(J/mol) ( kJ/mole)
S
e/K fG
e)
f
H
 
(kJ/mole (J/mol) )
S
e/K
2808.8 2974.8 175.4 2808.8
(0) 2974.8 (0) 174.5 (0.5)
NaAlSi2O6 (dehydrated a n a lc ime)
Na2Al2Si3O10·H2O (na trolite) 5316.6 5718.6 359.7 5 5
NaAlSi2O6·H2O (analcime) 3068.3 3291.1 234.3 3 3
Na4Al4Si20O48·3H2O (dachiardite) [9]24724.0 26723.0 1947 2 2
316.6
(0) 718.6 (0) 356.9 (1.0)
074.7
(0.2) 301.5 (0.3) 210.3 (11.4)
4687.3
(0.1) 6760.6 (0.1) 1919 (1.5)
able 2. The standard enthalpies of formation from elements(kJ/mole) of some hydrous and anhydrous forms of T fH 
sodalite family materials [10]. In brackets—mistakes of calculations on (6).
Chemical formula f
H
  kJ/mole Calculated on (6)
Na7.82(OH H2O)3.27 14)1.84[Al5.98Si6.02O24]( 275.4 14239.9 (0.2)
Na7.82(OH)1.84[Al5.98Si6.02O24] 13181.7 13171.5 (0.1)
Na7 3.00
.60(OH)1.64[Al5.96Si6.04O24](H2O) 14093.2 14076.2 (0.1)
Na7.60(OH)1.64[Al5.96Si6.04O24] 13085.1 13095.9 (0.1)
able 3. The standard thermodynamic potentials of some natural Na-zeolites calculated on (6,7).
Chemical formula (mineral) kJ/mole
T
fG,f
H
 , kJ/mole , J/mole/K
S
Na8Al6624 2 2 13281417Si O(OH) ·3HO (cancrinite)2.5 0.7 1205.8
Na8Al8Si16O48·22H2O (gmelinite) 28321.3 30947.0 2120.6
NaAlSi5O12·3H2O (mordenite) 6105.3 6649.6 449.8
N)
N)
N
a2Al2Si3O10·3H2O (paranatrolite5582.5 6037.9 390.3
Na8Al6Si6O24(OH)2 (sodalite) 12484.7 13212.7 908.7
a5Al5Si11O32·1H2O (gobbinsite17802.3 19364. 2 1298.4
a1.6Al1.5Si36O72·18H2O (heulandite)35278.2 38653.1 2788.2
NaAlSi3O8·3.5H2O ( z e o l i t e ) 4572.5 5008. 4 348.4
Figure 1. The enthalpy potentials of chemical elements for Na-zeolites.

O. V. YERIOMIN 230
entary analysis. F
calculation standard thermodynamic
o
nd G. Cao, “A New Method of Estimating
supplemor our example, the handbook
[7] contains one enthalpy value for natrolite, and differ-
ence between maximum and minimum values for dehy-
drated analcime is 15.4 kJ/mole (~0.6%). For this un-
certainty the divergence of estimations on (6) for all con-
sidered in article substances does not exceed 1%.
Evaluation on (6) gives for all Na-zeolites enthalpy
value of zeolitic water equal to –326.7 kJ/mole. This
energy of water is more preferable that ones in structure
of ice-I –292.7 kJ/mole [12].
The considered method has restriction in the applica-
bility only to Na-zeolites, but can be extended by addi-
tion in system of calcium, magnesium and (or) others
chemical elements.
. Conclusions 4
The method of
p- tentials for Na-zeolites has been proposed on the
basis of dual solutions of linear programming problems.
The re- sults of estimation have acceptable accuracy with
pub- lished experimental and predicted data. The pre-
sented method does not demand any information about
crystal structure of zeolites and can be applied to any of
their stoichiometric presentation.
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