Calculation of Standard Thermodynamic Potentials for Na-Zeolites with the Use of Linear Programming Problems

doi:10.4236/ijg.2011.23024

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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

E-mail: [email protected]

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

has a full

rank. Then for it is possible to write down the unique

chemical reaction [5]. For example, for

S = {NaAlSi2O6·H2O (analcime); NaAlSi2O6 (dehy-

drated analcime); Na2Al2Si3O10·H2O (natrolite); NaAl-

SiO4 (nefeline); 33 102

aAl SiO(OH) (paragonite); }

reaction can be written: 2

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

—values of standard enthalpy formation

from elements for substances from ,

O. V. YERIOMIN

228

- vector of their quantity,

b - vector of chemical elements mass balance.

 is a nonsingular so lution of (2), then:

xby







, (3)

where

—solution of equivalent to (2) dual problem

[6]:

max ,f

byAyH

 , (4)

where - index of transposing.

 contains nonzero quantity of any zeolite

, then from (3) follows:

x





zAzy





, (5)

where



z



—the potential of zeolite at equi-

librium, z

z—vector-column of

matrix, corre-

sponding to stoichiometric formula of zeolite .

Equality (5), namely the



value can be used for

calculation of unknown



z for Na-zeolites by

means a formula:

 

zYzy





, (6)

where —stoichiometric vector of zeolite .



Yz z

Let’s consider a numerical example.

The f

 values for component according to

their written sequence are taken from [7]:

 = – (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:

= (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

(NaAl SiO13HO)

Y(NaAl SiO13HO)y

(4,4,20,61,26)

(32157, 380213, 26705, 428889, 51073)

=26760655 (J/mole).















Similarly we’ll evaluate the values in (2). For

data [7]: fG

f = – (3291100, 2974800, 5718600, 2073800,

5943500, 0) J/mole,

G

solution (2):



= (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

Gibbs potentials and enthalpies are connected with

data consistency equion of “Selektor” software [3] for

calculation of S:

S





GTL



 



, (7)

where -erature (298.15 K),

T temp





, LYzs

(,,,,)

NaAlSi O H

sssss



= (51.2, 28.3, 18.8,) 102.5, 65.0

J/moes of le/K. Valu

 - entropies of chemical ele-

ments are taken from [8

Results of calculations o

].n (6) and (7) and comparison

. Results and Discussion

he set S defihe forms of problems (2). The choice

lculation of un-

hermodynamic poten-

tia

can be considered as the chemical ele-

calculations on (6) depends on vari-

ith published data [7] are presented in Table 1.

Tnes t

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





( kJ/mol(J/mol) ( kJ/mole)

S

e/K fG

 

(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

  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

fG,f

 , 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

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

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.

. References 6

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