Giddings, J. C., "Dynamics of Chromatography, Part I, Principles and Theory," pp 26-35, Marcel Dekker, New York, N.Y., 1965. Pigford. R. L.. Baker, B., Blum. D. E., Ind. €ng. Chem., Fundam., 8, 144 (1969). Sweed, N. H., Wilhelm, R. H.. lnd. Eng. Chem., Fundam., 8.221 (1969). Wankat, P. C., Separat. Sci., 8, 473 (1973). Wankat, P. C., J . Chromatog.. 88. 211 (1974a). Wankat, P. C.. Separat. Sci., 9, 85 (1974b). Wilhelm, R. H., Rice, A . W., Bendelius, A. R., Ind. Eng. Chem., Fundam., 5, 141 (1966).
Zhukhovitskii, A . A , , in "Gas Chromatography-1960,'' W. Scott, Ed., Butterworths. London, 1960.
pp 293-300, R. P.
Received for review J a n u a r y 31, 1974 A c c e p t e d N o v e m b e r 8, 1974 &search was p a r t i a l l y s u p p o r t e d by N a t i o n a l Science F o u n d a t i o n G r a n t No. GK-32681.
The Independent Reactions in Calculations of Complex Chemical Equilibria Pehr H. Bjornbom The ROyal lnsfitute of Technology. Department of Chemical Technology, S- 700 44 Stockholm, Sweden
From the atom matrix infinite many sets of linearly independent reactions can be constructed, a priori, which can describe all t h e composition changes d u e to t h e complex reaction. However, these sets cannot automatically be used for equilibrium calculations. T h e independent reactions for the latter purposes might be fewer than those from t h e atom matrix and t h e y are defined by t h e following requirements: ( 1 ) t h e reactions are linearly independent; ( 2 ) t h e y can describe all the available composition changes of t h e system; (3) all t h e extents of reaction can vary independently. T h e number of reactions, defined in t h i s way, must be found from f u r t h e r experimental information than is stored in the atom matrix. This n u m b e r decreases when passivation of molecular processes occurs; for example, a group in t h e molecules being inert. In such cases m a n y complex equilibrium calculation methods, proposed in t h e literature, will fail, especially free energy minimization with element material balances constraints.
Introduction The concept of independent (or distinct) chemical reactions has been introduced in the discussion of complex chemical equilibria (Jouguet, 1921; Aris and Mah, 1963). The equilibrium composition of a chemically reacting system can be obtained by analytical or computational minimization of the Gibbs free energy (1)
G = G(T,P,nj)
where n, (i = I, 2, . . . N ) refers to the number of moles of species i in the system. Zeleznik and Gordon (1968) have reviewed the field of complex equilibria. Let the following T reactions be the independent reactions C b i j B i= 0 i
( j = 1,2,.
.. , T )
(2)
where B, refers to the compound i and b,, is the stoichiometric coefficient of B , in reaction number j. By definition these reactions are linearly independent. According to reactions 2, the composition changes in the system must satisfy the following equations
[,
where refers to the extent of reaction for the j t h reaction. In matrix notation eq 3 becomes An = BAE
where B refers to the column vector sition). An refers 5r1.v)~ and 5E to 102
(4 )
the matrix, the j t h column of which is ( b ~ , ,bz,, . . . b , ~ , () T~ denotes transpoto the column vector ( A n i , An2, . . . , (551,5 5 2 , . . ., 5 5 ~ )B~ is . called the
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
stoichiometric matrix of the set of reactions 2. The equations 3 define the constraints under which (1) is to be minimized. According to Brinkley (1946), a set of T = N-R reactions can be constructed from the formula matrix (see below), where R refers to the rank of this matrix. This would indicate that there are N-R degrees of freedom in the An, values, if the 55' values are chosen independently. As Zeleznik and Gordon (1968) have pointed out, Brinkley's approach is equivalent to minimizing ( l ) ,using the material balance over the elements in the system as the constraints. A different view is held by Aris and Mah (1963), who implied that the number of independent chemical reactions is an empirical entity; they proposed a method for experimental determination of this number. The purpose of the present paper is to clarify this contradiction in the literature. The Stoichiometric Behavior of a Complex Chemical Reaction Restrictions Due to the Conservation of Atoms. The assumption that a complex reaction stoichiometrically can be described by the reactions 2 must be shown by showing that there are extents of reaction which satisfy eq 3 for all composition changes. This means that it is equivalent to say that (3) can be satisfied for all composition changes and that reactions 2 describe the composition changes. Assume that there are M species of atoms in the system denoted AI, Az, . . . , A.w. The stoichiometric coefficients of a reaction must satisfy the material balances over the atoms partaking in this reaction. Thus if the reaction is xbiBi = 0 i
(5)
we have the following material balances over the M atoms
(k = 1,2,..
Cakibi= 0 i
where aAL = the number of atoms notation this is Ab = 0
Ah
. ,M )
(6)
in &. In matrix
(7)
where b = the column vector ( b l , b2, . . . , b.v)T and A is the matrix, the ith column of which is the column vector (al,, a2i, . . . , a,vi)T. This column vector is called the formula vector of B,. A is called the formula matrix or the atom matrix. Equations 6 and 7 mean that b is a solution of the equation system
AX
(8)
= 0
where x is a column vector of unknowns (XI, x2, . . . , x N ) T . Moreover, every solution of (8) is a column vector of stoichiometric coefficients for some chemical reaction. There is a theorem in linear algebra (see, e.g., Shilov, 1961), which says that the equation system (8) has a t most N-R linearly independent solutions, where R is the rank of A. Furthermore, every solution of (8) is a linear combination of any set of N-R linearly independeht solutions. Now choose an arbitrary set of N-R linearly independent chemical reactions CbijBi = 0
( j = 1,2,
i
...,N-R)
(91
The corresponding column vectors of stoichiometric coefficients, bl, bp, . . . , b,v-R, are solutions of (8). Let us further note that An is a solution of (8), since the following equation A h = O (10) simply is the matrix notation of the material balances over the atoms. Thus An is a linear combination of bl, b2, . . . , b.v-H An = ffibi f ff2b2 + + ffH-RbN-R (11) In matrix notation this is An = BCY
. ..
where N = ( ~ 1 ,N Z , . . . , N ~ v - R ) ~Thus . cy satisfies eq 4, which means that ~ 1 CY^, , . . . , N N - H are the - 1 E values corresponding to the reactions 9. We have thus shown that a n arbitrary set of N-R linearly independent reactions can describe t h e composition changea in t h e s y s t e m . Restrictions Due to Molecular Processes. The result obtained above is founded on the single assumption that the material balances over the atoms are satisfied. Therefore this result must be generally valid and independent of the other factors, which influence the stoichiometric behavior of the system. However, the fact that the composition changes can always be described by N-R reactions does not mean that they cannot be described by fewer than N-R linearly independent reactions. The latter assumption would mean that all the composition changes, which are permitted without violation of the conservation of atoms, do not take place, but the molecular processes actually taking place in the system restrict the composition changes. Thus, the composition changes would be possible to describe both by the use of N-R reactions and by some set with less than N-R reactions. Let S denote the smallest number of linearly independent reactions, needed to describe the composition changes of the system. We will assume the general case that S is less than N-R. Therefore T in eq 2 can be equal to N-R for one set of reactions, but equal to S for another set. But this would indicate that the number of An, values, which can be arbitrarily chosen, is equal to N-R and '3 in the same time. However, as we will see, this is
not so, since when T = N-R, it is not possible to choose all of the A t Lvalues arbitrarily, but only S of them. Now consider two different sets, constituted of N-R linearly independent chemical reactions. The column vectors of stoichiometric coefficients are denoted by bl, b2, . . . , bLv-Rand c1, c2, . . . , C ~ - Rand the corresponding stoichiometric matrices by B and C. Then, according to eq 4 An = B A S = C A q
(12) where f refers to the extent-of-reaction column vector corresponding to B, and 1) refers to the extent column vector corresponding to C. The transformation of C to B can be written B = CZ (13) where Z refers to the nonsingular transformation matrix. Thus C(ZA[) = CAV (14) and since the extents of reaction variations are unique AQ = Z A t (15) Equation 15 shows how the extents of reaction variations transform, when B is transformed into C. Now consider S linearly independent reactions, with the corresponding column vectors d l , dp, . . . , ds. Thus eq 3 can be written An = diAOi 4 d2A8, + + d,AO, (16) where 81, 02, . . . , Os refer to the extents of reaction. d l , d2, . . . , d \ constitute a set of solutions of (8). According to a theorem in linear algebra, this set can be extended by N-R-S further solutions, to become a set of N-R linearly independent solutions, d l , d2, . . . , d%-R.Then
...
...
An = diAOi f dzAO2 + f dN-RA6N-R (17) However, due to the uniqueness of the extents of reaction, A&, 1 0 2 , . . . , ABS must have the same values in (17) as in (16) while AOs+l = . . . = A 8 . v . ~= 0. Now consider the transformation of B into D, where D refers to the stoichiometric matrix, the columns of which are d l , dp, . . . , d,\ -R . B = DZ (18) Thus AB = ZAE (19) The last N-R-S equations of (19) can be written AO, = 0 = X z j i A [ i i
(j =
s +
1,
..,, N - R ) (20)
where z,, is the element of Z in the intersection of t h e j t h row and the ith column. Equations 20 are N-R-S linearly independent equations, which the A t Lvalues must satisfy. Thus we have shown that all of t h e A t L ualues cannot be chosen arbitrarily since N-R-S of t h e m are determined b y (20)w h e n S of t h e m have been chosen On the other hand, there are no such equations like (20), which the A8 values must satisfy. Otherwise one such equation could be used to eliminate one of the -18's in (16), which would indicate that the smallest number of reactions needed to describe the composition changes is less than S The Independent Reactions The results above show that the number T of reactions 3 can have different values: N-R or '3. However, if AV-R reactions are chosen, the corresponding extents of reaction cannot be varied independently, but only S of them. Therefore, eq 3 must be completed by equations like (20), otherwise the n, values will be varied over a range of states of the system, some of which cannot exist. On the other hand, if T = S reactions should be chosen, all the extents Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
103
could be varied independently. Therefore, a set of S linearly independent reactions should be chosen as the independent reactions. In summary a set of independent reactions is defined by the following requirements: (1) the reactions are linearly independent; (2) they can describe all the available compositions changes of the system; (3) all the extents of reaction can vary independently. The number of independent reactions, S, must be determined empirically. The method proposed by Aris and Mah (1963) can be used. In this method the matrix of experimentally found sets of AnL values (not less than N-R sets) is studied. The rank of this matrix equals S . However, it is also possible to transform the experimental values into A t c values, corresponding to N-R reactions. The rank of the matrix formed by these values also equals S. This method has the advantage that less data are handled in the matrix and that not all the Anc values need be experimentally determined. The set of N-R reactions, which is needed for the latter variation to determine S, can be constructed as suggested by Brinkley (1946). Alternatively, a set of N-R linearly independent solutions of (8) can be constructed, since the solutions directly correspond to chemical reactions. The problem of finding these solutions is solved in linear algebra (see Shilov, 1961). Examples One question which calls for example is if nontrivial cases where S is less than N-R really occur in practice. Recently this question has been treated in the literature. In the liquid-phase oxidation of propane N-R = 7 but 6 reactions are sufficient to describe the system (Bjornbom, 1974) and in the sulfonation of a mixture of isomeric cresols the real number of independent reactions is also less than N-R (Whitwell and Dartt, 1973). Two examples will now be examined in order to illustrate the ideas brought out in this paper. Example 1. This is the hydrodealkylation of toluene to benzene. Schneider and Reklaitis (1975) have discovered that S = 1 < N-R in this case. The hydrodealkylation can be described by one reaction C,H,CH,
+ Hz = C,H,3 + CH,
(21)
for experiments have shown that benzene and methane form in practically equimolar amounts (Fowle and Pitts, 1962). In this system N-R = 2. Therefore, any set of two linearly independent reactions can be used to describe the composition changes, for example the following set CgHjCH, + 10H2 = 7CH4 9C,HjCH3 = lOC,H, + 3CH4
(22)
(23) However, since one single reaction can describe all the available compositions changes, S = 1 in this system. Consequently the reaction extent variations At22 and A t 2 3 are not independent. They must satisfy the following equation At22 = A523 (24) which can easily be verified numerically. For example, if 10 mol of toluene react to 10 mol of benzene and 10 mol of methane, the extent variation for reaction (21) A721 = 10 mol. But this can also be explained by setting A t 2 2 = At23 = 1mol since An =
(2 5) 104
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
The two reactions (21) and (22) are also linearly independent and can consequently describe the composition changes of the system. The two reaction extent variations, A721 and A722, must be dependent. This dependence is expressed by the equation A7722 = 0 (26) since, according to the definition of linear independence, eq 26 means that A721 and A7722 are linearly dependent. Equation 12 applied to the two reaction sets (21), (22) and (22), (23) gives
From this equation is obtained A7721 = 1 0 A h A7722 = A522
-
(28)
ht23
This is eq 15 applied to this example. Now equals zero so eq 29 takes the following form
A722
(29) always (24)
At22 = A t 2 3
Thus we have derived eq 24. Z in eq 15 takes the following value 0 10 = (1
-1)
According to eq 13, Z transforms the matrix of reactions 21 and 22 into the matrix of reactions 22 and 23. Testing this we obtain the expected result
[;';)(;
)!
=(j -1
-9
l$
In summary, we find that the hydrodealkylation of toluene can be explained by the single reaction 21, by the two reactions 21 and 22, or by the two reactions 22 and 23. In the two last cases there are, however, restrictions for the extent variations. In the first case this means that there is no conversion according to (22); in the second case the extent variations must satisfy eq 24. Let us now turn to the calculation of equilibrium of this reaction. Since S = 1 only reaction 21 should be considered in the equilibrium calculation. A set of two reactions, such as (21) and (22) or (22) and (23), must not be used since the extents cannot be varied independently. Neither must the free energy be minimized, using the element material balances as constraints, since this is equivalent to using two linearly independent reactions. As an illustration, the equilibrium has been calculated for the conditions used by Fowle and Pitts (1962): 980 K and 43 atm. The calculation was made both using reaction 21 as the single reaction and using the set of reactions 21 and 22. The standard free energies for reactions 21 and 22 a t 980 K was estimated from a diagram with standard free energies of formation of hydrocarbons at different temperatures (Hougenet al., 1962). AGO^^ = -10,ooo cal mol" AGO22 = -49,000 cal mol-' The calculations were made for two mole ratios between hydrogen and toluene: 4 and 6. Calculation with only reaction 21 shows practically complete conversion to benzene a t equilibrium. The composition of the equilibrium mixture resulting from calculation with both reaction 21 and 22 is 0.12 mol of toluene, 0.07 mol of hydrogen, 0.56 mol of benzene, and 2.91 mol of methane per mol of en-
tering toluene and the mole ratio 4. For the mole ratio 6 the corresponding result is 0.08, 0.12, 0.37, and 4.23 mol/ mol of entering toluene. The experimental results show that the equilibrium state calculated with only eq 21 is approached, increasing the conversion, rather than the state calculated with two reactions. The results with two reactions indicate that it is favorable to use a small hydrogen/toluene ratio. This is also in conflict with experiments (Silsby and Sawyer, 1956). Example 2. This is the cresol sulfonation example suggested by Whitwell and Dartt (1973). A mixture of o-cresol and m-cresol reacts with HzS04 to the corresponding o-cresolsulfonate and m-cresolsulfonate. In this case N-R = 3 and therefore all the composition changes can be described by three linearly independent reactions (if one column is included in the atomic matrix for each isomer the mathematics developed above is applicable) such as o-C7H80
+
H2SOd = O-C7H$O,
w - C ~ H ~+ O HzSO,
r
O-C7H8S04 =
+
w-CYHBSO, ???
-C 7H~S04
+
H@ H,O
(30) (31) (32)
However, experimental results show that 0- and m-sulfonate form in the same ratio as the ratio of 0 - and m-cresol in the feed. This means that no isomerization reaction ( 3 2 )can occur and that S = 2. In this reaction the cresols evidently are completely converted at equilibrium if more than a n equimolar amount of H2S04 is added. If the equilibrium were calculated by use of all the three reactions, the equilibrium mixture would contain 0 - and m-sulfonate in a constant ratio, not varying with the reactant isomer ratio, determined by the equilibrium of reaction 32. The same result would be obtained by minimizing the free energy with element material balance constraints. This result is not correct, but the equilibrium mixture must be calculated using reactions 30 and 31 only. Different Cases of Molecular Restrictions At least four different cases, resulting in S being less than N-R, can be recognized. The two cases with no reactions at all or with inert components in the reaction mixture are trivial, only to mention that Peneloux (1949) has suggested that these cases are the only forms of reaction passivation which can occur. The examples show that this is wrong. All the examples, including the liquid phase oxidation of propane, belong to the third case. The molecular processes are restricted because groups in some of the molecules are inert. In the hydrodealkylation case the aromatic nucleus is inert, and in the sulfonation case the two isomeric nuclei with their attached methyl- and hydroxy groups. In the liquid phase oxidation of propane two carbon atoms from the carbon chain of propane form an inert group. In the fourth case the reaction mechanism causes the restrictions, but the phenomenon cannot be attributed to the existence of inert groups. The metal-catalyzed decomposition of sec-butylhydroperoxide is an example of this. In pentane solution the products are sec-butanol, methyl ethyl ketone, oxygen, and water. The alcohol and the ketone form in the mole ratio 2:l (Hiatt et al., 1968). This means that S = 1, although N-R = 2. This cannot be attributed to inert groups in some of the molecules. Discussion The neglect to consider reaction passivation in designing equilibrium computation methods is surprising, since Gibbs (1876) has comprehensively discussed this problem
in his great paper on equilibrium. Using Gibbs’ term, the mixture obtained by minimizing the free energy with the element material balances as constraints is a phase of dissipated energy. In contrast to this state transitory states may also exist. Characteristic for the latter is that transformations of one set of molecules to another set of different molecules, viz. reactions, which are possible according to the composition of the molecules, do not occur. The equilibrium is determined only by those transformations which take place. Gibbs also suggested that a homogeneous body, in which molecular changes take place slowly, successively may pass transitory states before it is brought to the dissipated energy condition. Applied to toluene hydrodealkylation, for example, this means that an equilibrium with equimolar amounts of benzene and methane would be obtained, but this state then slowly transforms into the tworeaction equilibrium mixture. It might be said about the hydrodealkylation example that it is too simplified compared to reality. Examination of Fowle and Pitts’ results show that more methane is formed than equimolar to benzene and other aromatic products form but benzene. The aromatic byproducts do not change anything in principle, since they simply can be looked upon as being formed of initially produced benzene. However, the amount of methane, although only slightly too big, shows that there is a slow degradation of the aromatic nucleus. The resistance to degradation is high but the latter is not prevented as the isomerization in the sulfonation case, for example. Therefore, it would be wrong to calculate the one-reaction equilibrium, since now all the states only restricted by the element material balances are possible. In principle, this reasoning is correct, since only states which are prevented must be neglected according to Gibbs. However, the experimental results show that the state of this system a t first changes not at all into the direction of the two-reaction equilibrium but rather approaches the hypothetical transitory state, the one-reaction equilibrium. This means that the state of the system passes close to the transitory state, although the latter cannot be attained. This can be expressed such that the transitory state ‘seems to attract the state of the system due to the high resistance to changes in the direction defined by reaction 22. This idea can be supported by the argument that there is actually no difference in principle between this system with high resistance to a certain change and other systems where a change is said to be prevented. Considering the concept of activation energy, one can rather say that in systems such as in the sulfonation example, or a mixture of hydrogen, oxygen, and water vapor a t room temperature, there is a very high resistance to certain changes. The latter, however, are not completely prevented but take place so slowly that they cannot be detected. In summary, the resistance to isomerization is so big in the sulfonation that the transitory state is practically obtained. In the hydrodealkylation the resistance to degradation is smaller but sufficiently big so that the hypothetical transitory state greatly influences the direction of the reaction. Conclusion Given a complex chemical reaction, the initial step is to determine the products, thus defining the atom matrix. After that it is possible to find a priori an arbitrary set of N-R linearly independent reactions, the extents of which can describe all the composition changes in the system. However, if equilibrium calculations are to be done, it must be further determined if the extents are independent. If they are so, the N-R reactions found are the indeInd. Eng. Chem., Fundam.. Vol. 14, No. 2, 1975
105
pendent reactions and can be used for the calculation as well as the element material balances. In the opposite case there are fewer independent reactions, S, than N-R, defined by the conditions given in this paper, and only such a set of reactions should be used. In simple cases, such as our examples, it might easily be determined that S is less than N-R and to find the real independent reactions. Thus it should be easily seen from experimental results that toluene is dehydroalkylated according to the only reaction CGHSCH3
+ Hz
= CGH,
+
CH,
(21)
and that there is no isomerization occuring in the sulfonation of cresols. In other cases S can be found in a systematic way from experimental results by use of the matrix method proposed by Aris and Mah (1963) or the modified version proposed in this paper. Final Remarks During the work with the revision of this paper the author was informed about the results of Schneider and Reklaitis (1975). Their discovery that S < N-R in the hydrodealkylation of toluene has been very valuable information for working out example 1. It is interesting to note that the theorem proved in section 2 in the paper by Schneider and Reklaitis includes the same result as in the paragraph "Restrictions due to the conservation of atoms." However, the results in the present paper have been obtained independent of the results of these authors. Acknowledgment The author is indebted to Olle Lindstrom, who encouraged the publication of this paper and made a critical review of the manuscript. The author also wants to thank Edward M. Rosen for his very helpful suggestions regarding this work. Nomenclature Ab = symbol for element h u k i = the number of atoms of element Ah in one molecule of Bi A = the atom or formula matrix B , = symbol for species i b, = the stoichiometric coefficient before B , in a reaction formula
106
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
b,, = the stoichiometric coefficient before B , in reactionj b, b,, c J , d, = column vectors of stoichiometric coefficients B, C, D = matrices with columns constituted of stoichiometric coefficients G = Gibbsfreeenergy M = number of elements N = number of species nr = number of moles of species B , n = the column vector (nl, n2, . . . , n . ~ ) ~ P = pressure R = rankofthematrixA S = the minimum number of reactions needed to describe the composition changes in the system T = temperature x , = unknownvariable x = column vector of unknowns Z = transformation matrix zJ, = the element of 2 in the intersection of the j t h row and the ith column
Greek Letters coefficients in linear combination of column vectors
CY, =
a = (al,CY29 . 9 CYN-R): E,, OJ = extents of reaction E , q,O = column vectors of extents of reaction
Literature Cited Aris. R., Mah, R. H. S.,Ind. Eng. Chem., Fundam., 2, 90-96 (1963). Bjornbom, P. H.,A.I.Ch.E. J., 20, 1026 (1974). Brinkley, S. R.. J. Chem. Phys., 14, 563-564 (1946). Fowle, M. J.. Pitts, P. M., Chem. Eng. Progr., 58 (4), 37-40 (1962). Gibbs. J. W., Trans. Conn. Acad., 3, 108-248 (1876). See "The Scientific Papers of J. Willard Gibbs," Vol. 1, pp 138-44, Dover Publications, Inc., NewYork, N.Y., 1961. Hiatt, R.. Irwin, K. C.. Gould, C. W., J. Org. Chem.. 33, 1430-1435 (1968). Hougen, 0. A., Watson, K. M., Ragatz, R. A., "Chemical Process Principles,'' Part 2, 2nd ed, p 989, Wiley, New York, N.Y., 1962. Jouget. M. E.. J. Ec. Polyt. (Paris) (2de serie, Pleme cahier), 62-69 (1921). Peneloux. M. A.. C. R. Acad. Sci. (Paris), 288. 1727-1729 (1949). Schneider. D. R.. Reklaitis, G. V., Chem. Eng. Sci., in press, 1975. Shilov. G. E., "An Introduction to the Theory of Linear Spaces," Prentice Hall, Englewood Cliffs, N.J., 1961 Silsby, R. I., Sawyer, E. W.. J. Appl. Chem. (London), 6, 347-356 (1956), Whitwell, J. C., Dartt, S. R.,A.I.Ch.E. J., 19, 1114-1120 (1973). Zeleznik. F. J., Gordon, S..Ind. Eng. Chem., 60 ( 6 ) , 27-57 (1968)
Receiued for review August 26, 1974 Accepted December 2, 1974
