Ind. Eng. Chem. Fundam. 1981, 20, 161-164
Literature Cited Blumberg, A. A. J . Phys. Chem. 1959, 63, 1129. Blumbwg, A. A.; Stavrinou, S. C. J . Phys. Chem. 1960, 64, 1438. Born, H.; Prigogine, M. J. CMm. Phys. 1979, 79, 540. Dickman, S. R.; Bray, R. H. So/iScl. 1841, 52. 283. Eills, A. J. J. Chem; SOC.1963, 4300. Fogler, H. S.; Lund, K.; McCune, C. C. Chem. Eng. Scl. 1975, 30, 1325. Gatewood, J. R.; Hail. B. E.; Roberts, L. D.; Laseter, R. M. J. Pet. Techno/. 1870. 22. 693. Grim, R.’E.; ;‘Clay Mineralogy”, 2nd ed.; McGraw Hili: New York. Heklm. Y.; Fogler, H. S. Chem. Eng. Sci. 1977, 32, 1. Iler, R. K. “The Chemistry of Silica”, Wiley: New York, 1979. Judge, J. S. J . Electrochem. SOC.1971, 718. 1772.
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Kline, W. E. Ph.D. Thesis, The Unhrerslty of Michigan, Ann Arbor, 1980. Kline, W. E.; Fcgler, H. S. J. Golbhl Interface. S d . 1961, in press. Korrlnga, J.; Seevers, D. 0.; Torrey, H. C. phys. Rev. 1962, 727, 1143. LabrM, J. C. Rev. Inst. Franc. Petrob Ann. Combust. Llquldes 1971, 26 I 885. Samson, H. R. Clay Mineral. Bull. 1952, 1 , 266. Semmens, 8. 6.; Meggy. A. B. J. Appl. Chem. 1966, 16, 122. Semmens, B. B.; Meggy, A. B. J . Appl. Chem. 1988, 16, 125. Strelko, V. V. Theor. Eksp. K h h . 1973, 10, 359. Turner, R. Ph.D. Thesls. Unhrerslty of Californla, Davis, 1964.
Received for review July 21,1980 Accepted February 23,1981
Components in Chemical Stoichiometry Pehr Bjornbom Department of Chemical Technology, Royal InstnUre of Technology, S-100 44 Stockholm, Sweden
Jouguet’s and Brinkiey’s definitions of components are compared. In restricted chemical stoichiometry, when all chemical changes permitted within the framework of the atomic material balances cannot take place, these definitions give different results. In such cases only Jouguet’s definition Is compatible with Gibbs’ phase rule. If Jouguet’s definition is adopted, rather than Brinkley’s, the constraints placed upon the composition changes in restricted chemical stoichiometry cases can be interpreted as material balances over components. A proposed definition of chemical stoichiometry is based on this resuit.
Introduction In chemical stoichiometry the concepts of components and independent reactions are very important. These concepts were introduced by Gibbs (1876) in the theory of chemical equilibrium and have subsequently been discussed by several authors. Literature references are given elsewhere (BjBrnbom, 1975,1977;Smith and Missen, 1979). As Smith and Missen (1979) recently have pointed out, the results of research in chemical stoichiometry have not been given due attention as a separate field. The reason for this is probably that chemical stoichiometry has not been discussed separately but in the context of chemical equilibrium and chemical kinetics. In a previous paper (Bjornbom, 1975))the concept of independent reactions was discussed. It was found that in order to avoid difficulties in treating restricted chemical equilibrium problems the definition of this concept by previous authors should be extended with the requirement of independently variable reaction extents. Definition 1. A set of chemical reactions are independent if and only if (1)the reactions are linearly independent; (2) they can describe all composition changes in the system; and (3) the corresponding reaction extents can vary independently. Basically it is not the equilibrium which is restricted, but the stoichiometric behavior of the system is restricted due to the absence of certain elementary reactions (Bjornbom, 1977). Therefore it is more correct to say restricted chemical stoichiometry than restricted equilibrium. In the literature after Gibbs, two different types of mathematical definitions of components exist. The first method due to Jouguet (1921) starts from the independent reactions. The second method due to Brinkley (1946) and recently used by Smith and Missen (1979) starts from the elemental balances. In many cases these definitions are equivalent. However, in restricted chemical stoichiometry problems they produce different results. 0196-4313/81/1020-0161$01.25/0
Of course, neither of these definitions can be wrong since neither of them contains inherent contradictions. However, according to the criterion chosen it is possible that one of the definitions is better than the other. The aim of the present paper is to investigate this and to elaborate on the component concept. Although the concept of components is a tool for describing the chemical stoichiometry this concept cannot be completely isolated from chemical equilibrium, for components play a very important role in the theory of chemical equilibrium and especially in Gibbs’ phase rule. Therefore the criterion used to compare Jouguet’s and Brinkley’s definitions of components will be compatibility with Gibbs’ phase rule. Remarks on Terminology There is some variation in the terminology in this field motivating some clarifications. We will consider a system composed by molecular species which are compounds of atomic species, the elements. Occasionally there are free atoms in the system. The molecular species and the free atomic species are components of the system. However, due to chemical reactions all these components are not independent, since some of them can be formed out of others. Thus only a subset of the Components are needed to form an arbitrary mixture of all the components. Such a subset occasionally is called a set of independent components. However, since a set of independent components consequently can be considered ultimate components of the system, independent is often dropped, so components will mean the same as independent components. This terminology will be adopted in this paper. The term substance will be used in this paper for those components which are not independent. Definition of Components Definition 2. A component is one of a set of C species of the chemical system, the number of which is the least 0
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number required to make up any compositional state of the system. So far all authors agree on this definition. The differences appear in applying it to systems with restricted chemical stoichiometry. Thus following Jouguet, stoichiometric behavior is defined from the independent reactions giving C = N - r components. Following Brinkley the number of components is C = R. This is because Brinkley in principle defines the stoichiometric behavior directly from the R linearly independent elemental balances without considering that not all chemical changes which are allowed within the framework of the elemental balances can take place in a restricted system. Since in the restricted case r < N - R, it follows from Brinkley's definition that C < N - r. Definition 2 involves one difficulty which is demonstrated by an example. Example 1. We have the reaction CPHSOH + CHSCOOH = CZHbOCOCHB + HzO or
A+B=C+D Choose A, B, and C as the components. Given an initial mixture of the components, by using reaction extent 5 (Henley and Rosen, 1969), we have nA = nAO -
t nc = ne0 + t nD = t nB
=
nBO
-
Solving for nAO,nBO,and nco gives nAO = nA + nD nBO = nB + nD nco = nc - nD Obviously we cannot from A, B, and C produce mixtures where nc < nDunless we introduce an imaginary negative amount of the component C, that is nco < 0. Negative amounts could be avoided by changing the choice of components. In example 1we could choose A, B, and D as components instead. However, in the general case such a method would make the choice of components a function of the calculated end result; that is the choice of one of several components would be an unknown variable in the. calculations. In this work we will adopt the use of negative amounts of components. As we will see this can be done without difficulties since the amounts of components have merely mathematical significance. By this method the components can be chosen independently of the end result of the calculations. In the following we will assume as a work hypothesis that Jouguet's approach is the better one. Therefore C will denote the number of components derived from a set of r independent reactions according to definition 1. This also means that C = N - r. Linear Equations in Ani Variables Let a set of r independent reactions have the stoichiometric matrix S of rank r. Choose a basis of the null space of ST.Let the vectors of this basis form a matrix AT which consequently must satisfy Rank (AT) = N - r = C. Now, by definition, all possible An values are described by the columns of S. A n = Sat (1)
Let us calculate AAn (where A = (AT)T). AAn = ASAE = @ATAE = 0
(2)
where we have used the identity S T A T = 0 since the columns of AT belong to the null space of ST.This has proven the following statement. Statement 1. The Ani variables satisfy at least C = N - r linearly independent linear equations A An = 0. Note that these equations have a form which is analogous to the elemental balances taken over the Ani values. Let us now assume that there are some more linear equations in An, values independent of A An = 0. Then we have a system of equations. A'An = 0 (3) the matrix of which has a rank Rank(A9 = C ' > C = N - r. Choose a basis of the null space of A'and let the vectors of this basis form the columns of a matrix S'. Then Rank(S9 = N - C ' < r = Rank(S). By a method applied in Bjornbom (1975) for an analogous problem one can prove that the columns of S'describe all the possible An values according to the equations An = S'At' (4) This gives the contradiction that the number of independent reactions would be N - C' rather than r. Therefore our initial assumption about some more equations must be false. This has proven another statement. Statement 2. The maximum number of linearly independent linear equations in Ani variables equals C = N r where r = the number of independent reactions according to definition 1. From the equations A An = 0 one can prove that from initially C components an arbitrary mixture of the N substances can be formed. The way of reasoning can be found in Smith and Missen (1979) but these authors have applied it to elemental balances. An Interpretation of Components as Compound-Forming Species I t has been proven by Aris (1965) that a stoichiometric matrix gives another stoichiometric matrix by a nonsingular transformation. Given a set of independent reactions and a convenient numeration of the substances involved, by use of elementary matrix algebra, the stoichiometric matrix can always be transformed by a nonsingular transformation into the form
(5)
Thus S corresponds to another set of independent reactions. Now let us form a matrix A of Rank (A) = C = N -r VI1 V I * ' . . ' V I r 1 0 . . . .o A = V z 1 vzz' ' " V 2 r 0 1 ' '0
i
"
)
......................... v c 1 VC2' . 'VCr 0 0 . . . .1/
(6)
Obviously A S = 0 which gives that A An = 0. The form of the equations AAn = 0 is similar to the form of the
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material balances on a mixture with N = C + r substances where the first r substances are compounds of C elements constituting the C remaining substances. Denote the components Al, A2...Ac and the remaining substances B1,B2...Br. Then on the basis of the above mentioned equation we can introduce the formula Bj = ( A ~ ) ” ~ ( A ~ ) ” ~ . . . ( A 0’ c )=~ ~1,, 2, .**, r )
Mathematically our way to derive the material balances over components is analogous to a derivation by Schott (1964). Schott derived from the independent reactions a set of C linearly independent constraining equations for use in general equilibrium computation methods in place of elemental balances in problems with restricted chemical stoichiometry.
which means that the substances B1,B2, ..., 23, are considered compounds of components analogous to compounds of elements. Therefore, the components from a mathematical point of view are equivalent to elements except in number and in the fact that a compound can contain a negative amount of component if the corresponding vij value happens to be negative. We can also form material balances over the components in a manner analogous to material balances over elements, except that a mixture can contain negative total amounts of compounds. Example 2. In hydrodealkylation of one of the isomers of xylene the following independent reactions can be derived from a postulated reaction mechanism (Bjornbom, 1977) X+H=T+M T+H=B+M where X = xylene, H = H2,T = toluene, B = benzene, and M = methane. Thus C = 3 while there are only R = 2 independent material balances. The stoichiometric matrix of these reactions can be transformed in the following way by a nonsingular matrix
Discussion of Negative Amounts Let us from example 2 study what the complications would be like if we wished to avoid negative amounts of components. This means that after having gotten a n e g ative amount from our calculation (for example an equilibrium computation) a new S matrix must be computed, for example after renumbering of the substances. Then a new set of independent reactions and a new set of material balances are determined. This gives the new choice of components. Finally the calculations must be repeated with the amounts of the new components. If all components now have a positive amount we can fiiish; otherwise, we must proceed with a new modification of the S matrix and repeat the whole algorithm. This trial-and-error process is completely unnecessary since it can be easily avoided simply by accepting negative amounts of components. The unnecessary changes of components do not change the calculated amounts of the substances in the final result. Another disadvantage of refraining from negative components is that the interpretation of the components as building blocks of the substances in the system, analogous to elements, cannot be made since this usually requires negative amounts of components in some of the substances. In example 2 we got, for example, that X = B1M2H+ In this case changing the components does not help. The negative amount of a component in a substance has no physical meaning and moreover the composition of a substance varies with the choice of independent reactions. In general the amount of a component in a substance cannot be measured, unlike the amounts of atomic species, and therefore has merely mathematical significance. Note that the amount of a component in the system can mean two different things. One is the measurable amount; the other is the total amount which is calculated by taking a material balance for the component over all the substances in the system. For example, in the mixture of 1 mol of X and 1 mol of H the measurable amount of H is 1 mol, but the total amount is -1 mol since 1 mol of X contains -2 mol of component H. Only in the case when a mixture of substances of given composition should be prepared in the laboratory by chemical reaction, starting from a mixture of components, do the total amounts of components have a physical meaning. Then the total amounts are equal to the initial amounts (compare Denbigh, 1955). In such cases the trial-and-error calculations mentioned above are motivated. However, this case is rare, so usually the total amounts of components have only mathematical significance. It may seem peculiar with negative amounts, but one must have in mind that we are here talking of pure mathematical concepts. These mathematical concepts are only used within the calculations while the real, the physically significant amounts of the substances, which come out as end results, always become positive. There are several analogous cases in science and engineering where physically insignificant mathematical concepts, often much more complicated than negative numbers, are used within calculations: for example, complex numbers in electrical and control engineering and the wave functions in quantum chemistry. Therefore, the use of
s=
0
1
-1 -1
The corresponding reactions can be written X = B + 2M - 2H T=B+M-H The corresponding A -matrix is X
A=
/ 1
2
\-2
T B M H 1 1 0 0 B 1 0 1 O ) M -1 0 0 1 H
Therefore B, M, and H can be chosen as components. The material balances over components AAn = 0 can be written AnE* = Anx + AnT + AnE = 0 AnM* = 2Anx AnT AnM = 0 AnH* = -2Anx - AnT + AnH = 0 where nk*denotes the total amount of component lz in the system. Obviously the total amount of each component is constant. The composition of the compounds X and T are X = BiM2H-z T = BlM1H-1
A mixture of 1 mol of X and 1 mol of H contains the following total amounts of components: 1mol of B, 2 mol of M, and -1 mol of H. This is an example of a negative amount of a component.
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negative amounts as a mathematical concept in this work should not be regarded as anything else than a normal application of mathematics. Gibbs’ Phase Rule Let us consider a system with N different substances, P phases, r independent reactions, and C components. At equilibrium dG 2 0 for all possible variations in the system a t constant T and p . P
C
r
dG = C( C ~ kdnkj j +- CP’Lj dn’ij) j=l k = l
i=l
(7)
where nkj = the amount of the kth component in the jth phase and n\j = the amount of the ith noncomponent in the j t h phase. However, the differentials dnkj and dntij cannot vary independently but the following C constraints apply from the material balances over the components P
r
C((CVki dnIij) 4- dnkj) = 0 j=1 i = l
( k = 1, 2, ..., c) (8)
By Lagrange’s multiplier method applied on eq 7 and 8, the following equilibrium conditions results hkj = ( k = 1, 2, c;j = 1, 2, P ) (9) ..a,
c h’ij =
-Cvkihk k=l
(i = 1, 2, ..., r; j = 1, 2, ..., P )
(10)
where hk is the kth Lagrange multiplier. The number of eq 9 and 10 equals P(C + r) = PN. The unknowns in these equations are P(N - 1)mole fractions, T,p , and C pieces of hk. Therefore the degrees of freedom are F = P ( N - 1) + 2 + c - PN = c - P + 2 If some of the phases in equilibrium miss one or more substances the number of equations and the number of mole fractions decrease in equal numbers leaving the same degrees of freedom. Note that the chemical potential for each substance has the same value in all phases at equilibrium. In Lagrange’s multiplier method it is essential to use all the equations which constrain the variation of the differentials. In cases with restricted chemical stoichiometry the number of constraints equals C rather than R. Therefore, the derivation above shows that only Jouguet’s definition of components is compatible with Gibb’s phase rule in such cases. Remark on Material Balance Computations Note the change in matrix (6) with changing degree of restrictions on chemical stoichiometry. In the least restricted case this matrix consists of one R X R unit matrix and N - R v columns. With increasing restrictions the unit matrix grows on the cost of the number of v columns. In the most restricted case, no reaction, we obtain one N X N unit matrix and no v columns. This systematic behavior suggests that component balances be conveniently used for general material balance computation algorithms covering nonreacting systems as well as reacting systems with any degree of restrictions on chemical stoichiometry. It should be kept in mind that the discussion above on component balances implies Jouguet’s definition of components.
Remark on the Definition of Chemical Stoichiometry Smith and Missen (1979) suggest that chemical stoichiometry be defined as the constraints placed on the composition of a closed chemical system by conservation of the amount of each elemental or atomic species in any physicochemical change in state occurring within the system. On the basis of the discussion of component balances above, adopting Jouguet’s definition of components, their suggestion can be generalized in the following way. Definition 3. Chemical stoichiometry: the constraints placed on the composition of a closed, chemical system by conservation of the amount of each component in any physicochemical change in state occurring within the system. Conclusion It was shown that the number of constraining equations in Ani variables equals C = the number of components according to Jouguet’s definition. From this it followed that only Jouguet’s definition is compatible with Gibbs’ phase rule in restricted chemical stoichiometry. Other advantages of Jouguet’s definition over Brinkley’s are the possibility to generally use component balances and the possibility to define chemical stoichiometry on the basis of component conservation. This means that the substances can be considered composed by the Components in a way analogow to how molecular species are composed by atomic species. This concept in general requires that negative amounts of components appear in the system. However, this can usually be accepted since these negative amounts usually have only mathematical significance. Acknowledgment The author wishes to thank Dr. Robert L. Pigford for encouraging him to write this paper. Thanks are due to the National Swedish Board for Energy Source Development for financial support. Nomenclature C = number of components according to Jouguet’s definition =N-r N = number of molecular species r = number of independent reactions according to definition 1
R = rank of atomic matrix Except for the listed entities IUPAC standard nomenclature has been used. Literature Cited Aris, R. Arch. Raflon. Mech. Anal. 1985, 19, 81. Bjdrnbom, P. H. I d . Eng. Chem. fundam. 1975, 14, 102. BJiKnbom,P. H. AIChE J. 1877. 23, 285. Brinkley, S. R. J. Chem. Phys. 1948. 14, 563. Denbigh, K. “The Principles of Chemical Equilibrium”, Cambridge, 1955; p 169. Gibbs, J. W. Trens. Conn. Acad. 1878, 3 , 108. See “The Sclentific Papers of J. Willard Gibbs”, Vol. 1, Dover Publications Inc.: New York, 1961, pp 138- 144. Heniey, E. J.; Rosen, E. M. “Materhl and Energy Balance Computations”, Wiley: New York, 1969. Jouguet, E. J . Ec. Po/ytech. (Park), Ser. 21921, 27, 61. Schott, G. L. J. Chem. Phys. 1984, 40, 2065. Smth, W. R.; Missen, R. W. Chem. €ng. Educ. 1878, 13, 28.
Received for review August 27, 1979 Accepted February 16,1981
