Convex Geometry and Stoichiometry

CONVEX GEOMETR Y AND STOICHIOMETR Y JER-CHIN (LUKE) CHUANG Abstract. W e demonstrate the beneﬁts of a con vex geometric persp ective for questions on chemical stoic hiometry . W e show that the balancing of c hemical equations, the use of “mixtures” to explain m ultiple stoic hiometry , and the half-reaction for balancing redox actions all yield nice con vex geometric inter- pretations. W e also relate some natural questions on reaction mechanisms with the enumeration of lattice p oints in polytop es. Lastly , it is known that a given reaction mec hanism imp oses linear constraints on observed stoichiometries. W e consider the inv erse question of deducing reaction mechanism consisten t with a given set of linear stoichiometric restrictions. Contents 1. The Algebraic Approac h 3 1.1. The Algebraic Approac h in a Nutshell 3 1.2. A Geometric In terpretation for the Result of the Algebraic Metho d 5 2. Geometric Approach to Balancing 6 3. P olytopal Explanation for Some Chemical Practices 12 3.1. Insp ection Metho ds 12 3.2. Stoic hiometric Restriction 12 3.3. Mixtures of Reactions 13 4. Balancing Equations In volving Charged Sp ecies 16 4.1. Comp onen t Indexing Charge 16 4.2. Sp ectator Ions 17 4.3. Half-Reaction Metho d 17 5. Reaction Mechanisms and Stoichiometric Restrictions 22 5.1. F rom Mec hanism to Linear Stoichiometric Relations 22 5.2. F urther Conclusions from a Given Mec hanism 23 5.3. Mec hanisms Consisten t with Observed Relations 27 5.4. F rom Subspaces N to Mechanisms 30 6. App endix: Mathematical Justiﬁcations for Geometric Viewp oint 32 References 33 A common question encountered in c hemistry is the balancing of a c hemical equation, and it has b een widely discussed in the chemical education literature (see for example the review article b y Herndon[12]). It is w ell-known in the literature (though p erhaps not among chemistry students) that the question admits a linear The author w ould like to thank the mathematics faculties at Georgia College & State Univ ersity and at St. Olaf College for the opp ortunities to presen t sev eral of these results at departmental colloquia. 1 2 JER-CHIN (LUKE) CHUANG algebraic form ulation. Here we examine an aspect previously unexplored, namely a con vex geometric approach. W e show that after sp ecifying the sp e cies (i.e. the atoms or comp ounds in volv ed in a reaction) on each side of a chemical equation, c onvex p olytop es (a b ounded in tersection of half-spaces) may b e used to provide a visual illustration for questions of existence and uniqueness of balancings, and though generally applicable, it is particularly eﬀective for chemical equations in- v olving only neutral sp ecies where no more than three or four elements in total are represen ted. More generally , we explore the utilit y of con vex geometric representa- tions in c hemical stoic hiometry , and this forms the principal theme of the current article. The conv ex geometric approach complements w ell the usual linear algebraic metho d whic h w e summarize in Section 1, with particular attention to issues regard- ing existence and uniqueness of solutions. That section concludes with a geometric in terpretation for the result of the computation. Section 2 discusses the aforemen- tioned geometric approac h to the balancing question and its connection with the algebraic approach. The section includes v arious examples with diagrams illustrat- ing the p otential visual appeal and p edagogical v alue of this approach. In Section 3 w e contin ue to emphasize the utilit y of the geometric approach b y explaining v ari- ous chemical practices in this geometric framework. In particular w e sho w that the practice of explaining chemical equations arising in con texts of multiple balances as mixtures of equations with unique balances is mathematically justiﬁed. Section 4 discusses adaptations of the geometric approach in the presence of charged sp ecies. There is an extended discussion of the “half-reaction metho d” commonly used to balance oxidation-reduction reactions. The geometric approach nicely elucidates the scope and limits of this practice. Section 5 turns atten tion from individual equa- tions to reaction mec hanisms. There are tw o ma jor considerations in this section: First, we sho w ho w natural existence and enumerativ e questions ab out mechanisms lead to the muc h-studied problem of counting lattice p oints in p olytop es. Second, w e inv estigate the mo duli of reaction mec hanisms consistent with a given ov erall reaction and observed stoic hiometric linear dep endencies. The latter complemen ts the “forward” analysis of Missen and Smith [18] in deducing stoichiometric linear dep endencies given a reaction mechanism for an ov erall reaction. Finally , the Ap- p endix (Section 6) includes a geometric approach to some of the observ ations in Section 2. Belo w are some highligh ts of the results and p ersp ectives of this pap er: • (Section 2) Questions of existence and uniqueness of balancings for a chem- ical equation may b e phrased in terms of the intersections of conv ex p oly- top es. This con vex geometric approac h is visually appealing, p erhaps p eda- gogically b eneﬁcial, and intimately connected to the kno wn linear algebraic form ulation of the problem. • (Prop osition 3.7) If a chemical equation admits multiple balancings, then an y such balance ma y be realized as a “mixture” of reactions admitting unique balances. • (Prop osition 4.11) If a redo x reaction admits a balance, then the “half- reaction metho d” will generate a balance for the equation. How ev er, in case of multiple balances, not all p ossible balances may b e obtainable via the “half-reaction” metho d. CONVEX GEOMETR Y AND STOICHIOMETR Y 3 • (Remark 5.3) In cases of m ultiple balances, minimizing the sum of co eﬃ- cien ts in the balance may not uniquely determine a balance. • (Subsection 5.3) It is known that an y given mechanism imp oses linear de- p endencies on observ ed quantities of species. The “inv erse” problem of deducing mechanisms consisten t with given observed linear dep endencies on sp ecies yields a collection of mechanisms parametrized by subspaces that can b e explicitly identiﬁed given kno wledge of reaction intermediates. This do es not completely solv e the inv erse problem since conv ex geometric and order considerations still need to b e imp osed. The latter is resolv ed but the former may b e computationally exp ensive. Computations on conv ex polytop es w ere done with Matthias F ranz’s Maple pack- age Convex [10] with the exception of lattice p oint en umeration which w as com- puted using V erdo olaege’s program bar vinok [24]. 1. The Algebraic Appro ach The ﬁrst subsection summarizes the linear algebraic approach, while the second geometrically interprets the result of the computation. 1.1. The Algebraic Approach in a Nutshell. W e b egin with an example il- lustrating the connection betw een balancing c hemical equations and linear algebra. F urther references and examples are provided by Missen and Smith[16], Blakley[4] and Alb erty[1] among many others. 1.1. Example. Suppose we w ant to know all p ossible wa ys to balance a reaction in volving hypothetical neutral sp ecies X Y , Y Z, X Y Z 2 where X , Y , Z are distinct elemen ts. The algebraic metho d asso ciates to each species a 3-vector, in this case v X Y =   1 1 0   v Y Z =   0 1 1   v X Y Z 2 =   1 1 2   where w e hav e tacitly ordered the elements X , Y , Z so that the ﬁrst comp onent indicates the num b er of atoms of X , the second the num b er of Y atoms, etc.. Mass-conserv ation is then reﬂected by ﬁnding r ational n umbers a i suc h that a 1 v X Y + a 2 v Y Z + a 3 v X Y Z 2 = 0 One just m ultiplies by the least common denominator of the fractions to obtain an integral solution. Thus, if a balancing exists, then a matrix M with the ab ov e v ectors as its columns has a non-trivial n ullspace NS( M ): that is, the equation M x = 0 do es not ha ve just the vector 0 = (0 , . . . , 0) T as a solution. F or example, w e ma y set M =   1 0 1 1 1 1 0 1 2   (The ordering of the columns do es not aﬀect non-triviality of the nullspace.) One computes that NS( M ) = 0 so that no balancing exists. Con versely , if N S ( M ) 6 = 0 , then since M has integer en tries, one may ﬁnd a basis for N S ( M ) having only rational entries. Hence, a balancing exists iﬀ N S ( M ) 6 = 0 and it is furthermore unique (up to multiplicativ e factor) precisely if the nullspace is one-dimensional. 4 JER-CHIN (LUKE) CHUANG W e formalize the ab ov e as follows: Supp ose w e ha ve m neutral sp ecies inv olving a total of n elements which w e place in some order. T o each sp ecies w e asso ciate a n -v ector where the k -th comp onent is the multiplicit y of the k -th element in that sp ecies. Denoting these sp e cies ve ctors by v i , w e seek rational n umbers a i suc h that (1.2) m X i =1 a i v i = 0 Or equiv alent, let M = ( v 1 . . . v m ) be a n × m matrix with the v ectors v i as columns. Then, a balancing exists iﬀ dim(NS( M )) > 0 and the balancing is unique (up to m ultiplicative factor) iﬀ dim(NS( M )) = 1. Note that the metho d does not indicate whic h sp ecies are reactants and which pro ducts. Only mass-conserv ation is being enforced and no directionalit y is implied, though for a giv en solution to Equation (1.2) the sp ecies are separated into tw o collections dep ending on the sign of the co eﬃcients a i . (Those for which a i = 0 are not inv olv ed in the balancing.) F or conv enience, we will refer to these tw o groups as “reactants” and “pro ducts” without implying any dir e ctionality . No w, supp ose w e stipulate that certain species are “reactan ts” and the remaining “pro ducts.” In the case where dim(NS( M )) = 1, the balancing is unique and the sp ecies are partitioned into tw o groups uniquely . Ho wev er, for dim(NS( M )) > 1, certain partitions ma y not b e realizable and even if realizable, the balancing ma y not b e unique (ev en up to m ultiplicative factor). Mathematically , we seek rational co eﬃcients a i in Equation (1.2) such that the signs of those asso ciated to “reactan ts” are opposite of those ass ociated to “pro ducts.” Computationally , one ma y proceed as illustrated in Blakley[4] b y computing a basis { b j } for the n ullspace. Since now the comp onents of b j index the sp ecies, we seek linear combinations of b j suc h that comp onents indexing “reactants” hav e signs opp osite those indexing “pro ducts.” 1.3. Example. Supp ose we hav e sp ecies X Y , X Z , Y Z, X Y Z, X 5 Y 5 Z 2 partitioned in to “reactants” { X , Y , X Y Z } and “pro ducts” { X Z , Y Z, Z 5 Y 5 Z 2 } . Ordering the elemen ts X, Y , Z in that order, and deﬁning (1.4) M =   1 0 1 1 0 5 0 1 1 0 1 5 0 0 1 1 1 2   w e compute that dim(NS( M )) = 3 with the vectors (1.5) b 1 = (0 , 1 , − 1 , 1 , 0 , 0) T b 2 = (1 , 0 , − 1 , 0 , 1 , 0) T b 3 = ( − 3 , − 3 , − 2 , 0 , 0 , 1) T as one p ossible basis for the n ullspace. Note that the comp onents of b i index the sp ecies X , Y , X Y Z , X Z , Y Z , Z 5 Y 5 Z 2 in this or der . Cho osing the “reactants” to ha ve non-p ositiv e co eﬃcients, we th us seek linear com binations c 1 b 1 + c 2 b 2 + c 3 b 3 suc h that the ﬁrst three comp onents are non-p ositive and the remaining non-negative: c 2 − 3 c 3 ≤ 0 c 1 ≥ 0 c 1 − 3 c 3 ≤ 0 c 2 ≥ 0 − c 1 − c 2 − 2 c 3 ≤ 0 c 3 ≥ 0 CONVEX GEOMETR Y AND STOICHIOMETR Y 5 W e further require that all resulting six comp onents are rational n umbers (see fol- lo wing discussion). A geometric approach for this balancing is presen ted in Example 2.5 b elow. W e may formalize the preceding as follo ws: Supp ose we partition the m neutral sp ecies in to r “reactants” and p “pro ducts.” After p ossible re-ordering, we ma y assume vectors v 1 , . . . , v r corresp ond to the “reactants” and v r +1 , . . . , v m to the “pro ducts,” where r + p = m . Without loss of generalit y we ma y c ho ose “reactan ts” to hav e non-p ositiv e co eﬃcients. Then, each balancing corresp onds to a collection of rational num b ers a i suc h that P m i =1 a i v i = 0 now with the added stipulation that a 1 , . . . , a r b e non-positive and a r +1 , . . . , a m non-negativ e. Let M = ( v 1 . . . v m ) and { b j } b e a basis for the nullspace. Since M is a matrix of integers, we ma y assume that the comp onents of the basis v ectors b j are all rational. Then, the desired rational a i corresp ond precisely to r ational linear combinations since the b j constitute a basis for NS( M ) and hence are linearly indep endent: (1.6)    a 1 . . . a m    = k X j =1 c j b j = ( b 1 . . . b k )    c 1 . . . c k    a 1 , . . . , a r ≤ 0 a r +1 , . . . , a m ≥ 0 where k = dim(NS( M )) and the c j are rational. 1.2. A Geometric In terpretation for the Result of the Algebraic Metho d. W e provide a geometric interpretation for the system of inequalities in Equation (1.6). Each comp onent a i deﬁnes a linear inequality in the v ariables c j . Geomet- rically , this represen ts a half-space in R k passing through the origin, where recall k = dim(NS( M )). Th us, simultaneous solutions to the system of inequalities in Equation (1.6) are geometrically represented by the intersection lo cus Q of m half- spaces in R k . Such an ob ject is called a p olyhe dr on in R k . Since w e w ant rational c j , the p ossibilities are precisely the rational p oints (i.e. p oints where all co ordinates are rational num b ers) contained in the p olyhedron where tw o such rational p oints represen t the same balancing (up to multiplicativ e factor) if they are on the same line through the origin. Th us, the “moduli” of balancings with giv en “reactants” and “pro ducts” is realizable as the image of Q ⊆ R k under its pro jectivization, hence a subset of rational pro jective space P k Q . W e will return to a discussion of Q near the end of the next section. In summary , we hav e a geometric representation for all possible balancings of a chemical equation when species for b oth sides are initially sp eciﬁed. 1.7. Example. Consider the oxidation of nitric oxide ( N O ) to nitrogen dioxide ( N O 2 ). The reactants are N O , O 3 and the pro ducts are N O 2 , O 2 . Note that this example inv olves tw o allotrop es of oxygen. W e order the elemen ts by O , N in that order. Deﬁning (1.8) M =  1 3 2 2 1 0 1 0  w e compute that dim(NS( M )) = 2 with the vectors (1.9) b 1 = (0 , − 2 , 0 , 3) T b 2 = ( − 3 , − 1 , 3 , 0) T as one p ossible basis for the n ullspace. Note that the comp onents of b i index the sp ecies N O , O 3 , N O 2 , O 2 in this or der . Cho osing the reactan ts to ha ve non-positive 6 JER-CHIN (LUKE) CHUANG co eﬃcien ts, we th us seek linear combinations c 1 b 1 + c 2 b 2 suc h that the ﬁrst tw o comp onen ts are non-p ositive and the remaining non-nonegative: − 3 c 2 ≤ 0 3 c 2 ≥ 0 − 2 c 1 − c 2 ≤ 0 3 c 1 ≥ 0 The resulting p olyhedron Q is the set { ( c 1 , c 2 ) | c i ≥ 0 } and each rational p oint within Q indexes a balancing. A geometric approach for this balancing is presented in Example 2.7 b elow. 2. Geometric Appro ach to Balancing No w we examine the balancing question anew but guided b y geometric con- siderations. Supp ose as b efore we partition m neutral species into r “reactan ts” and p “pro ducts” and that these are represen ted by the collections v 1 , . . . , v r and v r +1 , . . . , v m resp ectiv ely , where r + p = m . Then, balancings are precisely equali- ties of a non-negative rational linear com bination of the former set of vectors with a non-negative rational linear com bination of the latter set: (2.1) r X i =1 r i v i = m X j = r +1 p j v j where the co eﬃcien ts r i , p j ≥ 0 are rational. Recall that the set of all non-negative scalings of a v ector is geometrically rep- resen ted b y a ray based at the origin in the direction of the vector. Non-negative linear combinations of a set of vectors determine a mathematical ob ject called the (p olyhe dr al) c one spanned by the vectors with the origin as the vertex of the cone. Since we only care ab out the rational num bers r i , p j up to m ultiplicative factor, we slice b oth cones simultaneously b y a hyperplane intersecting all the p ositive x i -axes for i = 1 , . . . , n . This h yp erplane then intersects eac h cone transversely and the in tersection lo ci are bounded p olyhedra, i.e. (c onvex) p olytop es , whic h we may call the “ r e actant ” and ” pr o duct ” p olytop es . Indeed, eac h is the conv ex h ull of the p oints giv en by the intersection of the v arious deﬁning rays (either those asso ciated with “reactan ts” { v 1 , . . . , v r } or those with “pro ducts” { v r +1 , . . . , v m } ) with the slicing h yp erplane. Without loss of generality , we may assume the hyperplane is describ ed b y an equation of the form n · x = h where n , h consist entirely of rational n umbers. Then, given a choice of “reactants” and “pro ducts,” a necessary condition for the existence of a balancing is that the in tersection of these t wo polytop es is non-empt y . The intersection is another p olytop e, which we will call the interse ction p olytop e , and if non-empty , it necessarily contains a rational point. Con versely , one can show that each rational point in the intersection polytop e corresp onds to some balancing of a c hemical equation with the sp eciﬁed grouping of sp ecies. (See the discussion at the end of this section or for a geometric argument see the app endix.) Since w e assume that the co eﬃcients r i , p j ≥ 0 are non-negative, all the compo- nen ts of the scaled vectors r i v i , p j v j are non-negativ e, and a canonical choice for the hyperplane is P n l =1 x l = 1. Notice that this hyperplane intersects eac h positive x i -axis for i = 1 , . . . , n as required. The intersection of this canonical h yp erplane with the region { ( x 1 , . . . , x n ) | x i ≥ 0 } is a ( n − 1)-dimensional “triangle,” called a ( n − 1)- simplex . The co ordinates ( x 1 , . . . , x n ) sum to unit y and are called b aryc entric c o or dinates . Each sp ecies v ector v i corresp onds to a p oint with barycen tric coordi- nates sp ecifying the prop ortions of each element within the species. F or example, if CONVEX GEOMETR Y AND STOICHIOMETR Y 7 Figure 1. Case where no balancing exists for any grouping of reactan ts and pro ducts X, Y , Z are the only elements in v olved and ordered thus, then the barycen tric co or- dinates for the sp ecies X 2 Y 3 Z 4 is (2 / 9 , 1 / 3 , 4 / 9). T aking a non-c hemical example, the RGB-system of colors describ es colors in terms of prop ortions ( r , g , b ) of red, green, and blue, resp ectively , where r + g + b = 1. Note that rational p oints on the boundary of the in tersection polytop e corre- sp ond to balancings in which not every sp ecies is presen t. Hence, if w e require that all sp ecies b e present, then if an equation can b e balanced then it can b e balanced in inﬁnitely man y distinct wa ys unless the in tersection p olytop e is a single p oint. Ho wev er, this is only a necessary and not a suﬃcient condition for the uniqueness of balance. W e will see later that additionally w e need the generating v ectors for the reactan t and product cones eac h to be linearly independent (see Equation 2.18). F or example, this is satisﬁed if the reactant and product polytop es are line segmen ts meeting at a p oint in the relative in terior of each. W e no w illustrate this p ersp ective with several examples: 2.2. Example (Non-Existence of Balancing) . Suppose w e hav e “reactants” { X Y , Y Z } and “pro duct” { X Y Z 2 } where X , Y , Z are distinct elements and all sp ecies are neutral. If we order the elemen ts as X, Y , Z , then the barycen tric co ordinates of the “reactant” sp ecies are (1 / 2 , 1 / 2 , 0) T and (0 , 1 / 2 , 1 / 2) T , and that of the “pro d- uct” sp ecies is (1 / 4 , 1 / 4 , 1 / 2) T . The “reac tan t” p olytop e is the conv ex hull of the barycen tric co ordinates for the “reactant sp ecies,” namely a line segmen t joining (1 / 2 , 1 / 2 , 0) T and (0 , 1 / 2 , 1 / 2) T whereas the “pro duct p olytop e” is just a single p oin t. See Figure 1. Since the line segment and point are disjoint, we conclude that no balancing exists for that particular choice of “reactants” and “pro ducts.” In fact, from the diagram we can easily see that there is no wa y to partition the sp ecies into “reactants” and “pro ducts” to obtain a situation where a balancing exists. This concurs with the conclusion in Example 1.1 obtained via the algebraic metho d. 8 JER-CHIN (LUKE) CHUANG Figure 2. Case where uniqueness of balancing dep ends on choice of reactants and pro ducts 2.3. Example (Unique and Non-Unique Balancings) . Consider the neutral c hemical sp ecies H 2 , H 2 O , C H 4 , C O 2 , C O . Ordering the elemen ts H , O , C in that order, the barycen tric co ordinates are resp ectively , (2.4)   1 0 0   ,   2 / 3 1 / 3 0   ,   4 / 5 0 1 / 5   ,   0 2 / 3 1 / 3   ,   0 1 / 2 1 / 2   See Figure 2. As eviden t from the diagram, to obtain a non-empty intersection p olytop e, we need to hav e at least tw o “reactants” and tw o “pro ducts” each. By the comments ab ov e, we hav e an unique balancing precisely when four of the ﬁve sp ecies are in volv ed and are grouped pairwise, and the lines so determined in tersect uniquely . One easily chec ks from the diagram that this yields the groupings: { C H 4 , C O 2 } { H 2 , C O } { H 2 , C O } { C H 4 , C O 2 } { H 2 , C O } { C H 4 , H 2 O } { H 2 O , C O } { C H 4 , C O 2 } { H 2 O , C O } { H 2 , C O 2 } (Compare with cases ( α )-(  ) in Example 6 of Blakley[4]). 2.5. Example (Non-Unique Balancing) . Consider “reactan ts” { X , Y , X Y Z } and “pro ducts” { X Z , Y Z, X 5 Y 5 Z 2 } where X, Y , Z are distinct elements ordered thus as in Example 1.3. The barycentric coordinates are (2.6)   1 0 0   ,   0 1 0   ,   1 / 3 1 / 3 1 / 3   ,   1 / 2 0 1 / 2   ,   0 1 / 2 1 / 2   ,   5 / 12 5 / 12 1 / 6   CONVEX GEOMETR Y AND STOICHIOMETR Y 9 Figure 3. Case where inﬁnitely-many distinct balancings exist for giv en c hoice of reactants and pro ducts Figure 4. Case where allotrop es are inv olved resp ectiv ely . See Figure 3. The intersection p olytop e is a quadrilateral, and hence there are inﬁnitely man y distinct balancings under this grouping of sp ecies. F or example, the rational points (3 / 8 , 3 / 8 , 1 / 4) T and (2 / 5 , 2 / 5 , 1 / 5) T are in the in terior of the in tersection p olytop e with 1 8   1 0 0   + 1 8   0 1 0   + 1 4   1 1 1   =   3 / 8 3 / 8 1 / 4   = 1 16   1 0 1   + 1 16   0 1 1   + 1 16   5 5 2   1 5   1 0 0   + 1 5   0 1 0   + 1 5   1 1 1   =   2 / 5 2 / 5 1 / 5   = 1 40   1 0 1   + 1 40   0 1 1   + 3 40   5 5 2   yielding distinct balancings: 2 X + 2 Y + 4 X Y Z = X Z + Y Z + X 5 Y 5 Z 2 8 X + 8 Y + 8 X Y Z = X Z + Y Z + 3 X 5 Y 5 Z 2 resp ectiv ely . 2.7. Example (Allotrop es) . Consider the oxidation of nitric o xide ( N O ) to nitrogen dio xide ( N O 2 ) as discussed earlier in Example 1.7. The reactants are N O , O 3 and the pro ducts are N O 2 , O 2 . Ordering the elements b y O , N in that order, b oth 10 JER-CHIN (LUKE) CHUANG allotrop es are represented by the barycentric coordinates (1 , 0) T whereas those for N O , N O 2 are (1 / 2 , 1 / 2) T and (2 / 3 , 1 / 3) T resp ectiv ely . The intersection p olytop e is a line segment { (1 − t, t ) T | 0 ≤ t ≤ 1 / 3 } . See Figure 4. W riting (1 − t, t ) T for a p oin t in the intersection p olytop e, we ha ve the representations:  1 − t t  = 2 t  1 / 2 1 / 2  + (1 − 2 t )  1 0  = t  1 1  + 1 − 2 t 3  3 0  (2.8)  1 − t t  = 3 t  2 / 3 1 / 3  + (1 − 3 t )  1 0  = t  2 1  + 1 − 3 t 2  2 0  (2.9) (2.10) relativ e the reactan t and pro duct cones, respectively . Thus, w e hav e a one-parameter family of distinct balancings: (2.11) tN O + 1 − 2 t 3 O 3 = tN O 2 + 1 − 3 t 2 O 2 indexed b y r ational t suc h that 0 ≤ t ≤ 1 / 3. If we insist that all reactan ts and pro ducts b e present, then we hav e strict inequalities. F or example, setting t = 1 / 4 and clearing denominators yields the balancing: 6 N O + 4 O 3 = 6 N O 2 + 3 O 2 . The case where t = 1 / 5 corresponds to the usual equation for the o xidation: N O + O 3 → N O 2 + O 2 . 2.12. R emark. In practice, one knows the relative amoun ts of reactan ts and pro d- ucts. Exact knowledge of either determines an unique ray in the corresp onding cone. Hence, if a solution exists, this information is suﬃcien t to iden tify an unique p oin t in the intersection p olytop e. Because of measurement error and uncertaint y , w e ha ve instead a neighborho o d of a ray (mathematically , a neigh b orho o d of a p oin t in pr oje ctive sp ac e ), and hence possibly inﬁnitely-many balancings. In prac- tice, c hemists choose balancings with “small” co eﬃcients, though a mathematical form ulation for this “rule-of-th umb” is unclear. See also Remark 5.3.. 2.13. R emark. Note that we can scale the intersection p olytop e so that it has in teger vertices. Let d b e its dimension and write ν ( n, d ) for the num b er of p oints in the intersection polytop e with denominator (in lo west terms) no greater than n . Analogously , let ν 0 ( n, d ) denote the n umber of suc h in the interior of the p olytop e. By conv ention, we set ν (0 , d ) = 1 = ν 0 (0 , d ). It is kno wn that b oth ν, ν 0 are p olynomials of degree d in n (see Stanley [21]). In particular, knowledge of d other v alues determine either ν or ν 0 completely . F or example, w e can scale the quadrilateral that is the intersection polytop e in Figure 3 to hav e integer v ertices (15 , 15 , 6) T (16 , 10 , 10) T (10 , 16 , 10) T (12 , 12 , 12) T Then, with regard to this p olytop e, we can coun t that in its interior there are 16 integer p oints and 33 p oints with denominator at most tw o so that ν 0 ( n, 2) = n 2 + 14 n + 1. How ev er, relating denominators to the least in tegral co eﬃcients of a balancing is not direct, as Example 2.7 shows. W e will return to en umerative questions in Subsection 5.2. No w, w e describe the connection b etw een the geometric and algebraic approac hes: Recall that the algebraic approach parametrizes p ossible balancings by rational CONVEX GEOMETR Y AND STOICHIOMETR Y 11 p oin ts of an unbounded p olyhedron Q in R k . Consider the linear transformations B : Q ( b 1 ... b k ) − → R m (2.14) C r : R r + ( v 1 ... v r ) − → R n (2.15) C p : R p + ( v r +1 ... v m ) − → R n (2.16) where R r + denote non-negative v alues and w e ha ve interpreted matrices as linear maps. Deﬁne R = C r ◦ π r and P = C p ◦ ( − π p ) where π r (resp. π p ) denotes pro jection onto the ﬁrst r (resp. last p ) co ordinates in R m . Then, B : Q → R m is the centralizer of the maps R, P : R m → R n (in particular, they coincide on the image of B ), and RB , P B : Q → R n eac h map the p olyhedron Q in a linear fashion on to the intersection cone. These maps ﬁt into a commutativ e diagram: (2.17) R r + C r ! ! B B B B B B B B Q B / / ˜ R > > } } } } } } } } ˜ P A A A A A A A R m R,P / / π r O O − π p   R n R p + C p = = | | | | | | | | where ˜ R = π r ◦ B and ˜ P = − π p ◦ B . Let R , P b e the reactan t and pro duct cones, resp ectively and I = R ∩ P the in tersection c one . Note that R = im C r and P = im C p . Deﬁne the matrix M = ( v 1 . . . v m ) whose columns are those of C r , C p resp ectiv ely . Note that NS( M ) = im( B ) so that dim( N S ( M )) = dim( Q ) the dimension of the un b ounded polyhedron in the algebraic approach. By a fact from linear algebra (see Section 6), one has (2.18) dim( Q ) = dim(NS( M )) = dim( I ) + dim(k er( C r )) + dim(k er( C p )) so that of the p olyhedron Q from the algebraic approach is not in general linearly isomorphic to the in tersection cone. It is so iﬀ C r , C p are injectiv e, that is, the reac- tan t and product sp ecies vectors are each linearly independent sets. One example where this fails is a reaction in volving either more reactant or pro duct sp ecies than elemen ts in volv ed. In tuitively , the nullspace NS( M ) measures linear relations among the reactan t and pro duct species vectors. These may b e among reactant vectors only , among pro ducts v ectors only , or among v ectors drawn from b oth reactan t and product sp ecies, hence the three terms in Equation 2.18. Letting s, r, p b e the n umber of sp ecies, reactants, and pro ducts resp ectively , we hav e in sp ecies space S = R s the ( s − 1)-simplex ∆ s − 1 = ∆ r − 1 ∗ ∆ p − 1 where ∆ r − 1 , ∆ p − 1 are m utual cofaces deﬁned by the reactant and pro duct vectors, respectively . The sp ecies vectors deﬁne partial maps ∆ r − 1 , ∆ p − 1 ⇒ E whose images are the reactant and pro duct p olytop es, respectively . The bijectiv ely image of the mo duli polyhedron Q in species space S is p ossibly “partially collapsed” under the “elemental prop ortion” map M in to the intersection cone in elemen t space E due to p ossible linear dependencies among generating rays for the reactant cone and also for the pro duct cone. In summary , under the geometric approach, w e gain often a visual determination of existence issues at the expense of p ossibly losing count of the algebraic dimension 12 JER-CHIN (LUKE) CHUANG of p ossible multiple balancings and hence a sligh tly more in volv ed criterion for uniqueness. 3. Pol ytop al Explana tion for Some Chemical Practices In this section we examine several common practices in chemical stoichiometry and show that they all hav e simple p olytopal explanations. 3.1. Insp ection Metho ds. In the c hemical education literature, there are n umer- ous “insp ection metho ds” for balancing chemical equations (for example [23],[15]). Mathematically , these often amoun t to establishing an ad ho c elimination order on the elements, or in geometric terms, matching the pro jections of the reactan t and product p olytop es onto v arious axial directions. F or example, the ﬁrst t wo of “Ling’s Rules for Balancing Redox Equations by Insp ection” as presented in Kolb[15] are: • “Step 1: Lo cate any elements that m ust hav e the same co eﬃcien t in the balanced equation, those app earing only onc e on each side of the equation and in e qual numb ers on b oth sides. Mark these terms with arrows.” • “Step 2: Lo cate an y elements that app ear only onc e on each side of the equation but ha ve une qual num b ers of atoms. Balance these elemen ts ﬁrst.” This is illustrated in the following example: 3.1. Example. [15] In balancing S + H N O 3 → S O 2 + N O + H 2 O b y step 1, we see that S, S O 2 ha ve iden tical co eﬃcents and so also H N O 3 , N O . By step 2, we balance hydrogen, which forces the balance for nitrogen, then sulfur, then oxygen yielding: 3 2 S + 2 H N O 3 → 3 2 S O 2 + 2 N O + H 2 O Kolb’s article[15] even presents a case where Ling’s metho d do es not work, and she resorts to linear algebra, the prop er algebraic framework. 3.2. Stoic hiometric Restriction. Sometimes one kno ws from exp eriment that certain species react in particular ratios, usually from kinetic and mechanistic con- siderations (see Section 5 for details). Missen and Smith[17] provide an example in volving a p ermanganate-p eroxide reaction in acidiﬁed aqueous solution: H 2 S O 4 + K M nS O 4 + H 2 O 2 → O 2 + H 2 O + K 2 S O 4 + M nS O 4 (un balanced) One chec ks (for example via Convex [10] or Pol ymake [11]) that the intersection p olytop e is 1-dimensional. How ev er, it is known from exp eriment that K M nO 4 reacts with H 2 O 2 in a 2 : 5 ratio. Algebraically , this may b e reﬂected by augmenting an additional r ow to the matrix M mapping sp ecies to elements. F or example, ordering the sp ecies as they app ear left-righ t in the abov e equation, the augmented matrix is  M 0 − 2 5 0 0 0 0  Geometrically , we incorp orate the restriction b y replacing the generating v ertices v K M nS O 4 , v H 2 O 2 with a linear combination (2 / 7) v K M nS O 4 + (5 / 7) v H 2 O 2 . One can compute that this alteration reduces the dimension of the reactan t polytop e. Either CONVEX GEOMETR Y AND STOICHIOMETR Y 13 Figure 5. Reaction as Mixture of Reactions w ay , w e compute that the in tersection p olytop e is now 0-dimensional so that there is an unique balance under this stoic hiometric restriction. Similarly , if v i is the generating vertex indexed b y the i -th sp ecie, and it is known exp erimentally that reactan ts R i 1 , . . . , R i k react in a ratio of r i 1 : · · · : r i k , then we replace the asso ciated generating vertices with the single vertex giv en as a weigh ted linear combination: (3.2) v = k X j =1 r i j r v i j r = k X j =1 r i j A similar observ ation applies if certain pro ducts are know to b e pro duced in par- ticular ratios. The in tersection from the resulting p olytop es ma y or may not b e of lo wer dimension. More generally , if there is linear relation among the sp ecies, it deﬁnes a hyper- plane H that interesects the simplex ∆ s − 1 ⊆ S . Intersecting the image M ( H ∩ ∆ s − 1 ) ⊆ E with the intersection polytop e then yields the new mo diﬁed in tersection p olytop e. Algebraically , one just adds these relations as additional rows to the matrix deﬁning the elemental comp osition map. 3.3. Mixtures of Reactions. In the c hemical education literature, reactions ad- mitting multiple balancings are often explained as a mixture of several other re- actions. This is tantamoun t to writing a balance as a rational combination of balancings relative subp olytop es of the reaction and pro duct p olytop es. 3.3. Example. [14] Kolb explains the reaction: 3 H C lO 3 → H C lO 4 + C l 2 + 2 O 2 + H 2 O as a mixture of tw o reactions: 7 H C lO 3 → 5 H C lO 4 + C l 2 + H 2 O 4 H C lO 3 → 2 C l 2 + 5 O 2 + 2 H 2 O 14 JER-CHIN (LUKE) CHUANG Figure 6. Reaction as Mixture of Reactions where the ﬁrst plus t wice the second yields the desired ov erall reaction. F rom the p olytopal diagram (see Figure 5), w e see that H C l O 3 is in the interior of the pro duct p olytop e, and hence the reaction admits multiple balancings. The ﬁrst reaction expresses the reactan t via the subp olytop e spanned by pro ducts H C lO 4 , C l 2 , H 2 O whereas the second via that spanned by products C l 2 , O 2 , H 2 O . 3.4. Example. [14] Kolb also explains the reaction: 3 S O 2 + 7 C → C S 2 + S + 6 C O as a mixture of the reactions S O 2 + 2 C → S + 2 C O 2 S O 2 + 5 C → C S 2 + 4 C O Again, from the polytopal diagram (see Figure 6), the existence m ultiple balancings is clear. Note that as in the preceding example, the particular choices for component reactions each admit unique balancings. 3.5. Example. [14] In the equation N aC lO 2 + N aC l O → N aC l O 3 + N aC l c hlorine app ears in four diﬀerent oxidation states. Hence, application of the usual half-reaction metho d is somewhat confusing (see discussion in Subsection 4.3 b e- lo w). Thankfully , the equation as written is already balanced, though it also admits m ultiple balancings. F rom the p olytopal diagram (see Figure 7), we see that the sp ecies are actually collinear, since each has N a, C l in a 1 : 1 ratio. Kolb explains this example as a combination of the reactions 3 N aC lO 2 → 2 N aC l O 3 + N aC l 3 N aC lO → N aC l O 3 + 2 N aC l CONVEX GEOMETR Y AND STOICHIOMETR Y 15 Figure 7. Reaction as Mixture of Reactions In all the preceding examples, the c hoice of component reactions w as c hemically- motiv ated but also happened to admit unique balancings. Indeed, one ma y sa y they w ere chosen in part to be uniquely determined, since the paradigm is to explain m ultiple-balancings in terms of what are p erceived to be “true” reactions, namely those admitting unique balancings. In fact, any reaction can b e written as a ra- tional com bination of reactions with the reactants (resp. pro ducts) b eing subsets of the original reactants (resp. pro ducts) though as demonstrated in Example 3.4, w e cannot require the subsets alw ays to b e prop er. This follows from a simple observ ation: let con v () denote conv ex hull. 3.6. Lemma. Supp ose conv( X ) = con v( R ) ∩ conv( P ) for ﬁnite sets R, P in some ﬁxe d Euclide an sp ac e. Then, ther e exist subsets R i ⊆ R and P i ⊆ P such that the interse ctions conv( R i ) ∩ con v( P i ) = { x i } ar e singletons and any p oint x ∈ con v ( X ) is a c onvex c ombination of the p oints x i . We c annot r e quir e R i , P i to b e simultane ously pr op er. Pr o of. W e may assume that X is the set of generating v ertices for the intersection con v ( R ) ∩ con v ( P ). In general, for each x i ∈ X , let R i , P i b e minimal subsets of R, P resp ectively suc h that x i ∈ conv( R i ) ∩ con v ( P i ). If x i is an interior p oint of either conv( R ) , con v ( P ) then a set of generating v ertices for one of the tw o hulls ma y be needed. Hence, we cannot require R i , P i to b e necessarily prop er subsets. If x i is a boundary p oint for b oth conv( R ) , conv( P ), then it is the intersection of a line segment of one hull and a face of the other hull. In either case the in tersections con v ( R i ) ∩ conv( P i ) are singletons. Since the x i generate con v( X ), we are done.  T aking R , P to b e the sets of reactants and pro ducts, resp ectiv ely and conv( X ) to b e the intersection p olytop e for the reaction, the we obtain the following: 3.7. Prop osition. Any chemic al e quation c an b e written as a r ational c ombination of chemic al e quations admitting unique b alanc e, with the r e actants (r esp. pr o ducts) 16 JER-CHIN (LUKE) CHUANG b eing subsets of the original r e actants (r esp. pr o ducts) though we c annot r e quir e the subsets always to b e pr op er. Pr o of. Refer to the pro of of the preceding lemma. All that remains to sho w is that eac h x i enco des an unique balance. Note that an y p oin t of a ﬁnitely generated con vex set is in the interior of a simplex generated b y some (necessarily) minimal subset of the generating v ertices. This simplex need not be a face of the p olyhedron. If x i is an interior p oin t, then the preceding observ ation sho ws that x i is in the in terior of some simplex. If x i is a b oundary p oint, then it is the intersection of a line segment and a face so that again b y the op ening observ ation, we may think of x i as the intersection of a line segment with a simplex. In either case, since the v ertices of a simplex are linearly indep endent, by Equation (2.18), x i represen ts an unique balance.  This thus justiﬁes the c hemical paradigm of expressing chemical equations as mixtures of those admitting unique balance. 4. Balancing Equa tions Invol ving Char ged Species If charged sp ecies are inv olved, there are several wa ys we may pro cede: we ma y in tro duce (a) an additional comp onen t to index the charge or (b) ﬁctitious “sp ectator ions” that allo w us to balance the equation as if only neutral sp ecies are present. One imp ortant class of chemical reactions inv olving charged ions are oxidation-r e duction r e actions . Though these may b e balanced by either of the t wo preceding metho ds, there is a commonly used “half-reaction metho d” that merits separate mathematical in vestigation since it is structurally diﬀerent from the preceding metho ds. 4.1. Comp onen t Indexing Charge. W e index charges as an additional comp o- nen t so that if n elements are in volv ed, then the sp ecies vectors are in R n +1 no w. W e use the con ven tion that the charge is alwa ys indexed last and note that it ma y b e negative. F or example, the sp ecies C a 2+ w ould ha ve sp ecies vector (1 , 2) T whereas C l − w ould hav e (1 , − 1) T . How ever, balancings still corresp ond to equali- ties of non-negative rational linear combinations, and almost everything ab o ve for b oth the algebraic and geometric methods remain v alid. Evidently , the co domain of R , P will b e R n +1 so that the comm utative diagram (2.17) should b e altered accordingly , but the subtlet y is that since w e now allow negative en tries in the last comp onen t, w e no longer hav e a canonical choice for the slicing hyperplane. If b oth lone + / − charges (i.e. not b ound with an element) are present, then no h yp erplane alw ays w orks b ecause an y h yp erplane in tersecting both the positive and negativ e x n +1 -ra ys contains the x n +1 -axis and hence passes through the origin. Of course, one can assume that the lone charges ha v e b een “canceled” so that only one kind remains. Alternativ ely , since in c hemistry lone p ositive c harges are denoted H + , we ma y assume only lone negative c harges are p ossibly present. Recall, that the choice P n i =1 x i = 1 is canonical if charges do not app ear b y themselv es without elemen ts, e.g. free electrons e − . When only one kind of free c harge is in volv ed (either p ositive or negativ e), we ma y b e tempted to consider as a canonical h yp erplane: ± x n +1 + P n i =1 x i = 1, where the sign dep ends on the parit y of the free charge. Unfortunately , as the follo wing example illustrates, this h yp erplane will not necessarily in tersect b oth the “reactant” and “pro duct” cones: CONVEX GEOMETR Y AND STOICHIOMETR Y 17 4.1. Example. Consider the reaction: H N O 2 → N O 2 + H + + e − Because a lone negativ e c harge is a specie, we use the h yp erplane − x 4 + P 3 i =1 x i = 1 (remem b er that c harge is indexed last). How ever, the ray corresp onding to H + is of the form ( k , 0 , 0 , k ) for real k > 0 and th us do es not intersect the hyperplane. Ev en without a canonical slicing hyperplane, there still exist (man y!) hyper- planes which slice b oth “reactan t” and “pro duct” cones of a given reaction. If w e insist on working with polytop es instead of cones, w e need only slice with any such h yp erplane, alb eit non-canonical. 4.2. Sp ectator Ions. One metho d chemists utilize for balancing equations in volv- ing charged sp ecies is to in tro duce ﬁctitious sp e ctator ions (e.g. Q + , X − ) which are formal symbols b ound with the charged sp ecies so that the equation no longer in volv es charged sp ecies. After a balance is found, the sp ectator ions are deleted. The following example illustrates this approach: 4.2. Example. Consider the unb alanc e d equation: Z n + N O − 3 + H + → Z n 2+ + N H + 4 + H 2 O (unbalanced) Using sp ectator ions Q + , X − w e instead balance Z n + N O 3 Q + H X → Z nX 2 + N H 4 X + H 2 O + QX Note the additional inclusion of QX among the pro ducts so that Q will b e present on b oth sides. This yields an unique balance: 4 Z n + N O 3 Q + 10 H X → 4 Z nX 2 + N H 4 X + 3 H 2 O + QX Hence, the desired balance to our original equation is 4 Z n + N O − 3 + 10 H + → 4 Z n 2+ + N H + 4 + 3 H 2 O The geometric explanation is simple: Let E b e the non-negative span of the elemen ts inv olved in the reaction. Then the image of the sp ecies simplex lies in E × R where the last co ordinate indexes charge. Thinking of R as the union of the non-negativ e and non-positive ra ys, w e ha v e E × R = E × ( R ≥ 0 ∪ R ≤ 0 ) as sets (simply b y t wisting one of the ra ys orthogonally). Identifying the latter tw o comp onents as the non-negative span of X , Q resp ectively , we see that the co domain ma y b e taken as the simplex spanned b y the elements together with the sp ectator ions. Eac h sp ecie is represented by a p oint in either the E Q, E X -hyperplanes or the p oint QX in the QX -plane, though the reactan t and pro duct cones are not restricted to these h yp erplanes. In fact, the in tro duction of QX is necessary in situations where the reactan t or pro duct p olytop e lie in diﬀerent b oundary faces of the orthorant of E × R ≥ 0 × R ≤ 0 or only one do es while the other is in the interior. See Figure 8. 4.3. Half-Reaction Metho d. One important class of chemical reactions are called oxidation-r e duction reactions. They are characterized by changes in the oxidation states of elements and often inv olv e H + , O H − , H 2 O as sp ecies. In fact, these sp ecies are sometimes not explicitly sp eciﬁed until an equation is balanced. F or c hemical details, see an y general c hemistry textb o ok (e.g. Olmsted[19]). One com- mon approach to balancing oxidation-reduction reactions is the “half-reaction” or “ion-electron” metho d. This method ﬁrst divides the reactant and pro duct sp ecies 18 JER-CHIN (LUKE) CHUANG Figure 8. Geometric Explanation for Sp ectator Ions Figure 9. Geometry of Balancing Redo x Half-Reaction: Steps (3) and (4). in to tw o “half-reactions” corresp onding to an “oxidation” and a “reduction” reac- tion. This selection is determined b y chemical considerations. The half-reactions are separately balanced using H + , O H − , H 2 O , e − and then linearly com bined so as to cancel unbound elections e − . The chemical considerations assure that unbound electrons in the t wo half-reactions will o ccur on opp osite sides of the equation and hence are amenable to an unique cancellation via linear com bination. Thus, the result of the half-reaction method is unique iﬀ each half-reaction admits an unique balance. Eac h half-reaction is balanced using the following recip e: (1) balance all elements except H , O , and the charge (2) balance oxygen O with H 2 O (3) balance hydrogen H with H + ions CONVEX GEOMETR Y AND STOICHIOMETR Y 19 (4) (for basic solutions): cancel H + using the equalit y H 2 O = H + + O H − (5) balance charge with electrons e − The geometry of steps (3) and (4) are illustrated in Figure 9. Symbols X , Y repre- sen t the complexes (i.e. each individual side of the equation) after step (2), shown b y their pro jections in directions O,H only . The diagram shows the case where complex X is deﬁcien t in H and hence balanced with H + resulting in V . Symbol W denotes water H 2 O and Z is the result of (4) (recall b oth charge directions are pro jected out). Line segmen ts H + W and V Z are parallel. The remaining step (5) balances the tw o lifts of Z (as reactant and pro duct complexes) from the face spanned b y the elements in to its join with the charge (i.e. spectator ion) directions. Notice that the ab ov e recip e pro vides an unique balance to a half-reaction iﬀ the pro jection onto the orthogonal complemen t of the subspace indexing H , O , and c harge admits an unique balancing. All steps subsequent to the ﬁrst are uniquely determined and provide a sequence of lifts of the partial solution into the subspaces indexed by O , H , and charge, resp ectiv ely . The following example illustrates the “half-reaction metho d”: 4.3. Example. [19, Ex 17.7, p827] Consider the following reaction in b asic solution : Au + O 2 + C N − → [ Au ( C N ) 2 ] − + H 2 O 2 (redo x reaction) By chemical principles one iden tiﬁes that gold Au is oxidized b y cyanide ions C N − (4.4) Au + C N − → [ Au ( C N ) 2 ] − and oxygen gas O 2 is reduced to hydrogen p ero xide H 2 O 2 : (4.5) O 2 → H 2 O 2 The oxidation half-reaction (Equation 4.4) is readily balanced as Au + 2 C N − → [ Au ( C N ) 2 ] − + e − and the reduction half-reaction (Equation 4.5) as 2 e − + 2 H + + O 2 → H 2 O 2 = ⇒ 2 e − + 2 H 2 O + O 2 → H 2 O 2 + 2 O H − since the reaction is taking place in basic solution. Hence, the ov erall reaction is 2 Au + 4 C N − + 2 H 2 O + O 2 → 2[ Au ( C N ) 2 ] − + H 2 O 2 + 2 O H − Ho wev er, the directness of the recip e b elies a subtlety: the in tersection cone asso ciated to the unb alanc e d reduction half-reaction H 2 O + O 2 + e − → H 2 O 2 + 2 O H − has dimension 2, so that b oth this half-reaction and the total reaction admit multi- ple balancings. The subtlety lies in the fact that for basic solutions the recip e does not balance using H 2 O , O H − directly as in the preceding equation, but instead uses H + and then conv erts via H 2 O = H + + OH − . F or the preceding example, this c hoice constrains the balanced reduction reaction to hav e H 2 O , O H − in a 1 : 1 ra- tio. In eﬀect, it slices the in tersection cone for the half-reaction with the h yp erplane deﬁned by equal co eﬃcients for H 2 O , O H − . Indeed, the one-parameter family of 20 JER-CHIN (LUKE) CHUANG balancings for the reduction action yields a one-parameter family of balancings for o verall reaction: (4.6) (1 − t ) Au + 2(1 − t ) C N − +  1 + t 2  H 2 O +  1 + t 4  O 2 → (1 − t )[ Au ( C N ) 2 ] − + tH 2 O 2 + (1 − t ) O H − for t ∈ [0 , 1]. The output of the “half-reaction” metho d corresp onds to t = 1 / 3. In particular, the reaction: (4.7) 2 Au + 4 C N − + 4 H 2 O + 2 O 2 → 2[ Au ( C N ) 2 ] − + 3 H 2 O 2 + 2 O H − from using the reduction half-reaction ( t = 3 / 5): 2 e − + 4 H 2 O + 2 O 2 → 3 H 2 O 2 + 2 O H − is not obtainable via our tw o chosen half-reactions since H 2 O , O H − are not in a 1 : 1 ratio. One ma y w onder if Equation 4.7 w ere obtainable using the “half-reaction” recipe relativ e a diﬀerent set of half-reactions. This is not the case: for our redox reaction, w e may classify half-reactions b y whether [ Au ( C N ) 2 ] − is a pro duct. If so, then Au, C N − need b e presen t as reactants to admit a balance. The only other possible pro duct combination is H 2 O 2 alone; since Au, C N − cannot b e included among the reactan ts without requiring [ Au ( C N ) 2 ] − among the pro ducts, w e hav e only four p ossible half-reactions: Un balanced Balanced Au + C N − → [ Au ( C N ) 2 ] − Au + 2 C N − → [ Au ( C N ) 2 ] − + e − Au + C N − + O 2 → [ Au ( C N ) 2 ] − 3 e − + 2 H 2 O + Au + 2 C N − → [ Au ( C N ) 2 ] − + 4 O H − Au + C N − → [ Au ( C N ) 2 ] − + H 2 O 2 Au + 2 C N − + 2 O H − → [ Au ( C N ) 2 ] − + H 2 O 2 + 3 e − O 2 → H 2 O 2 2 e − + 2 H 2 O + O 2 → H 2 O 2 + 2 O H − where eac h has been balanced using the recipe for balancing half-reactions. One c hecks that no pair of half-reactions yields Equation 4.7. Hence, the half-r e action metho d c annot ne c essarily ﬁnd al l p ossible b alanc es . Recall that the half-reaction metho d outputs a balancing iﬀ there exist t wo half- reactions for whic h step (1) abov e is realizable, i.e. the pro jection onto the orthogo- nal complemen t of the subspace indexing H , O , and charge admits a balancing. This orthogonal complemen t is equiv alently the span of all non H , O -elements. Supp ose a balancing for the ov erall reaction exists; then the pro jection onto the aforesaid complemen t is non-empt y . No w, denote the set of “reactan t” and “product” sp ecies b y R, P resp ectively . Hence, step (1) is realizable iﬀ there exists decompositions R = R 1 ∪ R 2 and P = P 1 ∪ P 2 suc h that con v( R i ) ∩ con v( P i ) 6 = ∅ for i = 1 , 2. Here, con v () denotes the conv ex hull of sets. The following example shows that we cannot require the subsets to b e disjoint: 4.8. Example. [14] Consider the follo wing unb alanc e d oxidation-reduction reaction: P 2 I 4 + P 4 + H 2 O → P H 4 I + H 3 P O 4 (un balanced) If w e order P b efore I in indexing elements, under the pro jection on to the span of P , I the co ordinates for the reactan ts are (2 , 4) T , (4 , 0) T and that for pro ducts (1 , 1) T , (1 , 0) T . A glance at the p olytop e diagram (see Figure 10) shows that w e cannot require b oth R 1 , R 2 and P 1 , P 2 to b e disjoin t pairs. CONVEX GEOMETR Y AND STOICHIOMETR Y 21 Figure 10. Cannot P artition Reactants and Products in to Dis- join t P airs The following lemma assures us that otherwise existence follows readily: 4.9. Lemma. L et con v ( X ) = con v ( R ) ∩ con v( P ) wher e conv( X ) has dimension c ≤ d for non-empty ﬁnite sets R , P , X ⊆ R d wher e we may assume e ach set individual ly to b e c onvex line arly indep endent. L et x ∈ conv( X ) . Then, ther e exist de c omp ositions R = R 1 ∪ R 2 and P = P 1 ∪ P 2 such that deﬁning H i = con v ( R i ) ∩ conv( P i ) for i = 1 , 2 we have x ∈ conv( H 1 ∪ H 2 ) . F urthermor e, if c is p ositive then we may r e quir e H i to b e non-empty. Pr o of. W e ﬁrst provide an intuitiv e argument. Since conv( X ) is conv ex, excepting the trivial case when con v( X ) is a singleton, we may see x b y p eering through an y b oundary facet of conv( X ). By shifting ourselves slightly , we can place x in the in terior of a bac kground facet of conv( X ). Then, x is in the conv ex h ull of these t wo facets of con v ( X ). T o formalize the ab ov e argumen t, we place con v ( X ) in a c -ball (where c is the dimension of conv( X )) such that the center of the ball is in the interior of con v( X ). T aking the radial pro jection out wards from the cen ter p oint yields a cellular decom- p osition of the b oundary sphere. Cho ose any cell. By appropriate choice of p oint p within this cell, the line through p, x meets the sphere at another p oint q interior to some other cell on the sphere (here we use the fact that conv( X ) is p ositiv e- dimensional). These tw o cells identify t wo facets H 1 , H 2 of conv( X ) such that their con vex hull includes x . Since eac h face of con v( X ) is the in tersection of some faces of con v( R ) , conv( P ) there exist subsets R i , P i suc h that H i = con v ( R i ) ∩ conv( P i ). Finally , note that since H i are facets of con v ( X ), at least one of R i , P i is a prop er subset for eac h i = 1 , 2.  4.10. R emark. In Example 4.3, the pro jection onto the complement of H , O and c harge yields only the species Au, C N , Au ( C N ) 2 . The pro jected reactan t and prod- uct p olytop es meet at a singleton. Hence, we are not assured tw o non-trivial half- reactions in the space spanned by Au, C , N . 4.11. Prop osition. If an oxidation-r e duction r e action admits a b alanc e, then ther e exist de c omp ositions of the r e actants and pr o ducts, e ach into two (not ne c essarily disjoint) gr oups so that applic ation of the half-r e act ion metho d yields a b alanc e r elative the given r e actants and pr o ducts. By appr opriate choic e of de c omp ositions, this b alanc e c an b e made to r eﬂe ct the elemental pr op ortions of al l non-O,H elements in a given initial b alanc e. Pr o of. As mentioned previously , we need only sho w that w e can decomp ose the reaction in to half-reactions for which step (1) of the recip e is realizable. Since the reaction admits a balance, the in tersection cone remains non-empt y after pro jecting out the H , O and charge-directions. Hence, the pro jections of the reactant and 22 JER-CHIN (LUKE) CHUANG pro duct p olytop es satisfy the hypotheses of the lemma. W e let R i , P i b e as assured b y the lemma. Then, H i = R i ∩ P i is non-empty so that step (1) is realizable for eac h half-reaction.  The half-reaction method has b een criticized by some chemists b ecause some- times the choice of o xidation/reduction reactions is unclear and even chemically ﬁctitious[15]. As discussed abov e, the metho d has mathematical dra wbacks in that the recip e may obscure the presence of m ultiple balancings and ma y not capture all p ossible balances. Ho wev er, at least the preceding prop osition assures us that if a balancing exists, then the “half-reaction metho d” can generate a v alid balance, though the half-reactions may b e chemically ﬁctitious. 5. Reaction Mechanisms and Stoichiometric Restrictions In this section we discuss the relationship b etw een reaction mechanisms and stoic hiometric restrictions. Given a c hemical reaction, a reaction mechanism is a sequence of steps that together elucidate how the reactan ts of the ov erall reac- tion are conv erted into the products of the ov erall reaction. Each step is called an elementary chemic al r e action and represents the most fundamental molecular transformations. 5.1. F rom Mec hanism to Linear Stoic hiometric Relations. A giv en reaction mec hanism has implications for observed stoic hiometries of the inv olved sp ecies as illustrated in the following example: 5.1. Example. [22] Let S 1 , S 2 , S 3 b e sp ecies and consider the following reaction mec hanism: S 1 → S 2 + S 3 S 1 + S 2 → 2 S 3 Letting [ S i ] denote (time-dependent) concentrations, the theory of chemical kinetics implies that d [ S 1 ] dt = − X 1 − X 2 d [ S 2 ] dt = X 1 − X 2 d [ S 3 ] dt = X 1 + 2 X 2 where the X i dep end only on the i -th elementary reaction in the mechanism. Elim- inating the X i yields the single relation: d dt (3[ S 1 ] + [ S 2 ] + 2[ S 3 ]) = 0 whic h constrains the observ ed stoichiometries. W e ma y formalize this as follows: Order the sp ecies inv olved and let N b e a matrix with columns indexed b y the steps of the mechanism and the rows indexed b y sp ecies. W e will follow the conv en tion in the literature of negative en tries for CONVEX GEOMETR Y AND STOICHIOMETR Y 23 reactan t species and positive entries for product species. F or the preceding example, w e w ould ha ve N =   − 1 − 1 1 − 1 1 2   Then, elimination of the X i is tan tamount to ﬁnding elemen ts of N S ( N T ) whic h we ma y think of as the space of stoichiometric restrictions. Each v ector in this subspace deﬁnes a linear relation on the concen tration of sp ecies that in tegrates to a stoi- c hiometric restriction which in the chemical literature is called a mass-c onservation e quation . (The terminology arises from the assumption of constant volume during the course of the reaction.) In particular, mass-conserv ation equations are deﬁned with resp ect to a given set of reactions. Note that though d [ S i ] /dt = X i , since we do not assume a particular form of X i w e need not assume mass-action kinetics. This sho ws that for given initial concentrations [ S i ], the resulting ﬂo w is constrained to lie in an aﬃne space linear isomorphic to N = CS( N ), the orthogonal comple- men t to N S ( N T ). F urther information can b e obtained if we assume mass-action kinetics, whence w e obtain an ODE system of p olynomials. The literature on these systems is v ast (see for example either Horn and Jac kson[13] or ´ Erdi and T´ oth[9]). In addition, chemists also deﬁne element c onservation e quations . These are de- ﬁned for any given single reaction. Given the elemen tal prop ortion matrix M for a reaction, since the ro ws index elemen ts, we may require their conserv ation in the course of a reaction. Thus, the set of all elemen t-conserv ation equations corresp onds to the ro w space RS( M ). Up to now our discussion has not assumed an y elemental composition for the sp ecies nor need there b e one. The existence of such implies a (non-negative) el- emen tal prop ortion matrix M suc h that M N = 0 b ecause each reaction indexed b y N is balanced. In particular, we ma y assume the ro ws of M index only ele- men ts present so that the row space RS( M ) must contain a p ositive vector. Since NS( N T ) ⊇ RS ( M ), a necessary condition for the existence of M is that NS( N T ) con tains a p ositive vector. In the chemical literature, suc h matrices N are kno wn as c onservative [18]. The suﬃciency of this condition is readily c heck ed. F urther- more, given an o verall reaction and a reaction mechanism for it, since the ro ws of M index elements, the relation RS( M ) ⊆ NS( N T ) shows that an y elemental- conserv ation equation is a mass-conserv ation equation but not vice-versa (see also Smith and Missen[18]). Letting X , S , E denote the spaces spanned by elementary reactions, sp ecies, and elements resp ectively and thinking of N , M as linear maps, w e ha ve X N → S M → E and the discrepancy b etw een mass- and elemen t-conserv ation equations is described b y the homology (k er M ) / (im N ) at S . In a reaction, there are usually short-lived sp ecies called interme diates that are diﬃcult to observe. Under a steady-state h yp othesis, or if we consider only com- pleted reactions, or just assuming that they cannot b e detected, the space of ob- serve d linear dependencies is instead the orthogonal pro jection of NS( N T ) on to the span of observ able sp ecies. W e denote this space π K NS( N T ). 5.2. F urther Conclusions from a Given Mechanism. In this subsection we sho w how sev eral chemically in teresting problems lead to the problem of counting 24 JER-CHIN (LUKE) CHUANG lattice p oints in polytop es, a m uc h-studied and activ e research area in combina- torics. 5.2.1. R e actions Consistent with Me chanism. Let N b e an integer matrix associated with a collection of (elementary) reactions, and write N = CS ≥ 0 ( N ) for the c one determined by its non-negative span. Let K b e the set of known (or observ able) sp ecies and K their span. F or a c hemical equation c ∈ K we will say that c is c onsistent with N or that N is a me chanism for c provided c ∈ N . F or a given N , the set of consistent chemical equations is thus the set of lattice p oin ts in N ∩ K mo dulo dilation, or alternatively the points of the pro jective subset P Q ( N ∩ K ). Supp ose the cone N ∩ K is non-trivial. Then, P Q ( N ∩ K ) is either a singleton represen ting an integer lattice point or con tains inﬁnitely-many such represen tatives. One can justify the latter claim by noting the inﬁnitude of rational v ectors in any non-trivial pro jective neigh b orho o d of an integral vector. Alternativ ely , via Ehrhart theory the n um b er of lattice points in in tegral dilations of a rational d -polytop e P are coun ted b y a quasi-polynomial E P called the Ehrhart quasi-p olynomial (see [7],[8]). That is, there exists some N > 0 and degree d - p olynomials f 0 , . . . , f N − 1 suc h that E P ( t ) = f i ( t ) for t ≡ i mo d N . Lattice p oints in the relativ e in terior are coun ted b y another quasi-p olynomial E P ( t ) where the tw o quasi-p olynomials are related b y E P ( t ) = ( − 1) d E P ( − t ) where d = dim P . Hence, if d > 1, then E P gro ws at least quadratically so that P Q ( N ∩ K ) contains inﬁnitely- man y p oints. Chemically , this implies that there is either an unique chemical equation consisten t with a given mechanism or there are inﬁnitely many . Ho wev er, since w e do not exp ect to observe ov erall reactions built from man y instances of eac h elemen tary step, in practice we see but ﬁnitely-many consisten t equations. Supp ose we wan t to consider ov erall reactions con taining at most t sp ecies (coun t- ing multiplicities). Then, w e can count the lattice p oints within the polytop e formed b y bounding the cone N ∩ K with the cross-p olytop es C d t = { z ∈ R d | P | z i | = t } . This is computationally feasible since there exist p olynomial-time algorithms to compute the num b er of lattice points in polytop es. See deLoera[7] or Barvinok[2],[3] and the programs La ttE [6] and bar vinok [24]. (The output of the latter is tec hni- cally a piece-wise step-polynomial, a generalization of quasi-polynomials.) How ever, w e should keep in mind that the counts from lattice p oin t enumeration in S ma y include multiples of a giv en balance. In light of the ab ov e discussion, we ma y b e tempted to deﬁne a partial order on the in terior p oints of the cone N ∩ K via the total num ber of known sp ecies in volv ed (counting m ultiplicit y), namely P K s i . How ever, there exist lattice p oints on arbitrarily large cross-p olytop es such that their con vex h ull with the origin does not include any interior lattice p oint. F or example, taking i, j, k unique indices, let x = t e i − e j and y = t e i − e k for p ositive in teger t . The conv ex hull con v( { 0 , x , y } ) con tains no in terior p oint. Thus, relative the l 1 -norm there ma y not be a minimum consisten t reaction. 5.2. R emark. W e may also wan t to consider consistent reactions with mechanisms in volving no more than a ﬁxed num b er of reactions. W e then would hav e to work in reaction space X by coun ting lattice p oin ts in the intersection of cross-p olytop es and X ≥ 0 ∩ N − 1 ( N ∩ K ) where X ≥ 0 is the non-negativ e orthoran t in X . 5.3. R emark. This observ ation on lattice p oints of cross-p olytop es also applies to studying the p olyhedron Q ⊆ S that parametrizes balances of a given c hemical CONVEX GEOMETR Y AND STOICHIOMETR Y 25 equation. Hence, using the sum of the coeﬃcients in a balance ma y not be suﬃcient to single out a unique “simplest” balance. How ev er, we ma y use the Ehrhart theory to estimate the minimal total num b er of species inv olved in the o verall reaction (coun ting m ultiplicity) b y in tersecting Q with cross-p olytop es C s t in S . 5.4. R emark. W e express Craciun and Pan tea’s result on c onfoundable reaction net works[5] using our framework. Tw o reaction netw orks are confoundable if they yield the same kinetic diﬀerential equations under the assumption of mass-action kinetics. Let N b e the equations of a reaction net work expressed as a matrix (columns indexing reactions, rows sp ecies), S the vector of sp ecie concentrations, and k i the forw ard rate-constant for the i -th reaction. Let x be a vector of dimension equal to the num b er of elementary reactions in the mechanism, and suppose that the entries of x dep end on a function of the reactan t (i.e. source) complexes alone. Then, d S /dt = N diag ( k i ) x . W e may assume the comp onents of x and the columns of N are sorted by reactant complex so that N diag( k i ) has only as many columns as distinct source complexes and the dimension of x is the n umber of distinct reactant complexes. (Just add together columns of N diag ( k i ) indexed by reactions having iden tical reactan t complex so eac h column of N diag( k i ) is a cone on N parametrized b y the k i of reactions with that reactan t complex.) Thus, a mec hanism associates to eac h reactant complex a cone. Then, tw o netw orks yield the same kinetic diﬀern tial equations provided they hav e identical set of reactant complexes and for a ﬁxed reactan t complex, the cone so indexed for each of the tw o mec hanisms in tersect non-trivially . This is Craciun and Pan tea’s result, though without assuming mass- action kinetics. 5.5. Example (Decomposition of Azomethane) . W e consider the decomp osition of azomethane: (5.6) 5 C 2 H 6 N 2 → 3 N 2 + C H 4 + C 2 H 6 + C 3 H 8 N 2 + C 4 H 12 N 2 A reaction mechansim is given b y Missen and Smith[18] which inv olves intermedi- ates that w e will lab el X , Y , Z . Under the ordering C 2 H 6 N 2 , N 2 , C H 4 , C 2 H 6 , C 3 H 8 N 2 , C 4 H 12 N 2 , X , Y , Z the mechanism has asso ciated matrix (5.7) N =               − 1 − 1 0 0 − 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 − 1 − 2 − 1 − 1 − 1 0 1 0 − 1 0 0 0 0 0 0 1 − 1               T o count consistent ov erall reactions where no more than a ﬁxed num b er of sp ecies are present in the equation, we consider the intersections (cone( 0 , N ) ∩ K ) ∩ C 9 t where C 9 t are cross-p olytop es. F or example, the in tersection with C 9 6 is a 6- p olytop e in S = R 9 ha ving 14 vertices. By bar vinok we ﬁnd that this p olytope has 35 lattice points so that there are 35 consistent o verall reactions where there are no more than 6 sp ecies (counting m ultiplicity). 26 JER-CHIN (LUKE) CHUANG Figure 11. Relations among subspaces of S 5.2.2. Algebr aic R epr esentations of a R e action. Let c ∈ K b e a given chemical equation with mechanism N . W e will call an y non-negativ e integral solution to N x = c an algebr aic r epr esentation for c . Algebraic representations do not nec- essarily corresp ond to c hemical path wa ys within a given mechanism b ecause the presence of intermediates in elementary reactions means that the order of reactions in a mec hanism is imp ortan t. F or example let I 1 , I 2 b e in termediates, and supp ose the following pair of equations: S 1 + I 1 = S 2 + I 2 S 3 + I 2 = S 4 + I 1 comes from some mechanism. Then their sum is an algebraic representation for S 1 + S 3 = S 2 + S 4 , but the t wo reactions alone cannot constitute a c hemically viable mec hanism unless one of the intermediates is present at the start of the reaction. F urther discussion of this restriction is postp oned un til Subsection 5.4. F or now we note that the num b er of algebraic representations is given by the num b er of non- negativ e lattice p oints in the polyhedron determined by N x = c . As mentioned ab o ve, this is computationally feasible to determine. This n umber is ﬁnite iﬀ NS( N ) has no non-negative element, or equiv alently that 0 is not in the con vex hull of the columns of N . If inﬁnite, we ma y still study represen tations in volving no more than t -steps in total by slicing the non-negative solution cone of N x = c in X with the h yp erplane P x i =1 x i = t where x = dim X the num b er of columns of N . The case of unique algebraic representation corresp onds to p olytop es { x | N x = c } with unique interior lattice p oint. Because we are working ov er Z , this may happ en ev en if ker N 6 = 0. F or example, the p olytop e corresp onding to the intersec- tion of the hyperplane { ( x, y ) | x + y = 2 } with the non-negative quadrant is a line segmen t with an unique interior lattice point. Polytopes in R 2 with unique interior lattice p oint hav e b een classiﬁed up to unimo dular equiv alence in Rabinowitz[20], but seems unknown in higher dimensions. Already , there exist lattice simplices in R d with unique interior lattice p oint but ha ving more than 2 2 d − 1 b oundary lattice p oin ts (see Zaks, Perles, and Wills[25]). 5.8. Example. Again, consider the mechanism for the decomp osition of azomethane giv en in Example 5.5. The conv ex hull of the columns of N determines a conv ex 5-p olytop e in S = R 9 . Since this polytop e does not con tain 0 , there are only ﬁnitely- man y non-negative solutions to N x = c for an y c ∈ S . Using c = ( − 5 , 3 , 1 , 1 , 1 , 1) T from the reaction (5.6), we ﬁnd that the p olytop e parametrizing non-negativ e so- lutions is actually just a point, namely (3 , 1 , 1 , 1 , 1 , 1) T . Th us, this is the unique non-negativ e in tegral comb ination of elementary reactions of the mechanism N yielding the o verall reaction (5.6). CONVEX GEOMETR Y AND STOICHIOMETR Y 27 Figure 12. Relation among subspaces of K . The p ossibilit y of non-trivial in tersection betw een O and Z = ( π K RS( M )) ⊥ has been suppressed in the Diagram. 5.3. Mec hanisms Consistent with Observed Relations. W e no w consider an in verse problem. Supp ose w e kno w all the species (including in termediates) in volv ed in a reaction with unkno wn mechanism; denote their span by S . A set of linear dep endencies are observed which span a space O (obtained via multiv ariate linear regression of data, for example). What can w e sa y about mec hanisms N compatible with these restrictions and in particular of the image space N = im N ? (Note here N represents a linear space contra the cone in the preceding subsection.) W e know that for a given c hemical equation c , the mo duli of mechanisms for c is given by the set of all integer matrices N suc h that c ∈ N . Figures 11 and 12 summarize the relations among the subspace now to be deﬁned. Let K , U b e the sets of known and unknown (intermediates) sp ecies, resp ectiv ely and S = K ∪ U . W rite K , U , S for the spaces they span and equip S with the canonical inner pro duct. Then S = K ⊥ U . Now, write ker M = N ⊥ H where we think of H as the homology at N → S M → . Then, S = N ⊥ H ⊥ RS( M ) so that H ⊥ RS( M ) = NS( N T ) = π K NS( N T ) ⊥ π U NS( N T ) where π K , π U denote orthogonal pro jection onto K , U resp ectively . W rite O for the space of observed dep endencies. W e will not assume that this constitutes all p ossible linear dep endencies, hence we ha ve (5.9) π K H ⊥ π K RS( M ) = π K NS( N T ) = O ⊥ W for some subspace W . In particular, we see that (5.10) dim π K NS( N T ) ≥ dim( O + π K RS( M )) where the right-hand-side ma y b e computed without know le dge of any interme di- ates . If we sp ecify intermediates, we can do muc h b etter. F rom Equation (5.9) we 28 JER-CHIN (LUKE) CHUANG ha ve (5.11) [ O + W ] ⊆ [ π K H ] ⊆ K π K RS( M ) and since K = π K NS( M ) + π K RS( M ) we ha ve (5.12) π K NS( M ) π K NS( M ) ∩ π K RS( M ) ∼ = K π K RS( M ) ∼ = ( π K RS( M )) ⊥ = : Z ⊂ π K NS( M ) so that w e mu st choose a subspace X such that (5.13) pro j Z O ⊆ X = : pro j Z H where we choose a prop er sup erset when w e believe that W 6 = 0, i.e. that O do not represen t all p ossibly observ able stoichiometric restrictions. Regardless, we ha ve π K H = X + ( π K NS( M ) ∩ π K RS( M )) (5.14) ⊇ pro j Z O + ( π K NS( M ) ∩ π K RS( M )) Since H ⊆ NS( M ), to lift π K H back up to NS( M ) we need to study the kernel of the restriction of π K to NS( M ). W e denote this map ϕ = π K : NS( M ) → K . W riting M = ( M K M U ) where M K , M U are the submatrices of M indexed b y kno wn and intermediate sp ecies, resp ectively , we see that NS( M ) is the cen tralizer of the maps M i K π K , − M i U π U so that k er ϕ = i U NS( M U ). Th us, H = π − 1 X + π − 1 ( π K NS( M ) ∩ π K RS( M )) + i U NS( M U ) (5.15) ⊇ π − 1 pro j Z O + π − 1 ( π K NS( M ) ∩ π K RS( M )) + i U NS( M U ) In the case NS( N T ) = O , we ha ve X = pro j Z O so that the p ossible choices for H (and hence N since N = ( H ⊕ RS( M )) ⊥ ) are parametrized by subspaces of ( π K NS( M ) ∩ π K RS( M )) ⊥ i U NS( M U ) with dimension bounded b y dim( O / Z ). W e illustrate these ideas with tw o examples: 5.16. Example. Again, consider the decomp osition of azomethane giv en b y Equation (5.6) of Example 5.5 ab ov e. With the sp ecies ordered as in that example and ordering the elements C , H , N in that order, the asso ciated elemen tal prop ortion matrix is M =   2 0 1 2 3 4 1 2 3 6 0 4 6 8 12 3 5 9 2 2 0 0 2 2 0 2 2   where the in termediates X , Y , Z hav e molecular formulae C H 3 , C 2 H 5 N 2 , C 3 H 9 N 2 resp ectiv ely . Supp ose the space O ⊂ K = R 6 of observed stoic hiometric restrictions has a basis ( − 1 , − 3 , 1 , 2 , 0 , 1) T (2 , 4 , 0 , − 2 , 0 , 0) T (0 , 0 , − 1 , 0 , 1 , 0) T Let Z = ( π K RS ( M )) ⊥ K . W e compute that dim( O / Z ) = 0. Thus, assuming O includes all p ossible stoichiometric restrictions, we ma y set H = 0. Also, we ﬁnd that dim π K RS ( M ) ∩ π K N S ( M ) = 3 and that NS( M U ) = 0 so that the p ossible N are parametrized by N = (span( s ) + RS( M )) ⊥ s ∈ π − 1 ( π K RS ( M ) ∩ π K N S ( M )) CONVEX GEOMETR Y AND STOICHIOMETR Y 29 5.17. Example (Oxidation of F ormaldehyde) . Missen and Smith pro vide a mech- anism in [18] for the oxidation of formaldeh yde. If we order the elements b y H , C , O, C o, + and the sp ecies by C H 2 O , C o 3+ , C o 2+ , H + , C O , H 2 O , C H 2 O 2 , H 2 , C H 2 O C o 3+ , C H O , O H , H where the last four sp ecies are intermediates, we hav e the asso ciated elemen tal prop ortion matrix (5.18) M =       2 0 0 1 0 2 2 2 2 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 2 0 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 3 2 1 0 0 0 0 3 0 0 0       The given mec hanism has asso ciated matrix (5.19) N =                     − 1 0 0 1 0 − 1 − 1 − 1 − 1 0 − 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 − 1 − 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 − 1 0 0 0 0 0 0 0 1 − 1 − 1 − 1 1 0 1 0 0 0 1 0 − 1 − 1 0 0 0 0 0 1 0 1 − 1                     so that the space O = π K NS( N T ) ⊂ K = R 8 has a basis (1 , 0 , 0 , 0 , 1 , 0 , 0 , 1) T (0 , 0 , 0 , 1 , − 2 , 0 , 0 , 0) T (0 , 1 , 0 , 0 , 2 , 0 , 0 , 0) T (0 , 0 , 0 , 0 , 0 , 1 , 0 , 1) T (0 , 0 , 1 , 0 , − 2 , 0 , 0 , 0) T (0 . 0 , 0 , 0 , 0 , 0 , 1 , − 1) T Supp ose we started knowing only the sp ecies and the abov e space O . Set Z = ( π K RS( M )) ⊥ K and let pro j Z denote the orthogonal pro jection on to Z . W e ﬁnd that dim( O / Z ) = 1 where pro j Z O = span  1 , − 12 23 , 12 23 , 12 23 , − 49 23 , − 26 23 , 26 23 , − 29 23  T Assuming all stoichiometric restrictions are included in O , i.e. O = π K NS( N T ), w e ha ve π K H = pro j Z O . A basis for π K NS( M ) ∩ π K RS( M ) is given by (2 , − 1 , 0 , 0 , 0 , 0 , 0 , 2) T (1 , 0 , 0 , 0 , 1 , 0 , 1 , 0) T ( − 1 , 0 , 0 , 0 , − 1 , 1 , 0 , 0) T (0 , 2 , 1 , 1 , 0 , 0 , 0 , 0) T and this represen ts the uncertain ty in determining π K H . One c hecks that π K is injectiv e on NS( M ) so that pro j Z O + ( π K NS( M ) ∩ π K RS( M )) lifts uniquely to NS( M ). Hence, compatible subspaces N = ( H ⊥ RS( M )) ⊥ are parametrized by 30 JER-CHIN (LUKE) CHUANG the 4-dimensional space π K NS( M ) ∩ π K RS( M ). Setting π K H = pro j Z O yields H = span  40 23 , 26 23 , 1 , 1 , − 29 23 , − 14 23 , 26 23 , − 26 23 , − 49 23 , 12 23 , − 12 23 , 1  T Ho wev er, using the mechanism N giv en ab o ve b y Missen and Smith yields H = span  1 , − 47 64 , 15 32 , 15 32 , − 75 32 , − 5 128 , 7 4 , − 101 128 , 17 64 , − 43 64 , − 219 128 , 315 128  T The diﬀerence of their pro jections into K is  761 12928 , − 1573 6464 , − 9 404 , − 9 404 , − 4379 12928 , 13249 12928 , 4435 6464 , 1285 3232  T whic h one veriﬁes is in the subspace π K NS( M ) ∩ π K RS( M ). The dimensions of imp ortant subspaces in Examples 5.16 and 5.17 are summa- rized in the table b elow: Space Ex 5.17 Ex 5.16 NS( M ) 7 6 RS( M ) 5 3 K 8 6 π K NS( M ) 7 6 π K RS( M ) 5 3 π K NS( M ) ∩ π K RS( M ) 4 3 O 6 3 O /π K RS( M ) 1 0 π U RS( M ) 5 3 5.4. F rom Subspaces N to Mec hanisms. Two tasks remain in obtaining reac- tion mechanisms consistent with observed stoichiometric restrictions. The ﬁrst is con vex-geometric: we seek spanning sets in N whose positive h ull con tains the o ver- all reaction. This ensures algebraic representabilit y . The second is order-theoretic: these algebraic representations must then b e sub ject to the constraint on the ap- p earance of in termediates discussed in Subsection 5.2.2. The ﬁrst task seems computationally exp ensive: Elementary reactions hav e usu- ally no more than tw o (or at most three) molecules of a given sp ecie so that we can restrict atten tion to those lattice p oin ts in N ∩ [ − 3 , 3] s where s = dim S having both p ositiv e and negative entries. How ever, this gro ws exp onentially with the n umber of species inv olv ed. Perhaps some savings may result from w orking pro jectiv ely and this may also b e helpful enumerating spanning sets, but on this the author is uncertain. Nev ertheless, as explained in Subsection 5.2.2, algebraic representativ es ma y b e obtained from the lattice p oints. The second task is more tractable. Let X ⊂ X b e a ﬁnite set of reactions, u the n umber of distinct intermediates among the reactionsin X , and ∆ u the u - dimensional simplex (as an abstract simplicial complex). W e ma y assume there are no redundacies in X . W e lab el the v ertices K, U 1 , . . . , U u . Deﬁne set maps R, P : X → ∆ u mapping a reaction to the simplex indexed by its reactants and pro ducts respectively , with kno wns collapsed to K . F or example, if w e ha ve kno wns K 1 , K 2 and in termediates U 1 , U 2 , the maps R, P take the reaction K 1 + U 1 → K 2 + U 2 to the 1-simplices [ K , U 1 ] and [ K, U 2 ] resp ectively . CONVEX GEOMETR Y AND STOICHIOMETR Y 31 Figure 13. Precedence relations among the reactions in the mec h- anism for the decomposition of azomethane. See Example 5.20 and subsequen t discussion. Let V ( · ) denote the set of vertices of a simplicial complex, and for each set A let ∆( A ) denote the simplex (as an abstract simplicial complex) with v ertices indexed b y the elements of A . F or any non-empty sub complex Y ⊆ ∆ u deﬁne ϕ ( Y ) : = (∆ ◦ V )( X ∪ P R − 1 ( Y )) The coface to the vertex K in the simplex ϕ ( Y ) indexes all intermediates inv olved (either as reactants or pro ducts) in the reactions indexed b y X . W e set ϕ ( {∅} ) = { K } . Since we assume X is ﬁnite, the iterates ( ϕ ◦ · · · ◦ ϕ )( {∅} ) ev entually stabilize. In particular the U -indices of this complex index the inter- mediates that can p ossibly arise through reactions in set X . Note also that if a reaction x ∈ X o ccurs then there is some sequence of reactions x 1 , . . . , x N where the reactan ts of x 1 are known sp ecies and the in termediate reactants of x j are among the intermediate pro ducts of x i for i < j . Thus, x ∈ ( R − 1 ◦ ϕ N +1 )( {∅} ) so that if a reaction o ccurs then it is an element of ( R − 1 ◦ ϕ i )( {∅} ) for some i > 0. The con verse is readily c heck ed. Finally , note that the sets ( R − 1 ◦ ϕ i )( {∅} ) for sucessiv e i ≥ 1 deﬁne a partial order on the elements in X the set of reactions. In particular, reactions not in volv ed in this partial order cannot o ccur. 5.20. Example. Consider again the decomp osition of azomethane from Example 5.5 where we denoted the in termediates X , Y , Z . Since N = NS( M ) w e may compute that there are 116 non-zero in tergal vectors in the in tersection NS( M ) ∩ [ − 1 , 1] 9 , or equiv alently 58 distinct lines. It seems computationally prohibitive to enumerate spanning sets for N among these without further heuristics. Nonetheless, choosing as a particular example the set of equations given by the mec hanism N in Equation (5.7), we ha ve Rxn P ( · ) R ( · ) 1 K [K,X] 2 [K,X] [K,Y] 3 X K 4 [X,Y] K 5 [K,M] Z 6 [X,Z] K 32 JER-CHIN (LUKE) CHUANG W e compute that ϕ ( K ) = ∆( { K, X } ) and that ϕ 2 ( K ) = ∆ 3 . The asso ciated p oset is given in Figure 13. Th us, we see that an algebraic representation is order realizable iﬀ it is equal to ( R − 1 ◦ ϕ i )( {∅} ) for some i . 5.21. R emark. W e hav e only considered stoichiometric prop erties of reactions. In particular we hav e not incorp orated stereo chemical and thermo dynamic consider- ations and suc h may help reduce the search space in determining spanning sets for computing algebraic represen tatives. F urthermore, muc h of the discussion requires kno wledge of the intermediate sp ecies which would need to b e guessed by chemical in tuition. Regardless, it is hop ed that the ab ov e pro cedure provides an attempt at the computational disco very of reaction mec hanisms. 6. Appendix: Ma thema tical Justifica tions for Geometric Viewpoint In this section we justify several mathematical assertions from Section 2. First, here is the justiﬁcation of the linear algebraic fact in vok ed in the discussion relation the p olyhedron Q and the intersection cone I : Let S = CS( A n × a ) and T = CS( B n × b ) and consider the n × ( a + b ) matrix M with columns from A, B resp ectively . Then, S + T = CS( M ). 6.1. Prop osition. nullit y( M ) − dim( S ∩ T ) = nullit y ( A ) + nullit y( B ) . Pr o of. By the Rank-Nullity Theorem for M : dim(CS( M )) + dim(NS( M )) = a + b By choosing a basis for S ∩ T , separately extending to bases for S, T and then in voking inclusion-exclusion: dim( S + T ) + dim( S ∩ T ) = dim( S ) + dim( T ) Subtracting these tw o equations and using the Rank-Nullity Theorem for A, B yields n ullity( M ) − dim( S ∩ T ) = nullit y( A ) + nullit y( B )  Next, w e provide geometric justiﬁcations for the statemen ts from Section 2 that for a giv en c hoice of “reactants” and “pro ducts”: (1) A non-empty in tersection p olytop e necessarily contains a rational p oint. (2) Eac h rational p oint in the intersection p olytop e corresp onds to some bal- ancing of the chemical equation. Algebraic arguments essen tially follo w from “chasing” Diagram 2.17. The former follo ws readily from the following tw o results: 6.2. Prop osition. The interse ction of two p olytop es with r ational vertic es is a p olytop e with r ational vertic es. Pr o of. Since the polytop es hav e rational vertices, eac h supp orting h yp erplane ma y b e chosen to b e of the form P n i x i = h for rational n i , h . Now, each v ertex in the in tersection polytop e is given as the unique intersection lo cus of certain supp orting h yp erplanes, hence the unique solution of a system N x = h where N , h hav e rational entries. Thus, each v ertex is rational.  CONVEX GEOMETR Y AND STOICHIOMETR Y 33 6.3. Prop osition. A p ositive-dimensional p olytop e with r ational vertic es c ontains a r ational p oint in its r elative interior. Pr o of. Let d b e the dimension of the polytop e and { x i } an y d + 1 generating v ertices whose aﬃne hull is d -dimensional. Any conv ex linear com bination P d i =1 t i x i with t i rational and 0 < t i < 1 yields a p oint with the desired prop erties.  6.4. Corollary . A non-empty interse ction p olytop e ne c essarily c ontains a r ational p oint. If the interse ction p olytop e is p ositive-dimensional, we c an ﬁnd a r ational p oint in its r elative interior. Pr o of. The “reactant” and “pro duct” p olytop es hav e rational v ertices. Hence, the in tersection p olytop e has rational v ertices, and by the preceding prop osition, w e are done.  6.5. Prop osition. Each r ational p oint in the interse ction p olytop e c orr esp onds to some b alancing of the chemic al e quation. Pr o of. It suﬃces to show that a rational point in the con vex hull of rational p oints is expressible as a rational linear com bination of those p oints. W e pro ceed b y induction on the n umber of generating v ertices and may assume without loss of generalit y that the generators are a minimal set for the given h ull. The base case of a line segment determined by its tw o endp oin ts is evident. Supp ose the statement is true for p oin ts in the conv ex hull of at most d rational p oints. Let p b e a rational p oint in the conv ex hull of d + 1 rational p oints. W e may assume that the conv ex h ull is not generated by any proper subset of the d + 1 p oints. Let q b e one of the generating p oints. By using a rational frame based at q , we ma y assume the ambien t space is R d . The d generating p oin ts b esides q determine a ( d − 1)-hyperplane H of the form n · x = b where n , b are rational and n is normal to H . Then, n · p and n · q are rational num b ers so that by taking dot pro duct of n with the orthogonal decomp ositions (6.6) p = s + p n q = t + q n where s , t are in hyperplane H , we conclude that p, q 6 = 0 are rational. Finally , (6.7) p = s + p q ( q − t ) =  s − p q t  + p q q a rational linear combination as desired.  References [1] Rob ert A. Alb erty . Chemical equations are actually matrix equations. J. Chem. Educ. , 68:684, 1991. [2] Alexander Barvinok. Lattice points and lattice p olytop es. In Handb o ok of discr ete and c om- putational ge ometry , CR C Press Ser. Discrete Math. Appl., pages 133–152. CRC, Bo ca Raton, FL, 1997. [3] Alexander Barvinok and James E. Pommersheim. An algorithmic theory of lattice p oints in p olyhedra. In New persp e ctives in algebr aic combinatorics (Berkeley, CA, 1996–97) , vol- ume 38 of Math. Sci. R es. Inst. Publ. , pages 91–147. Cambridge Univ. Press, Cambridge, 1999. [4] G.R. Blakley . Chemical equation balancing. J. Chem. Educ. , 59:728–734, 1982. [5] Gheorghe Craciun and Casian Pan tea. Identiﬁabilit y of chemical reaction networks. J. Math. Chem. , 44(1):244–259, 2008. [6] J.A. De Loera, D. Ha ws, R. Hemmeck e, P . Huggins, J. T auzer, and R. Y oshida. A user’s guide for latte v1.1. 2003. 34 JER-CHIN (LUKE) CHUANG [7] Jes ´ us A. De Loera. The many asp ects of counting lattice points in polytop es. Math. Semester- b er. , 52(2):175–195, 2005. [8] Jes ´ us A. De Lo era, Raymond Hemmeck e, Jeremiah T auzer, and Ruriko Y oshida. 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What is the overall stoichiometry of a complex reaction? J. Chem. Educ. , 80(5):520–523, 2003. [23] Zolt´ an T´ oth. Balancing c hemical equations by inspection. J. Chem. Educ. , 74(11):1363–1364, 1997. [24] S. V erdo olaege. softw are pack age barvinok (2004), electronically av ailable at http://freshmeat.net/projects/barvinok/. [25] J. Zaks, M. A. Perles, and J. M. Wilks. On lattice p olytop es having interior lattice p oints. Elem. Math. , 37(2):44–46, 1982. E-mail address : chuang.jerchin@gmail.com

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