1 Background: Balancing Chemical Reactions

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1 MSE 35 Balancing Chemical Reactions Instructor: R.G. Erdmann In this project, worth -2 points, you will write software to automatically balance chemical reactions. Background: Balancing Chemical Reactions. Balancing a Single Reaction The basic requirement in writing a balanced chemical reaction is that the total number of elements of each element is conserved (and, for electrochemical reactions, that the total charge is conserved). Thus, the basic problem in balancing a chemical reaction is finding stoichiometric coefficients on the reacting species that satisfy these constraints. For example, consider the problem of writing a balanced chemical reaction involving the species H 2 O, H 2, and O 2 : xh 2 + yo 2 + zh 2 O =, In this convention, in which all species have been written on the same side of the equal sign, positive stoichiometric coefficients will be taken to be products, while negative coefficients represent reactants. It it seen that in this example there are three unknowns, but only two conservation equations can be written since there are only two elements present. Thus, the solution to this system is only specified to within a multiplicative constant. This is because any balanced chemical reaction is still balanced if each of the stoichiometric coefficients is multiplied by a common non-zero factor. Thus, to obtain a unique solution, one of the stoichiometric coefficients must be specified. In the example above, specifying that z = results in two equations for the two unknowns x, and y: 2x + y = 2 (conservation of H), () x + 2y = (conservation of O), (2) for which the solution is x = and y = /2. The negative values for x and y indicate that they are reactants while the chosen value of z = implies that it is a product. Thus, we have H O 2 = H 2 O as the final balanced reaction. To further understand the structure of the problem, we write () in matrix form: ( ) ( ) ( ) 2 x 2 =. 2 y It can be seen that the first column of the matrix, ( 2 ), corresponds to the number of atoms of H and O in the first chemical species, H 2. The second column of the matrix, ( 2 ), corresponds to the number of atoms of H and O in the second chemical species, O 2. Finally, the righthand side of (3), ( ) 2, corresponds to the negative of the number of atoms of H and O in the

2 specie for which we specified a stoichiometric coefficient of unity, H 2 O. Thus, in the general case, a chemical reaction can be written involving n species and n elements by specifying a value for one of the n stoichiometric coefficients and writing n conservation equations for the remaining n unspecified coefficients. The equations can be easily assembled in matrix form by stacking column vectors with entries corresponding to the numbers of elements in each species, with the right-hand side corresponding to the negative of the column vector for the species with a specified stoichiometric coefficient of unity. To illustrate further, consider the problem of writing a balanced chemical reaction among the following species: CO, H 2 O, CO 2, and CH 4. There are four species involving three elements, so a single chemical reaction can be written involving these species: xco + yh 2 O + zco 2 + wch 4 =. Ordering the elements alphabetically as C, H, O, then we have the following column vectors for each of the species: CO : H 2 O : 2 CO 2 : CH 4 : 4. 2 If we specify w = then the system of equations is easily assembled as x 2 y = 4, 2 z or, equivalently, Solving this system, we obtain x = 4, y = 2, and z = 3. Thus, Ax = b. (3) 4 CO + 2 H 2 O = CH CO 2..2 Writing Systems of Chemical Reactions In the general case in which the number of unique species is s and the number of elements is e, we can write a system of r = s e independent chemical reactions among the species. To see how this is done, consider the problem of writing reactions among the following species: CO, H 2 O, CO 2, CH 4, H 2, C. We have s = 6 chemical species and e = 3 elements, so we can write a set of r = 3 independent chemical reactions. Each of the three reactions will take the form u CO + v H 2 O + w CO 2 + x CH 4 + y H 2 + z C =, where some of the stoichiometric coefficients may, of course, be zero. Given that we have six unknowns but can still only write three conservation equations for any given reaction, we must specify three of the six unknown stoichiometric coefficients manually. 2

3 .2. Choosing which coefficients to specify In choosing which stoichiometric coefficients to solve for and which to specify manually, it is important to choose sets which will enable us to write balanced chemical reactions. Choosing the wrong set may result in equations that have no solution, as will be demonstrated in the following example. As an example of what can happen if the wrong set is chosen, suppose that we choose to manually specify u, v, and w and solve for x, y, and z. The simplest way manually set the coefficients is to set one of u, v, or w equal to unity and the others equal to zero. There are three ways to do this, each resulting in a different chemical reaction. Denoting the unknown stoichiometric coefficients for the nth balanced chemical reaction as x n, y n, and z n, then we now have three chemical reactions to balance: CO + H 2 O + CO 2 + x CH 4 + y H 2 + z C =, (4) CO + H 2 O + CO 2 + x 2 CH 4 + y 2 H 2 + z 2 C =, (5) CO + H 2 O + CO 2 + x 3 CH 4 + y 3 H 2 + z 3 C =. (6) Again ordering the elements alphabetically as C, H, O, then the column vectors containing the element content of each species are given as CO : H 2 O : 2 CO 2 : CH 4 : 4 H 2 : 2 C :. 2 In attempting to balance (4) or (5) or (6), we would write x 4 2 y = or 2 or, z 2 respectively. But none of these equations has a solution since the matrix is singular: 4 2 =, where the A denotes the determinant of the entries of matrix A. Thus, when we choose the coefficients for which to solve (and thereby choose which coefficients to set manually), we must choose a set that forms a non-singular matrix. As an example of an acceptable choice, suppose that we wish to solve for u, v, and w, and manually set x, y, and z. Taking the same approach as before in which one coefficient is set to unity and the others to zero, we again have three reactions to balance: u CO + v H 2 O + w CO 2 + CH 4 + H 2 + C =, u 2 CO + v 2 H 2 O + w 2 CO 2 + CH 4 + H 2 + C =, u 3 CO + v 3 H 2 O + w 3 CO 2 + CH 4 + H 2 + C =. 3

4 To balance these three reactions, we must solve three linear systems: u 2 v = 4, 2 w u 2 2 v 2 = 2, and 2 w 2 u 3 2 v 3 =. 2 w 3 These can be condensed into a single matrix equation by writing u u 2 u 3 2 v v 2 v 3 = w w 2 w 3 Denoting the left-hand coefficient matrix as A, the right-hand matrix as B, and the matrix of unknowns as U, then we have A U = B, (7) for which the solution is U = A B. (8) Solving this system numerically (in SciPy using scipy.linalg.solve(a,b)) gives u u 2 u U = v v 2 v 3 = 2. w w 2 w 3 3 Thus, the stoichiometric coefficients for reaction n appear in column n of U. Writing the reactions in standard form, we have 2 Problem Statement 4 CO + 2 H 2 O = 3 CO 2 + CH 4, CO + H 2 O = CO 2 + H 2, 2 CO = CO 2 + C. This project is worth or 2 points, depending on the level of difficulty you choose. You will design software to automatically balance and print one or more chemical reactions given a list of chemical species. The chemical species will be represented in string form with element names and subscripts all joined together. Thus, H 2 O is represented as H2O, CH 4 is represented as CH4, and so on. 4

5 2. Required for both - and 2-point options Implement the following regardless of whether you are implementing the - or 2-point options:. Parsing: Write a function using the pyparsing module which will take a string representing a chemical species and return a list of tuples consisting of elements and corresponding subscripts. In the absence of a subscript, the subscript should be. Example: calling your function with an input of H2SO4 should return an output of [( H, 2), ( S,), ( O,4)]. Note: To assist in this portion of the project, a link to a working chemical species parser is given on the projects web page. 2. Determining unique elements: Write a function which, when given a list of species, will return an alphabetized list of the unique elements contained in the set of species. Make use of the parser from the previous step. Example: calling your function with an input of [ CO, H2O, CO2, CH4 ] should return an output of [ C, H, O ] 3. Extracting elemental column vectors: Write a function which, when given a chemical species and an element list such as that returned from the last function, will return the elements of the column vector of element contents. Example: calling your function with inputs of CH4 and [ C, H, O ] should return an output of [, 4, ]. 4. Constructing the left-hand coefficient matrix: Write a function which, when given a list of chemical species and a list of elements, will return a 2-D array representing the right-hand coefficient matrix A in either (3) or (7). Example: calling your function with inputs of [ CO, H2O, CO2 ] and [ C, H, O ] should return an output of array([[,, ], [, 2, ], [,, 2]]). Hint: column vectors c, c2, c3 and c4 can be assembled into a matrix with c [c, c2, c3, c4] in SciPy. 5. Representing a balanced chemical reaction as a string: Write a function which, when given a list of chemical species and an array of stoichiometric coefficients for those species will return a string representation of the balanced chemical reaction. Reactants and products should be written on the left and right sides of the equal sign, respectively. Chemical species on one side of the equal sign should be separated by + characters. Species with stoichiometric coefficients of zero should not be included, and species with coefficients of one should be included without any coefficient. Example: calling your function with inputs of [ CO, H2O, CO2, CH4, H2, C ] and [-4., -2., 3.,,.,.] should return (not print) a string output of 4 CO + 2 H2O = 3 CO2 + CH Point option only (single chemical reactions) Implement the following only if you are choosing the -point option for this project: 5

6 . Balancing a single chemical reaction: Write a function which, when given a list of chemical species, will either (a) raise an exception if the number of species minus the number of elements is not one, or (b) return a string representing a balanced chemical reaction involving those species. Examples: (a) Calling your function with an input of [ CO, H2O, CO2, CH4 ] should return an output of 4 CO + 2 H2O = 3 CO2 + CH4 or an equivalent balanced reaction. (Note: this reaction was obtained by setting the coefficient on CH 4 to unity, but other equivalent reactions could be obtained by setting other coefficients instead.) (b) Calling your function with an input of [ CO, H2O, CO2, CH4, H2, C ] should raise an exception with an error message along the lines of RuntimeError: the number of species minus the number of elements is not. Cannot write a single balanced reaction for this system. 2. Testing and Demonstration: Include several tests of your code with several different sets of chemical species. Print the results in an easily-understood format point option only (systems of chemical reactions) Implement the following only if you are choosing the 2-point option for this project:. Finding and balancing a set of independent chemical reactions: Write a function which, when given a list of chemical species, will either (a) raise an exception if it is not possible to write a set of balanced chemical reactions with the given species, or (b) return a list of strings representing a complete independent set of balanced chemical reaction involving those species. Examples: (a) Calling your function with an input of [ CO, H2O, CO2, CH4, H2, C ] should return an output of [ 4 CO + 2 H2O = 3 CO2 + CH4, CO + H2O = CO2 + H2, 2 CO = CO2 + C ] or an equivalent set of three independent balanced reactions. (Note: these reaction were obtained by choosing coefficients as explained above, but other equivalent reactions could be obtained by setting other admissible subsets of coefficients instead.) (b) Calling your function with an input of [ C, H2, S, O2 ] should raise an exception with an error message along the lines of RuntimeError: A set of balanced reactions cannot be written among the given species. 6

7 Hints: (a) This will require finding a subset of chemical species among the given set such that the determinant of the coefficient matrix A for that subset is non-zero. This in turn will require the ability to iterate through the subsets of the given set of species. A link to a function for generating the length-n combinations drawn from a list of given elements is provided on the projects web page. For example, the supplied function, called combinations functions as follows when called to generate length-3 combinations from a length-6 list of species: >>> from pprint import pprint >>> pprint(list(combinations([ CO, H2O, CO2, CH4, H2, C ], 3))) [[ CO, H2O, CO2 ], [ CO, H2O, CH4 ], [ CO, H2O, H2 ], [ CO, H2O, C ], [ CO, CO2, CH4 ], [ CO, CO2, H2 ], [ CO, CO2, C ], [ CO, CH4, H2 ], [ CO, CH4, C ], [ CO, H2, C ], [ H2O, CO2, CH4 ], [ H2O, CO2, H2 ], [ H2O, CO2, C ], [ H2O, CH4, H2 ], [ H2O, CH4, C ], [ H2O, H2, C ], [ CO2, CH4, H2 ], [ CO2, CH4, C ], [ CO2, H2, C ], [ CH4, H2, C ]] Thus, the return from the combinations function be iterated over to find a set of elements for which the determinant of the corresponding element matrix A is non-zero. (b) Scipy s linalg.solve(a,b) can be used to solve a linear system in which A and B are matrices or 2-D arrays. (c) Scipy s linalg.det(a) can be used to calculate the determinant of a matrix or 2-D array A. (d) Testing and Demonstration: Include several tests of your code with several different sets of chemical species. Print the results in an easily-understood format. In your tests, include the example worked out in Sec

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