Chemical Stoichiometry Using MATLAB

In beginning chemistry courses, students are taught a variety of techniques for balancing chemical equations. The determination of the stoichiometric coefficients in a chemical equation is mathematically equivalent to solving a system of linear algebraic equations, a problem for which MATLAB is ideally suited. Using MATLAB, it is possible to balance equations describing all kinds of chemical transformations, including acid-base reactions, redox reactions, electrochemical half-reactions, combustion reactions, and synthetic reactions. MATLAB is especially convenient for balancing chemical equations that are difficult to treat by traditional methods. Introduction A number of different techniques have traditionally been taught in beginning chemistry courses for balancing chemical reactions. Three methods are commonly found in introductory textbooks [1–3]: • Inspection. In its simplest form, this method may be little more than intelligent guessing. For this reason, it is sometimes called the trial-and-error method. Students may be taught various rules of thumb that make the method more efficient. (e.g., “start with an element that appears in just one species on each side of the equation.”) • Half-Equation Method. This approach, also called the ion-electron method, is used for balancing oxidation-reduction (redox) reactions. The reaction is divided into two half reactions, one for oxidation, the other for reduction. These half reactions are balanced separately, then combined to form a balanced redox reaction. • Oxidation Number Method. This is another approach for balancing redox reactions. The first step is to assign oxidation numbers to the elements involved in the reaction. The oxidation numbers are balanced, after which the anions, cations, and remaining elements are balanced by inspection. There is a fourth method, which is arguably more general and powerful than the other three: • Algebraic Method. The unbalanced chemical equation is used to define a system of linear equations, which can then be solved to yield the stoichiometric coefficients [4, 5]. The algebraic method has traditionally been less popular than the alternatives, probably because of the inconveniences related to solving systems of equations. However, modern mathematics software handle such systems with ease, making the algebraic method much more attractive. A Simple Example The algebraic method is perhaps best grasped by way of an example. The combustion of methane in oxygen can be represented by the chemical equation x1CH4 + x2O2 → x3CO2 + x4 H2O Our task is to determine the unknown coefficients x1, x2, x3, and x4. There are three elements involved in this reaction: carbon (C), hydrogen (H), and oxygen (O). A balance equation can be written for each of these elements: Carbon (C): 1⋅x1 + 0⋅x2 = 1⋅x3 + 0⋅x4 Hydrogen (H): 4⋅x1 + 0⋅x2 = 0⋅x3 + 2⋅x4 Oxygen (O): 0⋅x1 + 2⋅x2 = 2⋅x3 + 1⋅x4 We write these as homogeneous equations, each having zero on its right hand side: x1 – x3 = 0 4x1 – 2x4 = 0 2x2 – 2x3 – x4 = 0 At this point, we have three equations in four unknowns. To complete the system, we define an auxiliary equation by arbitrarily choosing a value for one of the coefficients: