Chemical Reactions Represented on a 2D Simplex

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This Demonstration shows the connection between the equation for a balanced chemical reaction and its geometrical representation by a simplex: the figure formed from vertices in a space of dimensions, where is the number of chemical elements involved in the reaction. In two-dimensional space the simplex is a triangle, while in three-dimensional space it is a tetrahedron.


Reaction balancing requires a characteristic system of equations, but its coefficient matrix should be of rank . In fact, the coefficients are not fixed unambiguously but are proportional to each other. In this case, we choose the traditional convention of using the lowest possible positive integer values. Moreover, in order for the results to be physically valid, the coefficients in the chemical equation must be positive.

For example, the reaction

has ; for , , and for , (2 equations in 3 variables). The coefficient matrix is


By convention, the coefficients for the reactants are taken to be positive and for the products, negative; the matrix rank is equal to 2. The simplex is constructed from the columns of the coefficient matrix. In dimensions, the simplex has as vertices the columns of subscripts in the chemical reaction formula.

For the reaction to occur, the interior of the simplex must include the origin [1].

The reactions considered here involve two species (such as and in , , …), which generate a segment (a 1-simplex), or three species (such as , , in , ), which generate a triangle (2-simplex). As a result, in all possible cases, a balanced reaction is obtained.

Use the popup menus to select new chemical reactions or to view the given examples for two or three reactants and products.


Contributed by: D. Meliga and S. Z. Lavagnino (April 2017)
With additional contributions by: A. Chiavassa and G. Follo
Open content licensed under CC BY-NC-SA



As a graphical representation, we choose the subscripts of reactants as positive and the subscripts of products as negative; for this reason, points all fall in the first and third quadrants. Click on the selected points. Each axis represents an element, which can be selected from the popup menus "first element" and "second element." A large selection of elements is given; boron and other nonmetals are marked in red.

Snapshot 1: with "number of compounds" set to 2, the oxygen dissociation reaction () is possible because the 1-simplex passes through the origin

Snapshot 2: with "number of compounds" set to 3, the hydriodic acid reaction () is possible because the origin is inside the 2-simplex represented by the triangle

Snapshot 3: with "number of compounds" set to 3, a reaction does not occur because the origin is not inside the 2-simplex


[1] G. Follo and S. Z. Lavagnino "Bilanciamento delle Reazioni Chimiche e Sistemi Lineari," La Chimica nela Scuola, Anno XXXIII, n. 1, 2011 pp. 18–26.

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