WOLFRAM|DEMONSTRATIONS PROJECT

Chemical Reactions Represented via a 3D Simplex

​
reaction
hydroxide formation
combustion reaction
neutralization reaction
hydroxide formation
hydroxide
M
(OH)
M
(OH)
2
M
(OH)
3
M
(OH)
4
oxide
M
O
combustion reaction
C
1
2
3
4
5
H
1
2
3
4
5
neutralization reaction
hydroxide
M
(OH)
M
(OH)
2
M
(OH)
3
M
(OH)
4
acid
H
A
H
2
A
H
3
A
H
4
A
salt
M
A
M
A
2
M
A
3
M
A
4
hydroxide formation
Reaction
x
M
O
+ y
H
2
O → z M
(OH)
cannot occur
This Demonstration shows the connection between the equation for a balanced chemical reaction and its geometrical representation by a simplex: the figure formed from
n+1
vertices in a space of
n
dimensions, where
n
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
n
equations, but its coefficient matrix should be of rank
n-1
. In fact, the coefficients are determined only up to a proportionality factor. The traditional convention is to choose the lowest 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
x
AB
2
→y
AB
3
+z
A
2
B
3
has
n=2
; for
A
,
x=y+2z
, and for
B
,
2x=3y+3z
(2 equations in 3 variables). The coefficient matrix is

1
-1
-2
2
-3
-3

.
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
n
dimensions, the simplex has as vertices the
n+1
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 three elements in three chemical species. As a first example, consider the formation reaction of hydroxides (
M
x
O
y
+
H
2
O
M(OH)
z
). You can select the specific compounds with the "hydroxide" and "oxide" controls.
Second, a reaction with three elements in four chemical species is considered, namely, the combustion reaction of hydrocarbons (
C
x
H
y
+
O
2

CO
2
+
H
2
O
). You can choose the hydrocarbon molecule using the "C" and "H" controls.
Finally, we consider four elements in four chemical species, for example, a neutralization reaction (
M(OH)
x
+
H
y
A
M
y
A
x
+
H
2
O
). You can select the reaction components with the "hydroxide", "acid" and "salt" controls.
You can select one of the above types with the "reaction" control. If the chosen reaction is possible, a balanced reaction equation will be displayed.
The simplex theory here is applied to reactions with three or four chemical species, thus implying 3D graphics. The first graphic analyzes a reaction where three elements can be found arranged in three chemical species, which gives a plane in 3D space.
In the case of three elements arranged in four chemical species, for example, in standard combustion reactions, the simplex is a pyramid with a triangular base. As shown, once the vertices corresponding to
O
2
,
CO
2
and
H
2
O
have been established, notwithstanding the formula of hydrocarbon
C
x
H
y
, the origin will always be located within the simplex, and therefore, a reaction is always possible. For this reason, all the hydrocarbons contained in petroleum can always burn.
Finally, in the case of four elements arranged in four chemical species, as in the salification reactions, a problem would arise for representing a figure in a 4D space. However, the reactions, wherever possible, have systems with determinant zero, so it is possible to choose a new 3D system [1].
Snapshot 1: the hydroxide formation reaction is possible
Snapshot 2: the combustion reaction is always possible
Snapshot 3: the salt formation reaction is possible
Snapshot 4: the salt formation reaction is not possible