Application to Chemical Reactions

Chapter 1 • Section 1-6

"I told a chemistry joke, but there was no reaction... 🧪"

⚗️ Balancing Chemical Equations

Chemistry is all about balance! We need to ensure the number of atoms of each element is the same on both sides of the reaction.

Problem:

Balance the following reaction involving Tin ($Sn$), Hydrogen ($H$), and Oxygen ($O$):

$$ x \text{SnO}_2 + y \text{H}_2 \to z \text{Sn} + w \text{H}_2\text{O} $$

In a chemical equation, what do the variables $x, y, z, w$ represent?

We create an equation for each element to ensure conservation of mass.

Left: $1 \cdot x$

Right: $1 \cdot z$

$x = z \implies x - z = 0$

Left: $2 \cdot x$

Right: $1 \cdot w$

$2x = w \implies 2x - w = 0$

Left: $2 \cdot y$

Right: $2 \cdot w$

$2y = 2w \implies 2y - 2w = 0$

The system in matrix form ($x, y, z, w$ columns):

$$ \left[\begin{array}{rrrr|r} 1 & 0 & -1 & 0 & 0 \\ 2 & 0 & 0 & -1 & 0 \\ 0 & 2 & 0 & -2 & 0 \end{array}\right] $$

$$ \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & -1/2 & 0 \end{array}\right] $$

We have a free variable $w$. Let $w = t$.

$$ \begin{cases} x = \frac{1}{2}t \\ y = t \\ z = \frac{1}{2}t \\ w = t \end{cases} $$

Chemical equations usually use integers. Let's pick a value for $t$ to clear the fractions.

If we choose $t = 4$ (or $w=4$):

$$ x = 2, \quad y = 4, \quad z = 2, \quad w = 4 $$

Balanced Equation: $$ 2\text{SnO}_2 + 4\text{H}_2 \to 2\text{Sn} + 4\text{H}_2\text{O} $$

(Note: We could also divide by 2 to get $1, 2, 1, 2$, which corresponds to $t=2$. Both are valid, but simplest integers are preferred!)

Why did we choose $w=4$ (or $t=4$) instead of $w=1$?