Exercise 5

This exercise is about balancing chemical reactions.

1. Chemists denote a molecule of water as $\text{H}_2\text{O}$, which means it is composed of two atoms of hydrogen (H) and one atom of oxygen (O). The process by which hydrogen burns is described by the chemical reaction $$x\thinspace \text{H}_2 + y\thinspace\text{O}_2 \to z\thinspace \text{H}_2\text{O}$$ This means that $x$ molecules of hydrogen $\text{H}_2$ combine with $y$ molecules of oxygen $\text{O}_2$ to produce $z$ water molecules. The number of hydrogen atoms is the same before and after the reaction; the same is true of the oxygen atoms.

   1. In terms of $x$, $y$, and $z$, how many hydrogen atoms are there before the reaction? How many hydrogen atoms are there after the reaction? Find a linear equation in $x$, $y$, and $z$ by equating these quantities.

   2. Find a second linear equation in $x$, $y$, and $z$ by equating the number of oxygen atoms before and after the reaction.

   3. Find the solutions of this linear system. Why are there infinitely many solutions?

   4. In this chemical setting, $x$, $y$, and $z$ should be positive integers. Find the solution where $x$, $y$, and $z$ are the smallest possible positive integers.

2. Now consider the reaction where potassium permanganate and manganese sulfate combine with water to produce manganese dioxide, potassium sulfate, and sulfuric acid: $$x_1\thinspace \text{K}\text{Mn}\text{O}_4 + x_2\thinspace \text{Mn}\text{S}\text{O}_4 + x_3\thinspace \text{H}_2\text{O} \to x_4\thinspace \text{Mn}\text{O}_2 + x_5\thinspace \text{K}_2\text{S}\text{O}_4 + x_6\thinspace \text{H}_2\text{S}\text{O}_4.$$ As in the previous exercise, find the appropriate values for $x_1, x_2, \ldots, x_6$ to balance the chemical reaction.