This exercise is about balancing chemical reactions.

  1. Chemists denote a molecule of water as \(\text{H}_2\text{O}\text{,}\) which means it is composed of two atoms of hydrogen (H) and one atom of oxygen (O). The process by which hydrogen is burned is described by the chemical reaction

    \begin{equation*} x\thinspace \text{H}_2 + y\thinspace\text{O}_2 \to z\thinspace \text{H}_2\text{O} \end{equation*}

    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. How many hydrogen atoms are there before the reaction? How many hydrogen atoms are there after the reaction? Find a linear equation in \(x\text{,}\) \(y\text{,}\) and \(z\) by equating these quantities.

    2. Find a second linear equation in \(x\text{,}\) \(y\text{,}\) 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, \(x\text{,}\) \(y\text{,}\) and \(z\) should be positive integers. Find the solution where \(x\text{,}\) \(y\text{,}\) 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:

    \begin{equation*} 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. \end{equation*}

    As in the previous exercise, find the appropriate values for \(x_1, x_2, \ldots, x_6\) to balance the chemical reaction.