Error in formula for the solution to a chemical problem The problem in my textbook goes like this:


  Burning $0.68\ \mathrm{g}$ of an unknown substance, we obtained $1.28\ \mathrm{g}$ of sulfur oxide (IV) and $0.36\ \mathrm{g}$ of water. Find the chemical formula of the burned substance, providing that it was a complete combustion.


It's quite easy to see that $0.2\ \mathrm{mol}$ of both $\ce{SO2}$ and $\ce{H2O}$ have been produced, there are $0.04\ \mathrm{mol}$ of hydrogen in the water and $0.02\ \mathrm{mol}$ of sulfur in the oxide. Calculating the masses and summing them up, we get $0.68\ \mathrm{g}$, meaning the original substance contained only sulfur and hydrogen, and it's $\ce{H2S}$.

However, the solution volume offers this equation as part of the solution process:

$$\ce{S_{$x$}H_{$y$}O_{$z$} + \frac{x + y}{4} O2 -&gt; $x$ SO2 + \frac{$y$}{$z$} H2O}$$

The volume is full of typos (apparently a plus sign missing here), but still there must be some explanation for the coefficients which I don't get. Why do the coefficients for $\ce{O2}$ and $\ce{H2O}$ equal $(x+y)/4$ and $y/z$? If there's zero oxygen in the first term, won't we get division by zero?

How do we come up with coefficients of this sort? Maybe these particular coefficients are erroneous? (the solutions volume is quite low in quality)
The equation is a little dubious, as I mentioned in the comments.


  How do we come up with coefficients of this sort?


Let's go from scratch. We don't know the coefficients at this point so we'll just call them $a$, $b$, and $c$. Note that the coefficient of $\ce{S$_x$H$_y$O$_z$}$ is taken to be 1. It doesn't have to be 1; however, for the sake of simplicity you might as well take it to be 1. If you take it to be 2, then you just have to double all the other coefficients.

$$\ce{S$_x$H$_y$O$_z$ + $a\,$O2 -&gt; $b\,$SO2 + $c\,$H2O}$$

In the reactants, we have $x$ sulfur, $y$ hydrogen, and $z + 2a$ oxygen. In the products, we have $b$ sulfur, $2c$ hydrogen, and $2b + c$ oxygen.

Our aim is to express $a$, $b$, and $c$ in terms of $x$, $y$, and $z$. Equating all three, we get

$$\begin{align}
x &amp;= b \\
y &amp;= 2c \\
z + 2a &amp;= 2b + c
\end{align}$$

which immediately tells you that $b$ and $c$ are equal to $x$ and $y/2$ respectively. Substituting these into the last equation,

$$\begin{align}
z + 2a &amp;= 2x + \frac{y}{2} \\
a &amp;= \frac{4x + y - 2z}{4}
\end{align}$$

So the full equation is:

$$\ce{S$_x$H$_y$O$_z$ + $\frac{4x + y - 2z}{4}$O2 -&gt; $x\,$SO2 + $\frac{y}{2}\,$H2O}$$