# How do you balance Ca + P -> Ca_2P_3?

Dec 25, 2015

$\textcolor{red}{2} C a + \textcolor{red}{3} P \rightarrow \textcolor{red}{1} C {a}_{2} {P}_{3}$

#### Explanation:

Assume:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{x} C a + \textcolor{red}{y} P \rightarrow \textcolor{red}{z} C {a}_{2} {P}_{3}$

Based on $C a$
$\textcolor{w h i t e}{\text{XXX}} x - 2 z = 0$
Based on $P$
$\textcolor{w h i t e}{\text{XXX}} y - 3 z = 0$

In order to find the ratio of $\left(x : y : z\right)$ we will arbitrarily set one of our variables to $1$; later we may need to adjust our ratios to get integer results.

In order to see how this works in the general case I have chosen to use $x = 1$

From  with $x = 1$
$\textcolor{w h i t e}{\text{XXX}} z = \frac{1}{2}$
From  with $z = \frac{1}{2}$
$\textcolor{w h i t e}{\text{XXX}} y = \frac{3}{2}$

So the ratio of components is
$\textcolor{w h i t e}{\text{XXX}} \left(1 , \frac{3}{2} , \frac{1}{2}\right)$

Since we want integer factors we will multiply all terms by $2$ to clear all fractions, giving:
$\textcolor{w h i t e}{\text{XXX}} \left(x : y : z\right) = \left(2 : 3 : 1\right)$