# How do you balance NO_2 + H_2O → HNO_3 + NO?

Nov 25, 2017

$3 N {O}_{2} + {H}_{2} O \to 2 H N {O}_{3} + N O$

#### Explanation:

Given:

$N {O}_{2} + {H}_{2} O \to H N {O}_{3} + N O$

We are looking for something like:

$a N {O}_{2} + b {H}_{2} O \to c H N {O}_{3} + d N O$

for some integers $a , b , c , d$.

Equating the amounts of $N$, $O$ and $H$, we find:

$\left\{\begin{matrix}a = c + d \\ 2 a + b = 3 c + d \\ 2 b = c\end{matrix}\right.$

Subtracting the first equation from the second to eliminate $d$, we find:

$a + b = 2 c$

Then subtracting twice the third equation, we find:

$a - 3 b = 0$

So let's put $b = 1$, $a = 3$, $c = 2$ and $d = 1$ to find:

$3 N {O}_{2} + {H}_{2} O \to 2 H N {O}_{3} + N O$