# no2 h2o

To balance the equation NO2 + H2O = HNO3 + NO using the algebraic method step-by-step, you must have experience solving systems of linear equations. The most common methods are substitution/elimination and linear algebra, but any similar method will work.

### Step 1: Label Each Compound With a Variable

Label each compound (reactant or product) in the equation with a variable to tát represent the unknown coefficients.

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a NO2 + b H2O = c HNO3 + d NO

### Step 2: Create a System of Equations

Create an equation for each element (N, O, H) where each term represents the number of atoms of the element in each reactant or product.

```N:	1a	+	0b	=	1c	+	1d
O:	2a	+	1b	=	3c	+	1d
H:	0a	+	2b	=	1c	+	0d

```

### Step 3: Solve For All Variables

Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.

• 1a - 1c - 1d = 0
• 2a + 1b - 3c - 1d = 0
• 2b - 1c = 0

Use your graphing calculator's rref() function (or an online rref calculator) to tát convert the following matrix into reduced row-echelon-form:

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```[ 1	 0	-1	-1	0]
[ 2	 1	-3	-1	0]
[ 0	 2	-1	 0	0]
```

The resulting matrix can be used to tát determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.

Simplify the result to tát get the lowest, whole integer values.

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• a = 3 (NO2)
• b = 1 (H2O)
• c = 2 (HNO3)
• d = 1 (NO)

### Step 4: Substitute Coefficients and Verify Result

Count the number of atoms of each element on each side of the equation and verify that all elements and electrons (if there are charges/ions) are balanced.

3 NO2 + H2O = 2 HNO3 + NO

Reactants Products N 3 3 ✔️ 7 7 ✔️ 2 2 ✔️

Since there is an equal number of each element in the reactants and products of 3NO2 + H2O = 2HNO3 + NO, the equation is balanced.