# fe(no3)2 ra fe2o3

To balance the equation Fe(NO3)2 = Fe2O3 + NO2 + O2 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 lớn represent the unknown coefficients.

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a Fe(NO3)2 = b Fe2O3 + c NO2 + d O2

### Step 2: Create a System of Equations

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

```Fe:	1a	=	2b	+	0c	+	0d
N:	2a	=	0b	+	1c	+	0d
O:	6a	=	3b	+	2c	+	2d

```

### Step 3: Solve For All Variables

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

• 1a - 2b = 0
• 2a - 1c = 0
• 6a - 3b - 2c - 2d = 0

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

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```[ 1	-2	 0	 0	0]
[ 2	 0	-1	 0	0]
[ 6	-3	-2	-2	0]
```

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

Simplify the result lớn get the lowest, whole integer values.

• a = 4 (Fe(NO3)2)
• b = 2 (Fe2O3)
• c = 8 (NO2)
• d = 1 (O2)

### 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.

4 Fe(NO3)2 = 2 Fe2O3 + 8 NO2 + O2

Reactants Products Fe 4 4 ✔️ 8 8 ✔️ 24 24 ✔️

Since there is an equal number of each element in the reactants and products of 4Fe(NO3)2 = 2Fe2O3 + 8NO2 + O2, the equation is balanced.