To balance the equation Fe2O3 + HNO3 = NO + Fe(NO3)3 + H2O 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.

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

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

a Fe_{2}O_{3} + b HNO_{3} = c NO + d Fe(NO_{3})_{3} + f H_{2}O

### Step 2: Create a System of Equations

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

**Fe**: 2a + 0b = 0c + 1d + 0f
**O**: 3a + 3b = 1c + 9d + 1f
**H**: 0a + 1b = 0c + 0d + 2f
**N**: 0a + 1b = 1c + 3d + 0f

### Step 3: Solve For All Variables

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

- 2a - 1d = 0
- 3a + 3b - 1c - 9d - 1f = 0
- 1b - 2f = 0
- 1b - 1c - 3d = 0

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

[ 2 0 0 -1 0 0] [ 3 3 -1 -9 -1 0] [ 0 1 0 0 -2 0] [ 0 1 -1 -3 0 0]

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

Simplify the result đồ sộ get the lowest, whole integer values.

- a = 1 (Fe2O3)
- b = 6 (HNO3)
- c = 0 (NO)
- d = 2 (Fe(NO3)3)
- f = 3 (H2O)

### 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.

Fe_{2}O_{3} + 6 HNO_{3} = 0 NO + 2 Fe(NO_{3})_{3} + 3 H_{2}O

Fe | 2 | 2 | ✔️ |
---|---|---|---|

O | 21 | 21 | ✔️ |

H | 6 | 6 | ✔️ |

N | 6 | 6 | ✔️ |

Since there is an equal number of each element in the reactants and products of Fe2O3 + 6HNO3 = 0NO + 2Fe(NO3)3 + 3H2O, the equation is balanced.

The law of conservation of mass states that matter cannot be created or destroyed, which means there must be the same number atoms at the over of a chemical reaction as at the beginning. To be balanced, every element in Fe2O3 + HNO3 = NO + Fe(NO3)3 + H2O must have the same number of atoms on each side of the equation. When using the inspection method (also known as the trial-and-error method), this principle is used đồ sộ balance one element at a time until both sides are equal and the chemical equation is balanced.

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### Step 1: Count the number of each element on the left and right hand sides

Reactants (Left Hand Side) | Products (Right Hand Side) | |||||||
---|---|---|---|---|---|---|---|---|

Reactants | Products | |||||||

Fe_{2}O_{3} | HNO_{3} | Total | NO | Fe(NO_{3})_{3} | H_{2}O | Total | ||

Fe | 2 | 2 | 1 | 1 | ❌ | |||

O | 3 | 3 | 6 | 1 | 9 | 1 | 11 | ❌ |

H | 1 | 1 | 2 | 2 | ❌ | |||

N | 1 | 1 | 1 | 3 | 4 | ❌ |

### Step 2: Multiply coefficients for compounds đồ sộ balance out each element

For each element that is not equal, try đồ sộ balance it by adding more of it đồ sộ the side with less. Sometimes there may be multiple compounds with that element on one side, so sánh you'll need đồ sộ use your best judgement and be prepared đồ sộ go back and try the other options.

**H**is not balanced. Add**1**molecule of**HNO**đồ sộ the reactant (left-hand) side đồ sộ balance Hydrogen:_{3}

Fe_{2}O_{3}+ 2HNO_{3}= NO + Fe(NO_{3})_{3}+ H_{2}OReactants Products Fe 2 1 ❌ O 9 11 ❌ H 2 2 ✔️ N 2 4 ❌ **N**is not balanced. Add**2**molecules of**HNO**đồ sộ the reactant (left-hand) side đồ sộ balance Nitrogen:_{3}

Fe_{2}O_{3}+ 4HNO_{3}= NO + Fe(NO_{3})_{3}+ H_{2}OReactants Products Fe 2 1 ❌ O 15 11 ❌ H 4 2 ❌ N 4 4 ✔️ **O**is not balanced. Add**-1**molecules of**NO**đồ sộ the product (right-hand) side đồ sộ balance Oxygen:

Fe_{2}O_{3}+ 4HNO_{3}= 0NO + Fe(NO_{3})_{3}+ H_{2}OReactants Products Fe 2 1 ❌ O 15 10 ❌ H 4 2 ❌ N 4 3 ❌ **Fe**is not balanced. Add**1**molecule of**Fe(NO**đồ sộ the product (right-hand) side đồ sộ balance Iron:_{3})_{3}

Fe_{2}O_{3}+ 4HNO_{3}= 0NO + 2Fe(NO_{3})_{3}+ H_{2}OReactants Products Fe 2 2 ✔️ O 15 19 ❌ H 4 2 ❌ N 4 6 ❌ **H**is not balanced. Add**1**molecule of**H**đồ sộ the product (right-hand) side đồ sộ balance Hydrogen:_{2}O

Fe_{2}O_{3}+ 4HNO_{3}= 0NO + 2Fe(NO_{3})_{3}+ 2H_{2}OReactants Products Fe 2 2 ✔️ O 15 20 ❌ H 4 4 ✔️ N 4 6 ❌ **N**is not balanced. Add**2**molecules of**HNO**đồ sộ the reactant (left-hand) side đồ sộ balance Nitrogen:_{3}

Fe_{2}O_{3}+ 6HNO_{3}= 0NO + 2Fe(NO_{3})_{3}+ 2H_{2}OReactants Products Fe 2 2 ✔️ O 21 20 ❌ H 6 4 ❌ N 6 6 ✔️ **O**is not balanced. Add**1**molecule of**H**đồ sộ the product (right-hand) side đồ sộ balance Oxygen:_{2}O

Fe_{2}O_{3}+ 6HNO_{3}= 0NO + 2Fe(NO_{3})_{3}+ 3H_{2}OReactants Products Fe 2 2 ✔️ O 21 21 ✔️ H 6 6 ✔️ N 6 6 ✔️

### Step 3: Verify that the equation is balanced

Since there are an equal number of atoms of each element on both sides, the equation is balanced.

Fe_{2}O_{3} + 6HNO_{3} = 0NO + 2Fe(NO_{3})_{3} + 3H_{2}O

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