# fe2o3+hno3

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 Fe2O3 + b HNO3 = c NO + d Fe(NO3)3 + f H2O

### 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.

Fe2O3 + 6 HNO3 = 0 NO + 2 Fe(NO3)3 + 3 H2O

Reactants Products Fe 2 2 ✔️ 21 21 ✔️ 6 6 ✔️ 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)
ReactantsProducts
Fe2O3HNO3TotalNOFe(NO3)3H2OTotal
Fe2211
O33619111
H1122
N11134

### 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.

1. H is not balanced. Add 1 molecule of HNO3 đồ sộ the reactant (left-hand) side đồ sộ balance Hydrogen:
Fe2O3 + 2HNO3 = NO + Fe(NO3)3 + H2O

ReactantsProducts
Fe21
O911
H22✔️
N24
2. N is not balanced. Add 2 molecules of HNO3 đồ sộ the reactant (left-hand) side đồ sộ balance Nitrogen:
Fe2O3 + 4HNO3 = NO + Fe(NO3)3 + H2O

ReactantsProducts
Fe21
O1511
H42
N44✔️
3. O is not balanced. Add -1 molecules of NO đồ sộ the product (right-hand) side đồ sộ balance Oxygen:
Fe2O3 + 4HNO3 = 0NO + Fe(NO3)3 + H2O

ReactantsProducts
Fe21
O1510
H42
N43
4. Fe is not balanced. Add 1 molecule of Fe(NO3)3 đồ sộ the product (right-hand) side đồ sộ balance Iron:
Fe2O3 + 4HNO3 = 0NO + 2Fe(NO3)3 + H2O

ReactantsProducts
Fe22✔️
O1519
H42
N46
5. H is not balanced. Add 1 molecule of H2O đồ sộ the product (right-hand) side đồ sộ balance Hydrogen:
Fe2O3 + 4HNO3 = 0NO + 2Fe(NO3)3 + 2H2O

ReactantsProducts
Fe22✔️
O1520
H44✔️
N46
6. N is not balanced. Add 2 molecules of HNO3 đồ sộ the reactant (left-hand) side đồ sộ balance Nitrogen:
Fe2O3 + 6HNO3 = 0NO + 2Fe(NO3)3 + 2H2O

ReactantsProducts
Fe22✔️
O2120
H64
N66✔️
7. O is not balanced. Add 1 molecule of H2O đồ sộ the product (right-hand) side đồ sộ balance Oxygen:
Fe2O3 + 6HNO3 = 0NO + 2Fe(NO3)3 + 3H2O

ReactantsProducts
Fe22✔️
O2121✔️
H66✔️
N66✔️

### 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.

Fe2O3 + 6HNO3 = 0NO + 2Fe(NO3)3 + 3H2O

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