# hno3 ra cuno32

To balance the equation Cu + HNO3 = Cu(NO3)2 + NO2 + 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.

### 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 Cu + b HNO3 = c Cu(NO3)2 + d NO2 + f H2O

### Step 2: Create a System of Equations

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

```Cu:	1a	+	0b	=	1c	+	0d	+	0f
H:	0a	+	1b	=	0c	+	0d	+	2f
N:	0a	+	1b	=	2c	+	1d	+	0f
O:	0a	+	3b	=	6c	+	2d	+	1f

```

### Step 3: Solve For All Variables

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

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

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

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

• a = 1 (Cu)
• b = 4 (HNO3)
• c = 1 (Cu(NO3)2)
• d = 2 (NO2)
• f = 2 (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.

Cu + 4 HNO3 = Cu(NO3)2 + 2 NO2 + 2 H2O

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Reactants Products Cu 1 1 ✔️ 4 4 ✔️ 4 4 ✔️ 12 12 ✔️

Since there is an equal number of each element in the reactants and products of Cu + 4HNO3 = Cu(NO3)2 + 2NO2 + 2H2O, 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 Cu + HNO3 = Cu(NO3)2 + NO2 + 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 to tát balance one element at a time until both sides are equal and the chemical equation is balanced.

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

Reactants (Left Hand Side)Products (Right Hand Side)
ReactantsProducts
CuHNO3TotalCu(NO3)2NO2H2OTotal
Cu1111✔️
H1122
N11213
O336219

### Step 2: Multiply coefficients for compounds to tát balance out each element

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

1. H is not balanced. Add 1 molecule of HNO3 to tát the reactant (left-hand) side to tát balance Hydrogen:
Cu + 2HNO3 = Cu(NO3)2 + NO2 + H2O

ReactantsProducts
Cu11✔️
H22✔️
N23
O69
2. N is not balanced. Add 1 molecule of HNO3 to tát the reactant (left-hand) side to tát balance Nitrogen:
Cu + 3HNO3 = Cu(NO3)2 + NO2 + H2O

ReactantsProducts
Cu11✔️
H32
N33✔️
O99✔️
3. H is not balanced. Add 1 molecule of H2O to tát the product (right-hand) side to tát try to tát balance Hydrogen:
Cu + 3HNO3 = Cu(NO3)2 + NO2 + 2H2O

ReactantsProducts
Cu11✔️
H34
N33✔️
O910
4. H is not balanced. Add 1 molecule of HNO3 to tát the reactant (left-hand) side to tát balance Hydrogen:
Cu + 4HNO3 = Cu(NO3)2 + NO2 + 2H2O

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ReactantsProducts
Cu11✔️
H44✔️
N43
O1210
5. N is not balanced. Add 1 molecule of NO2 to tát the product (right-hand) side to tát balance Nitrogen:
Cu + 4HNO3 = Cu(NO3)2 + 2NO2 + 2H2O

ReactantsProducts
Cu11✔️
H44✔️
N44✔️
O1212✔️

### 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.

Cu + 4HNO3 = Cu(NO3)2 + 2NO2 + 2H2O