# cus2 + hno3

To balance the equation CuS2 + HNO3 = Cu(NO3)2 + H2SO4 + NO + 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 tướng represent the unknown coefficients.

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a CuS2 + b HNO3 = c Cu(NO3)2 + d H2SO4 + f NO + g H2O

### Step 2: Create a System of Equations

Create an equation for each element (Cu, S, 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	+	0g
S:	2a	+	0b	=	0c	+	1d	+	0f	+	0g
H:	0a	+	1b	=	0c	+	2d	+	0f	+	2g
N:	0a	+	1b	=	2c	+	0d	+	1f	+	0g
O:	0a	+	3b	=	6c	+	4d	+	1f	+	1g

```

### Step 3: Solve For All Variables

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

• 1a - 1c = 0
• 2a - 1d = 0
• 1b - 2d - 2g = 0
• 1b - 2c - 1f = 0
• 3b - 6c - 4d - 1f - 1g = 0

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

```[ 1	 0	-1	 0	 0	 0	0]
[ 2	 0	 0	-1	 0	 0	0]
[ 0	 1	 0	-2	 0	-2	0]
[ 0	 1	-2	 0	-1	 0	0]
[ 0	 3	-6	-4	-1	-1	0]
```

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

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

• a = 3 (CuS2)
• b = trăng tròn (HNO3)
• c = 3 (Cu(NO3)2)
• d = 6 (H2SO4)
• f = 14 (NO)
• g = 4 (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.

3 CuS2 + 20 HNO3 = 3 Cu(NO3)2 + 6 H2SO4 + 14 NO + 4 H2O

Reactants Products Cu 3 3 ✔️ 6 6 ✔️ 20 20 ✔️ 20 20 ✔️ 60 60 ✔️

Since there is an equal number of each element in the reactants and products of 3CuS2 + 20HNO3 = 3Cu(NO3)2 + 6H2SO4 + 14NO + 4H2O, 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 CuS2 + HNO3 = Cu(NO3)2 + H2SO4 + NO + 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 tướng 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
CuS2HNO3TotalCu(NO3)2H2SO4NOH2OTotal
Cu1111✔️
S2211
H11224
N11213
O33641112

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

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

1. S is not balanced. Add 1 molecule of H2SO4 to tướng the product (right-hand) side to tướng balance Sulfur:
CuS2 + HNO3 = Cu(NO3)2 + 2H2SO4 + NO + H2O

ReactantsProducts
Cu11✔️
S22✔️
H16
N13
O316
2. H is not balanced. Add 5 molecules of HNO3 to tướng the reactant (left-hand) side to tướng balance Hydrogen:
CuS2 + 6HNO3 = Cu(NO3)2 + 2H2SO4 + NO + H2O

ReactantsProducts
Cu11✔️
S22✔️
H66✔️
N63
O1816
3. N is not balanced. Add 3 molecules of NO to tướng the product (right-hand) side to tướng balance Nitrogen:
CuS2 + 6HNO3 = Cu(NO3)2 + 2H2SO4 + 4NO + H2O

ReactantsProducts
Cu11✔️
S22✔️
H66✔️
N66✔️
O1819
4. O is not balanced. Add 14 molecules of HNO3 to tướng the reactant (left-hand) side to tướng try to tướng balance Oxygen:
CuS2 + 20HNO3 = Cu(NO3)2 + 2H2SO4 + 4NO + H2O

ReactantsProducts
Cu11✔️
S22✔️
H206
N206
O6019
5. H is not balanced. Add 4 molecules of H2SO4 to tướng the product (right-hand) side to tướng balance Hydrogen:
CuS2 + 20HNO3 = Cu(NO3)2 + 6H2SO4 + 4NO + H2O

ReactantsProducts
Cu11✔️
S26
H2014
N206
O6035
6. S is not balanced. Add 2 molecules of CuS2 to tướng the reactant (left-hand) side to tướng balance Sulfur:
3CuS2 + 20HNO3 = Cu(NO3)2 + 6H2SO4 + 4NO + H2O

ReactantsProducts
Cu31
S66✔️
H2014
N206
O6035
7. Cu is not balanced. Add 2 molecules of Cu(NO3)2 to tướng the product (right-hand) side to tướng balance Copper:
3CuS2 + 20HNO3 = 3Cu(NO3)2 + 6H2SO4 + 4NO + H2O

ReactantsProducts
Cu33✔️
S66✔️
H2014
N2010
O6047
8. H is not balanced. Add 3 molecules of H2O to tướng the product (right-hand) side to tướng balance Hydrogen:
3CuS2 + 20HNO3 = 3Cu(NO3)2 + 6H2SO4 + 4NO + 4H2O

ReactantsProducts
Cu33✔️
S66✔️
H2020✔️
N2010
O6050

We could not determine all the steps needed to tướng balance the equation using trial and error. This may mean it requires too many steps and the chemical equation is too complex to tướng use this method, or that the equation is invalid (e.g. has 0 or non-integer coefficients). The algebraic method is likely better suited for balancing this equation.

If you would lượt thích to tướng attempt to tướng guess the next steps, the final element counts in the balanced equation should be:

3CuS2 + 20HNO3 = 3Cu(NO3)2 + 6H2SO4 + 14NO + 4H2O

ReactantsProducts
Cu33✔️
S66✔️
H2020✔️
N2020✔️
O6060✔️