# hno3 ra n2o

To balance the equation HNO3 + Ra = Ra(NO3)2 + N2O + 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 lớn represent the unknown coefficients.

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a HNO3 + b Ra = c Ra(NO3)2 + d N2O + f H2O

### Step 2: Create a System of Equations

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

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

```

### Step 3: Solve For All Variables

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

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

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

```[ 1	 0	 0	 0	-2	0]
[ 1	 0	-2	-2	 0	0]
[ 3	 0	-6	-1	-1	0]
[ 0	 1	-1	 0	 0	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 = 10 (HNO3)
• b = 4 (Ra)
• c = 4 (Ra(NO3)2)
• d = 1 (N2O)
• f = 5 (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.

10 HNO3 + 4 Ra = 4 Ra(NO3)2 + N2O + 5 H2O

Reactants Products H 10 10 ✔️ 10 10 ✔️ 30 30 ✔️ 4 4 ✔️

Since there is an equal number of each element in the reactants and products of 10HNO3 + 4Ra = 4Ra(NO3)2 + N2O + 5H2O, 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 kết thúc of a chemical reaction as at the beginning. To be balanced, every element in HNO3 + Ra = Ra(NO3)2 + N2O + 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 lớn 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
HNO3RaTotalRa(NO3)2N2OH2OTotal
H1122
N11224
O336118
Ra1111✔️

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

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

1. H is not balanced. Add 1 molecule of HNO3 lớn the reactant (left-hand) side lớn balance Hydrogen:
2HNO3 + Ra = Ra(NO3)2 + N2O + H2O

ReactantsProducts
H22✔️
N24
O68
Ra11✔️
2. N is not balanced. Add 2 molecules of HNO3 lớn the reactant (left-hand) side lớn balance Nitrogen:
4HNO3 + Ra = Ra(NO3)2 + N2O + H2O

ReactantsProducts
H42
N44✔️
O128
Ra11✔️
3. H is not balanced. Add 1 molecule of H2O lớn the product (right-hand) side lớn balance Hydrogen:
4HNO3 + Ra = Ra(NO3)2 + N2O + 2H2O

ReactantsProducts
H44✔️
N44✔️
O129
Ra11✔️
4. O is not balanced. Add 3 molecules of H2O lớn the product (right-hand) side lớn balance Oxygen:
4HNO3 + Ra = Ra(NO3)2 + N2O + 5H2O

ReactantsProducts
H410
N44✔️
O1212✔️
Ra11✔️
5. H is not balanced. Add 6 molecules of HNO3 lớn the reactant (left-hand) side lớn balance Hydrogen:
10HNO3 + Ra = Ra(NO3)2 + N2O + 5H2O

ReactantsProducts
H1010✔️
N104
O3012
Ra11✔️
6. N is not balanced. Add 3 molecules of Ra(NO3)2 lớn the product (right-hand) side lớn balance Nitrogen:
10HNO3 + Ra = 4Ra(NO3)2 + N2O + 5H2O

ReactantsProducts
H1010✔️
N1010✔️
O3030✔️
Ra14
7. Ra is not balanced. Add 3 molecules of Ra lớn the reactant (left-hand) side lớn balance Radium:
10HNO3 + 4Ra = 4Ra(NO3)2 + N2O + 5H2O

ReactantsProducts
H1010✔️
N1010✔️
O3030✔️
Ra44✔️

### 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.

10HNO3 + 4Ra = 4Ra(NO3)2 + N2O + 5H2O