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
| H | 10 | 10 | ✔️ |
|---|---|---|---|
| N | 10 | 10 | ✔️ |
| O | 30 | 30 | ✔️ |
| Ra | 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.
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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) | |||||||
|---|---|---|---|---|---|---|---|---|
| Reactants | Products | |||||||
| HNO3 | Ra | Total | Ra(NO3)2 | N2O | H2O | Total | ||
| H | 1 | 1 | 2 | 2 | ❌ | |||
| N | 1 | 1 | 2 | 2 | 4 | ❌ | ||
| O | 3 | 3 | 6 | 1 | 1 | 8 | ❌ | |
| Ra | 1 | 1 | 1 | 1 | ✔️ | |||
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.
H is not balanced. Add 1 molecule of HNO3 lớn the reactant (left-hand) side lớn balance Hydrogen:
2HNO3 + Ra = Ra(NO3)2 + N2O + H2OReactants Products H 2 2 ✔️ N 2 4 ❌ O 6 8 ❌ Ra 1 1 ✔️ N is not balanced. Add 2 molecules of HNO3 lớn the reactant (left-hand) side lớn balance Nitrogen:
4HNO3 + Ra = Ra(NO3)2 + N2O + H2OReactants Products H 4 2 ❌ N 4 4 ✔️ O 12 8 ❌ Ra 1 1 ✔️ H is not balanced. Add 1 molecule of H2O lớn the product (right-hand) side lớn balance Hydrogen:
4HNO3 + Ra = Ra(NO3)2 + N2O + 2H2OReactants Products H 4 4 ✔️ N 4 4 ✔️ O 12 9 ❌ Ra 1 1 ✔️ O is not balanced. Add 3 molecules of H2O lớn the product (right-hand) side lớn balance Oxygen:
4HNO3 + Ra = Ra(NO3)2 + N2O + 5H2OReactants Products H 4 10 ❌ N 4 4 ✔️ O 12 12 ✔️ Ra 1 1 ✔️ H is not balanced. Add 6 molecules of HNO3 lớn the reactant (left-hand) side lớn balance Hydrogen:
10HNO3 + Ra = Ra(NO3)2 + N2O + 5H2OReactants Products H 10 10 ✔️ N 10 4 ❌ O 30 12 ❌ Ra 1 1 ✔️ 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 + 5H2OXem thêm: nghị luận về lòng biết ơn thầy cô giáo
Reactants Products H 10 10 ✔️ N 10 10 ✔️ O 30 30 ✔️ Ra 1 4 ❌ Ra is not balanced. Add 3 molecules of Ra lớn the reactant (left-hand) side lớn balance Radium:
10HNO3 + 4Ra = 4Ra(NO3)2 + N2O + 5H2OReactants Products H 10 10 ✔️ N 10 10 ✔️ O 30 30 ✔️ Ra 4 4 ✔️
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
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