To balance the equation C2H4 + AgNO3 + NH3 = C3H4Ag2 + NH4NO3 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 lớn represent the unknown coefficients.

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a C_{2}H_{4} + b AgNO_{3} + c NH_{3} = d C_{3}H_{4}Ag_{2} + f NH_{4}NO_{3}

### Step 2: Create a System of Equations

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

C: 2a + 0b + 0c = 3d + 0fH: 4a + 0b + 3c = 4d + 4fAg: 0a + 1b + 0c = 2d + 0fN: 0a + 1b + 1c = 0d + 2fO: 0a + 3b + 0c = 0d + 3f

### Step 3: Solve For All Variables

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

- 2a - 3d = 0
- 4a + 3c - 4d - 4f = 0
- 1b - 2d = 0
- 1b + 1c - 2f = 0
- 3b - 3f = 0

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

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

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

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

- a = 3 (C2H4)
- b = 4 (AgNO3)
- c = 4 (NH3)
- d = 2 (C3H4Ag2)
- f = 4 (NH4NO3)

### 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 C_{2}H_{4} + 4 AgNO_{3} + 4 NH_{3} = 2 C_{3}H_{4}Ag_{2} + 4 NH_{4}NO_{3}

C | 6 | 6 | ✔️ |
---|---|---|---|

H | 24 | 24 | ✔️ |

Ag | 4 | 4 | ✔️ |

N | 8 | 8 | ✔️ |

O | 12 | 12 | ✔️ |

Since there is an equal number of each element in the reactants and products of 3C2H4 + 4AgNO3 + 4NH3 = 2C3H4Ag2 + 4NH4NO3, 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 C2H4 + AgNO3 + NH3 = C3H4Ag2 + NH4NO3 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 lớn 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) | |||||||
---|---|---|---|---|---|---|---|---|

Reactants | Products | |||||||

C_{2}H_{4} | AgNO_{3} | NH_{3} | Total | C_{3}H_{4}Ag_{2} | NH_{4}NO_{3} | Total | ||

C | 2 | 2 | 3 | 3 | ❌ | |||

H | 4 | 3 | 7 | 4 | 4 | 8 | ❌ | |

Ag | 1 | 1 | 2 | 2 | ❌ | |||

N | 1 | 1 | 2 | 2 | 2 | ✔️ | ||

O | 3 | 3 | 3 | 3 | ✔️ |

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

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

**Ag**is not balanced. Add**1**molecule of**AgNO**to lớn the reactant (left-hand) side to lớn balance Silver:_{3}

C_{2}H_{4}+ 2AgNO_{3}+ NH_{3}= C_{3}H_{4}Ag_{2}+ NH_{4}NO_{3}Reactants Products C 2 3 ❌ H 7 8 ❌ Ag 2 2 ✔️ N 3 2 ❌ O 6 3 ❌ **O**is not balanced. Add**1**molecule of**NH**to lớn the product (right-hand) side to lớn balance Oxygen:_{4}NO_{3}

C_{2}H_{4}+ 2AgNO_{3}+ NH_{3}= C_{3}H_{4}Ag_{2}+ 2NH_{4}NO_{3}Reactants Products C 2 3 ❌ H 7 12 ❌ Ag 2 2 ✔️ N 3 4 ❌ O 6 6 ✔️ **N**is not balanced. Add**1**molecule of**AgNO**to lớn the reactant (left-hand) side to lớn balance Nitrogen:_{3}

C_{2}H_{4}+ 3AgNO_{3}+ NH_{3}= C_{3}H_{4}Ag_{2}+ 2NH_{4}NO_{3}Reactants Products C 2 3 ❌ H 7 12 ❌ Ag 3 2 ❌ N 4 4 ✔️ O 9 6 ❌ **O**is not balanced. Add**1**molecule of**NH**to lớn the product (right-hand) side to lớn balance Oxygen:_{4}NO_{3}

C_{2}H_{4}+ 3AgNO_{3}+ NH_{3}= C_{3}H_{4}Ag_{2}+ 3NH_{4}NO_{3}Reactants Products C 2 3 ❌ H 7 16 ❌ Ag 3 2 ❌ N 4 6 ❌ O 9 9 ✔️ **H**is not balanced. Add**2**molecules of**C**to lớn the reactant (left-hand) side to lớn balance Hydrogen:_{2}H_{4}

3C_{2}H_{4}+ 3AgNO_{3}+ NH_{3}= C_{3}H_{4}Ag_{2}+ 3NH_{4}NO_{3}Reactants Products C 6 3 ❌ H 15 16 ❌ Ag 3 2 ❌ N 4 6 ❌ O 9 9 ✔️ **C**is not balanced. Add**1**molecule of**C**to lớn the product (right-hand) side to lớn balance Carbon:_{3}H_{4}Ag_{2}

3C_{2}H_{4}+ 3AgNO_{3}+ NH_{3}= 2C_{3}H_{4}Ag_{2}+ 3NH_{4}NO_{3}Reactants Products C 6 6 ✔️ H 15 20 ❌ Ag 3 4 ❌ N 4 6 ❌ O 9 9 ✔️ **Ag**is not balanced. Add**1**molecule of**AgNO**to lớn the reactant (left-hand) side to lớn balance Silver:_{3}

3C_{2}H_{4}+ 4AgNO_{3}+ NH_{3}= 2C_{3}H_{4}Ag_{2}+ 3NH_{4}NO_{3}Reactants Products C 6 6 ✔️ H 15 20 ❌ Ag 4 4 ✔️ N 5 6 ❌ O 12 9 ❌ **N**is not balanced. Add**1**molecule of**NH**to lớn the reactant (left-hand) side to lớn balance Nitrogen:_{3}

3C_{2}H_{4}+ 4AgNO_{3}+ 2NH_{3}= 2C_{3}H_{4}Ag_{2}+ 3NH_{4}NO_{3}Xem thêm: susan didn't apply for the summer

Reactants Products C 6 6 ✔️ H 18 20 ❌ Ag 4 4 ✔️ N 6 6 ✔️ O 12 9 ❌

We could not determine all the steps needed to lớn balance the equation using trial and error. This may mean it requires too many steps and the chemical equation is too complex to lớn 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 lớn attempt to lớn guess the next steps, the final element counts in the balanced equation should be:

3C_{2}H_{4} + 4AgNO_{3} + 4NH_{3} = 2C_{3}H_{4}Ag_{2} + 4NH_{4}NO_{3}

Reactants | Products | ||
---|---|---|---|

C | 6 | 6 | ✔️ |

H | 24 | 24 | ✔️ |

Ag | 4 | 4 | ✔️ |

N | 8 | 8 | ✔️ |

O | 12 | 12 | ✔️ |

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