To balance the equation C6H5OH + Na = C6H5ONa + H2 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.

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### Step 1: Label Each Compound With a Variable

Label each compound (reactant or product) in the equation with a variable vĩ đại represent the unknown coefficients.

a C_{6}H_{5}OH + b Na = c C_{6}H_{5}ONa + d H_{2}

### Step 2: Create a System of Equations

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

**C**: 6a + 0b = 6c + 0d
**H**: 6a + 0b = 5c + 2d
**O**: 1a + 0b = 1c + 0d
**Na**: 0a + 1b = 1c + 0d

### Step 3: Solve For All Variables

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

- 6a - 6c = 0
- 6a - 5c - 2d = 0
- 1a - 1c = 0
- 1b - 1c = 0

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

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[ 6 0 -6 0 0] [ 6 0 -5 -2 0] [ 1 0 -1 0 0] [ 0 1 -1 0 0]

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

Simplify the result vĩ đại get the lowest, whole integer values.

- a = 2 (C6H5OH)
- b = 2 (Na)
- c = 2 (C6H5ONa)
- d = 1 (H2)

### 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.

2 C_{6}H_{5}OH + 2 Na = 2 C_{6}H_{5}ONa + H_{2}

C | 12 | 12 | ✔️ |
---|---|---|---|

H | 12 | 12 | ✔️ |

O | 2 | 2 | ✔️ |

Na | 2 | 2 | ✔️ |

Since there is an equal number of each element in the reactants and products of 2C6H5OH + 2Na = 2C6H5ONa + H2, the equation is balanced.

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