C9H12 + KMnO4 + H2SO4 = C9H6O6 + K2SO4 + MnSO4 + H2O

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To balance the equation C9H12 + KMnO4 + H2SO4 = C9H6O6 + K2SO4 + MnSO4 + 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 represent the unknown coefficients.

a C9H12 + b KMnO4 + c H2SO4 = d C9H6O6 + f K2SO4 + g MnSO4 + h H2O

Step 2: Create a System of Equations

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

C: 9a + 0b + 0c = 9d + 0f + 0g + 0h H: 12a + 0b + 2c = 6d + 0f + 0g + 2h K: 0a + 1b + 0c = 0d + 2f + 0g + 0h Mn: 0a + 1b + 0c = 0d + 0f + 1g + 0h O: 0a + 4b + 4c = 6d + 4f + 4g + 1h S: 0a + 0b + 1c = 0d + 1f + 1g + 0h

Step 3: Solve For All Variables

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

Using Substitution or Elimination
  • 9a - 9d = 0
  • 12a + 2c - 6d - 2h = 0
  • 1b - 2f = 0
  • 1b - 1g = 0
  • 4b + 4c - 6d - 4f - 4g - 1h = 0
  • 1c - 1f - 1g = 0
Using Linear Systems / Algebra

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

[ 9 0 0 -9 0 0 0 0] [ 12 0 2 -6 0 0 -2 0] [ 0 1 0 0 -2 0 0 0] [ 0 1 0 0 0 -1 0 0] [ 0 4 4 -6 -4 -4 -1 0] [ 0 0 1 0 -1 -1 0 0] The resulting matrix can be used to determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients. Convert to RREF and Solve Step-by-Step

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

  • a = 5 (C9H12)
  • b = 18 (KMnO4)
  • c = 27 (H2SO4)
  • d = 5 (C9H6O6)
  • f = 9 (K2SO4)
  • g = 18 (MnSO4)
  • h = 42 (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.

5 C9H12 + 18 KMnO4 + 27 H2SO4 = 5 C9H6O6 + 9 K2SO4 + 18 MnSO4 + 42 H2O
Reactants Products
C4545✔️
H114114✔️
K1818✔️
Mn1818✔️
O180180✔️
S2727✔️
Since there is an equal number of each element in the reactants and products of 5C9H12 + 18KMnO4 + 27H2SO4 = 5C9H6O6 + 9K2SO4 + 18MnSO4 + 42H2O, the equation is balanced.

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