To balance the equation Fe2O3 + H2SO4 = Fe2(SO4)3 + 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 lớn represent the unknown coefficients.
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a Fe2O3 + b H2SO4 = c Fe2(SO4)3 + d H2O
Step 2: Create a System of Equations
Create an equation for each element (Fe, O, H, S) where each term represents the number of atoms of the element in each reactant or product.
Fe: 2a + 0b = 2c + 0d O: 3a + 4b = 12c + 1d H: 0a + 2b = 0c + 2d S: 0a + 1b = 3c + 0d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to lớn solve for each variable.
- 2a - 2c = 0
- 3a + 4b - 12c - 1d = 0
- 2b - 2d = 0
- 1b - 3c = 0
Use your graphing calculator's rref() function (or an online rref calculator) to lớn convert the following matrix into reduced row-echelon-form:
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[ 2 0 -2 0 0] [ 3 4 -12 -1 0] [ 0 2 0 -2 0] [ 0 1 -3 0 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 = 1 (Fe2O3)
- b = 3 (H2SO4)
- c = 1 (Fe2(SO4)3)
- d = 3 (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.
Fe2O3 + 3 H2SO4 = Fe2(SO4)3 + 3 H2O
Since there is an equal number of each element in the reactants and products of Fe2O3 + 3H2SO4 = Fe2(SO4)3 + 3H2O, the equation is balanced.