To balance the equation Al + Fe2O3 = Al2O3 + Fe 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 tát represent the unknown coefficients.
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a Al + b Fe2O3 = c Al2O3 + d Fe
Step 2: Create a System of Equations
Create an equation for each element (Al, Fe, O) where each term represents the number of atoms of the element in each reactant or product.
Al: 1a + 0b = 2c + 0d Fe: 0a + 2b = 0c + 1d O: 0a + 3b = 3c + 0d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 1a - 2c = 0
- 2b - 1d = 0
- 3b - 3c = 0
Use your graphing calculator's rref() function (or an online rref calculator) to tát convert the following matrix into reduced row-echelon-form:
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[ 1 0 -2 0 0] [ 0 2 0 -1 0] [ 0 3 -3 0 0]
The resulting matrix can be used to tát determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result to tát get the lowest, whole integer values.
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- a = 2 (Al)
- b = 1 (Fe2O3)
- c = 1 (Al2O3)
- d = 2 (Fe)
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 Al + Fe2O3 = Al2O3 + 2 Fe
Since there is an equal number of each element in the reactants and products of 2Al + Fe2O3 = Al2O3 + 2Fe, the equation is balanced.