To balance the equation Al + Ba(OH)2 + H2O = Ba(AlO2)2 + 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.
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 Ba(OH)2 + c H2O = d Ba(AlO2)2 + f H2
Step 2: Create a System of Equations
Create an equation for each element (Al, Ba, O, H) where each term represents the number of atoms of the element in each reactant or product.
Al: 1a + 0b + 0c = 2d + 0f Ba: 0a + 1b + 0c = 1d + 0f O: 0a + 2b + 1c = 4d + 0f H: 0a + 2b + 2c = 0d + 2f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 1a - 2d = 0
- 1b - 1d = 0
- 2b + 1c - 4d = 0
- 2b + 2c - 2f = 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 0 -2 0 0] [ 0 1 0 -1 0 0] [ 0 2 1 -4 0 0] [ 0 2 2 0 -2 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 (Ba(OH)2)
- c = 2 (H2O)
- d = 1 (Ba(AlO2)2)
- f = 3 (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 Al + Ba(OH)2 + 2 H2O = Ba(AlO2)2 + 3 H2
Since there is an equal number of each element in the reactants and products of 2Al + Ba(OH)2 + 2H2O = Ba(AlO2)2 + 3H2, the equation is balanced.