To balance the equation Ba + FeCl3 = BaCl2 + 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 tướng represent the unknown coefficients.
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a Ba + b FeCl3 = c BaCl2 + d Fe
Step 2: Create a System of Equations
Create an equation for each element (Ba, Fe, Cl) where each term represents the number of atoms of the element in each reactant or product.
Ba: 1a + 0b = 1c + 0d Fe: 0a + 1b = 0c + 1d Cl: 0a + 3b = 2c + 0d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 1a - 1c = 0
- 1b - 1d = 0
- 3b - 2c = 0
Use your graphing calculator's rref() function (or an online rref calculator) to tướng convert the following matrix into reduced row-echelon-form:
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[ 1 0 -1 0 0] [ 0 1 0 -1 0] [ 0 3 -2 0 0]
The resulting matrix can be used to tướng determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result to tướng get the lowest, whole integer values.
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- a = 3 (Ba)
- b = 2 (FeCl3)
- c = 3 (BaCl2)
- 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.
3 Ba + 2 FeCl3 = 3 BaCl2 + 2 Fe
Since there is an equal number of each element in the reactants and products of 3Ba + 2FeCl3 = 3BaCl2 + 2Fe, the equation is balanced.