To balance the equation CH3COOH + CaCO3 = (CH3COO)2Ca + H2O + CO2 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 CH3COOH + b CaCO3 = c (CH3COO)2Ca + d H2O + f CO2
Step 2: Create a System of Equations
Create an equation for each element (C, H, O, Ca) where each term represents the number of atoms of the element in each reactant or product.
C: 2a + 1b = 4c + 0d + 1f H: 4a + 0b = 6c + 2d + 0f O: 2a + 3b = 4c + 1d + 2f Ca: 0a + 1b = 1c + 0d + 0f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 2a + 1b - 4c - 1f = 0
- 4a - 6c - 2d = 0
- 2a + 3b - 4c - 1d - 2f = 0
- 1b - 1c = 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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[ 2 1 -4 0 -1 0] [ 4 0 -6 -2 0 0] [ 2 3 -4 -1 -2 0] [ 0 1 -1 0 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 = 2 (CH3COOH)
- b = 1 (CaCO3)
- c = 1 ((CH3COO)2Ca)
- d = 1 (H2O)
- f = 1 (CO2)
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 CH3COOH + CaCO3 = (CH3COO)2Ca + H2O + CO2
Since there is an equal number of each element in the reactants and products of 2CH3COOH + CaCO3 = (CH3COO)2Ca + H2O + CO2, the equation is balanced.