To balance the equation K2CO3 + HCl = H2O + CO2 + KCl 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 K2CO3 + b HCl = c H2O + d CO2 + f KCl
Step 2: Create a System of Equations
Create an equation for each element (K, C, O, H, Cl) where each term represents the number of atoms of the element in each reactant or product.
K: 2a + 0b = 0c + 0d + 1f C: 1a + 0b = 0c + 1d + 0f O: 3a + 0b = 1c + 2d + 0f H: 0a + 1b = 2c + 0d + 0f Cl: 0a + 1b = 0c + 0d + 1f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 2a - 1f = 0
- 1a - 1d = 0
- 3a - 1c - 2d = 0
- 1b - 2c = 0
- 1b - 1f = 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 0 0 0 -1 0] [ 1 0 0 -1 0 0] [ 3 0 -1 -2 0 0] [ 0 1 -2 0 0 0] [ 0 1 0 0 -1 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.
- a = 1 (K2CO3)
- b = 2 (HCl)
- c = 1 (H2O)
- d = 1 (CO2)
- f = 2 (KCl)
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.
K2CO3 + 2 HCl = H2O + CO2 + 2 KCl
Since there is an equal number of each element in the reactants and products of K2CO3 + 2HCl = H2O + CO2 + 2KCl, the equation is balanced.