To balance the equation MnO2 + HCl = MnCl2 + Cl2 + H2O 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 lớn represent the unknown coefficients.
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a MnO2 + b HCl = c MnCl2 + d Cl2 + f H2O
Step 2: Create a System of Equations
Create an equation for each element (Mn, O, H, Cl) where each term represents the number of atoms of the element in each reactant or product.
Mn: 1a + 0b = 1c + 0d + 0f O: 2a + 0b = 0c + 0d + 1f H: 0a + 1b = 0c + 0d + 2f Cl: 0a + 1b = 2c + 2d + 0f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator lớn solve for each variable.
- 1a - 1c = 0
- 2a - 1f = 0
- 1b - 2f = 0
- 1b - 2c - 2d = 0
Use your graphing calculator's rref() function (or an online rref calculator) lớn convert the following matrix into reduced row-echelon-form:
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[ 1 0 -1 0 0 0] [ 2 0 0 0 -1 0] [ 0 1 0 0 -2 0] [ 0 1 -2 -2 0 0]
The resulting matrix can be used lớn determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result lớn get the lowest, whole integer values.
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- a = 1 (MnO2)
- b = 4 (HCl)
- c = 1 (MnCl2)
- d = 1 (Cl2)
- f = 2 (H2O)
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.
MnO2 + 4 HCl = MnCl2 + Cl2 + 2 H2O
| Mn | 1 | 1 | ✔️ |
|---|---|---|---|
| O | 2 | 2 | ✔️ |
| H | 4 | 4 | ✔️ |
| Cl | 4 | 4 | ✔️ |
Since there is an equal number of each element in the reactants and products of MnO2 + 4HCl = MnCl2 + Cl2 + 2H2O, the equation is balanced.
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