To balance the equation KOH + MgSO4 = Mg(OH)2 + K2SO4 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 KOH + b MgSO4 = c Mg(OH)2 + d K2SO4
Step 2: Create a System of Equations
Create an equation for each element (K, O, H, Mg, S) where each term represents the number of atoms of the element in each reactant or product.
K: 1a + 0b = 0c + 2d O: 1a + 4b = 2c + 4d H: 1a + 0b = 2c + 0d Mg: 0a + 1b = 1c + 0d S: 0a + 1b = 0c + 1d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 1a - 2d = 0
- 1a + 4b - 2c - 4d = 0
- 1a - 2c = 0
- 1b - 1c = 0
- 1b - 1d = 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 0 -2 0] [ 1 4 -2 -4 0] [ 1 0 -2 0 0] [ 0 1 -1 0 0] [ 0 1 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.
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- a = 2 (KOH)
- b = 1 (MgSO4)
- c = 1 (Mg(OH)2)
- d = 1 (K2SO4)
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 KOH + MgSO4 = Mg(OH)2 + K2SO4
Since there is an equal number of each element in the reactants and products of 2KOH + MgSO4 = Mg(OH)2 + K2SO4, the equation is balanced.