To balance the equation Ba(HCO3)2 + Na2CO3 = BaCO3 + NaHCO3 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 Ba(HCO3)2 + b Na2CO3 = c BaCO3 + d NaHCO3
Step 2: Create a System of Equations
Create an equation for each element (Ba, H, C, O, Na) where each term represents the number of atoms of the element in each reactant or product.
Ba: 1a + 0b = 1c + 0d H: 2a + 0b = 0c + 1d C: 2a + 1b = 1c + 1d O: 6a + 3b = 3c + 3d Na: 0a + 2b = 0c + 1d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator lớn solve for each variable.
- 1a - 1c = 0
- 2a - 1d = 0
- 2a + 1b - 1c - 1d = 0
- 6a + 3b - 3c - 3d = 0
- 2b - 1d = 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] [ 2 0 0 -1 0] [ 2 1 -1 -1 0] [ 6 3 -3 -3 0] [ 0 2 0 -1 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 (Ba(HCO3)2)
- b = 1 (Na2CO3)
- c = 1 (BaCO3)
- d = 2 (NaHCO3)
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.
Ba(HCO3)2 + Na2CO3 = BaCO3 + 2 NaHCO3
Since there is an equal number of each element in the reactants and products of Ba(HCO3)2 + Na2CO3 = BaCO3 + 2NaHCO3, the equation is balanced.