To balance the equation Cu(NO3)2 + Ag = Cu + AgNO3 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 đồ sộ represent the unknown coefficients.
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a Cu(NO3)2 + b Ag = c Cu + d AgNO3
Step 2: Create a System of Equations
Create an equation for each element (Cu, N, O, Ag) where each term represents the number of atoms of the element in each reactant or product.
Cu: 1a + 0b = 1c + 0d N: 2a + 0b = 0c + 1d O: 6a + 0b = 0c + 3d Ag: 0a + 1b = 0c + 1d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator đồ sộ solve for each variable.
- 1a - 1c = 0
- 2a - 1d = 0
- 6a - 3d = 0
- 1b - 1d = 0
Use your graphing calculator's rref() function (or an online rref calculator) đồ sộ convert the following matrix into reduced row-echelon-form:
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[ 1 0 -1 0 0] [ 2 0 0 -1 0] [ 6 0 0 -3 0] [ 0 1 0 -1 0]
The resulting matrix can be used đồ sộ determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result đồ sộ get the lowest, whole integer values.
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- a = 1 (Cu(NO3)2)
- b = 2 (Ag)
- c = 1 (Cu)
- d = 2 (AgNO3)
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.
Cu(NO3)2 + 2 Ag = Cu + 2 AgNO3
Since there is an equal number of each element in the reactants and products of Cu(NO3)2 + 2Ag = Cu + 2AgNO3, the equation is balanced.