To balance the equation PB(OH)2 + HNO3 = PB(NO3)2 + 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 to tướng represent the unknown coefficients.
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a PB(OH)2 + b HNO3 = c PB(NO3)2 + d H2O
Step 2: Create a System of Equations
Create an equation for each element (P, B, O, H, N) where each term represents the number of atoms of the element in each reactant or product.
P: 1a + 0b = 1c + 0d B: 1a + 0b = 1c + 0d O: 2a + 3b = 6c + 1d H: 2a + 1b = 0c + 2d N: 0a + 1b = 2c + 0d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 1a - 1c = 0
- 1a - 1c = 0
- 2a + 3b - 6c - 1d = 0
- 2a + 1b - 2d = 0
- 1b - 2c = 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 -1 0 0] [ 1 0 -1 0 0] [ 2 3 -6 -1 0] [ 2 1 0 -2 0] [ 0 1 -2 0 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 (PB(OH)2)
- b = 2 (HNO3)
- c = 1 (PB(NO3)2)
- d = 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.
PB(OH)2 + 2 HNO3 = PB(NO3)2 + 2 H2O
Since there is an equal number of each element in the reactants and products of PB(OH)2 + 2HNO3 = PB(NO3)2 + 2H2O, the equation is balanced.