To balance the equation Al + HNO3 = Al(NO3)3 + N2 + 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 tát represent the unknown coefficients.
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a Al + b HNO3 = c Al(NO3)3 + d N2 + f H2O
Step 2: Create a System of Equations
Create an equation for each element (Al, H, N, O) where each term represents the number of atoms of the element in each reactant or product.
Al: 1a + 0b = 1c + 0d + 0f H: 0a + 1b = 0c + 0d + 2f N: 0a + 1b = 3c + 2d + 0f O: 0a + 3b = 9c + 0d + 1f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 1a - 1c = 0
- 1b - 2f = 0
- 1b - 3c - 2d = 0
- 3b - 9c - 1f = 0
Use your graphing calculator's rref() function (or an online rref calculator) to tát convert the following matrix into reduced row-echelon-form:
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[ 1 0 -1 0 0 0] [ 0 1 0 0 -2 0] [ 0 1 -3 -2 0 0] [ 0 3 -9 0 -1 0]
The resulting matrix can be used to tát determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result to tát get the lowest, whole integer values.
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- a = 10 (Al)
- b = 36 (HNO3)
- c = 10 (Al(NO3)3)
- d = 3 (N2)
- f = 18 (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.
10 Al + 36 HNO3 = 10 Al(NO3)3 + 3 N2 + 18 H2O
Since there is an equal number of each element in the reactants and products of 10Al + 36HNO3 = 10Al(NO3)3 + 3N2 + 18H2O, the equation is balanced.