al hno3 alno33 n2 n2o h2o

To balance the equation AL + HNO3 = AL(NO3)3 + N2O + 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 N2O + f H2O

Step 2: Create a System of Equations

Create an equation for each element (A, L, H, N, O) where each term represents the number of atoms of the element in each reactant or product.

A:	1a	+	0b	=	1c	+	0d	+	0f
L:	1a	+	0b	=	1c	+	0d	+	0f
H:	0a	+	1b	=	0c	+	0d	+	2f
N:	0a	+	1b	=	3c	+	2d	+	0f
O:	0a	+	3b	=	9c	+	1d	+	1f

Step 3: Solve For All Variables

Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.

  • 1a - 1c = 0
  • 1a - 1c = 0
  • 1b - 2f = 0
  • 1b - 3c - 2d = 0
  • 3b - 9c - 1d - 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]
[ 1	 0	-1	 0	 0	0]
[ 0	 1	 0	 0	-2	0]
[ 0	 1	-3	-2	 0	0]
[ 0	 3	-9	-1	-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 = 8 (AL)
  • b = 30 (HNO3)
  • c = 8 (AL(NO3)3)
  • d = 3 (N2O)
  • f = 15 (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.

8 AL + 30 HNO3 = 8 AL(NO3)3 + 3 N2O + 15 H2O

Reactants Products
A88✔️
L88✔️
H3030✔️
N3030✔️
O9090✔️

Since there is an equal number of each element in the reactants and products of 8AL + 30HNO3 = 8AL(NO3)3 + 3N2O + 15H2O, the equation is balanced.