Fe + HNO3 = Fe(NO3)3 + NO2 + H2O - Chemical Equation Balancer

To balance the equation Fe + HNO3 = Fe(NO3)3 + NO2 + 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 Fe + b HNO₃ = c Fe(NO₃)₃ + d NO₂ + f H₂O

Step 2: Create a System of Equations

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

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

Step 3: Solve For All Variables

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

1a - 1c = 0
1b - 2f = 0
1b - 3c - 1d = 0
3b - 9c - 2d - 1f = 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	0]
[ 0	 1	 0	 0	-2	0]
[ 0	 1	-3	-1	 0	0]
[ 0	 3	-9	-2	-1	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.

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a = 1 (Fe)
b = 6 (HNO3)
c = 1 (Fe(NO3)3)
d = 3 (NO2)
f = 3 (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.

Fe + 6 HNO₃ = Fe(NO₃)₃ + 3 NO₂ + 3 H₂O