To balance the equation HNO3 + O2 = NO + 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 đồ sộ represent the unknown coefficients.

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a HNO3 + b O2 = c NO + d H2O

Step 2: Create a System of Equations

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

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

Step 3: Solve For All Variables

Use substitution, Gaussian elimination, or a calculator đồ sộ solve for each variable.

  • 1a - 2d = 0
  • 1a - 1c = 0
  • 3a + 2b - 1c - 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	 0	-2	0]
[ 1	 0	-1	 0	0]
[ 3	 2	-1	-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 = 4 (HNO3)
  • b = -3 (O2)
  • c = 4 (NO)
  • 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.

4 HNO3 + -3 O2 = 4 NO + 2 H2O

Reactants Products

Since there is an equal number of each element in the reactants and products of 4HNO3 + -3O2 = 4NO + 2H2O, the equation is balanced.