To balance the equation PH3 + H2O = H3PO4 + H2 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 lớn represent the unknown coefficients.

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a PH3 + b H2O = c H3PO4 + d H2

Step 2: Create a System of Equations

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

P:	1a	+	0b	=	1c	+	0d
H:	3a	+	2b	=	3c	+	2d
O:	0a	+	1b	=	4c	+	0d

Step 3: Solve For All Variables

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

  • 1a - 1c = 0
  • 3a + 2b - 3c - 2d = 0
  • 1b - 4c = 0

Use your graphing calculator's rref() function (or an online rref calculator) lớn convert the following matrix into reduced row-echelon-form:

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[ 1	 0	-1	 0	0]
[ 3	 2	-3	-2	0]
[ 0	 1	-4	 0	0]

The resulting matrix can be used lớn determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.

Simplify the result lớn get the lowest, whole integer values.

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  • a = 1 (PH3)
  • b = 4 (H2O)
  • c = 1 (H3PO4)
  • d = 4 (H2)

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.

PH3 + 4 H2O = H3PO4 + 4 H2

Reactants Products

Since there is an equal number of each element in the reactants and products of PH3 + 4H2O = H3PO4 + 4H2, the equation is balanced.