To balance the equation HNO3 + AlCl3 = Al(NO3)3 + HCl 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 HNO3 + b AlCl3 = c Al(NO3)3 + d HCl
Step 2: Create a System of Equations
Create an equation for each element (H, N, O, Al, Cl) where each term represents the number of atoms of the element in each reactant or product.
H: 1a + 0b = 0c + 1d N: 1a + 0b = 3c + 0d O: 3a + 0b = 9c + 0d Al: 0a + 1b = 1c + 0d Cl: 0a + 3b = 0c + 1d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 1a - 1d = 0
- 1a - 3c = 0
- 3a - 9c = 0
- 1b - 1c = 0
- 3b - 1d = 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 0 -1 0] [ 1 0 -3 0 0] [ 3 0 -9 0 0] [ 0 1 -1 0 0] [ 0 3 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.
- a = 3 (HNO3)
- b = 1 (AlCl3)
- c = 1 (Al(NO3)3)
- d = 3 (HCl)
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.
3 HNO3 + AlCl3 = Al(NO3)3 + 3 HCl
Since there is an equal number of each element in the reactants and products of 3HNO3 + AlCl3 = Al(NO3)3 + 3HCl, the equation is balanced.