To balance the equation HNO3 + CaCO3 = Ca(NO3)2 + H2O + CO2 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 CaCO3 = c Ca(NO3)2 + d H2O + f CO2
Step 2: Create a System of Equations
Create an equation for each element (H, N, O, Ca, C) where each term represents the number of atoms of the element in each reactant or product.
H: 1a + 0b = 0c + 2d + 0f N: 1a + 0b = 2c + 0d + 0f O: 3a + 3b = 6c + 1d + 2f Ca: 0a + 1b = 1c + 0d + 0f C: 0a + 1b = 0c + 0d + 1f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 1a - 2d = 0
- 1a - 2c = 0
- 3a + 3b - 6c - 1d - 2f = 0
- 1b - 1c = 0
- 1b - 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:
[ 1 0 0 -2 0 0] [ 1 0 -2 0 0 0] [ 3 3 -6 -1 -2 0] [ 0 1 -1 0 0 0] [ 0 1 0 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 = 2 (HNO3)
- b = 1 (CaCO3)
- c = 1 (Ca(NO3)2)
- d = 1 (H2O)
- f = 1 (CO2)
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.
2 HNO3 + CaCO3 = Ca(NO3)2 + H2O + CO2
Since there is an equal number of each element in the reactants and products of 2HNO3 + CaCO3 = Ca(NO3)2 + H2O + CO2, the equation is balanced.