To balance the equation C6H5ONa + CO2 + H2O = C6H5OH + Na2CO3 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 C6H5ONa + b CO2 + c H2O = d C6H5OH + f Na2CO3
Step 2: Create a System of Equations
Create an equation for each element (C, H, O, Na) where each term represents the number of atoms of the element in each reactant or product.
C: 6a + 1b + 0c = 6d + 1f H: 5a + 0b + 2c = 6d + 0f O: 1a + 2b + 1c = 1d + 3f Na: 1a + 0b + 0c = 0d + 2f
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.
- 6a + 1b - 6d - 1f = 0
- 5a + 2c - 6d = 0
- 1a + 2b + 1c - 1d - 3f = 0
- 1a - 2f = 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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[ 6 1 0 -6 -1 0] [ 5 0 2 -6 0 0] [ 1 2 1 -1 -3 0] [ 1 0 0 0 -2 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.
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- a = 2 (C6H5ONa)
- b = 1 (CO2)
- c = 1 (H2O)
- d = 2 (C6H5OH)
- f = 1 (Na2CO3)
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 C6H5ONa + CO2 + H2O = 2 C6H5OH + Na2CO3
Since there is an equal number of each element in the reactants and products of 2C6H5ONa + CO2 + H2O = 2C6H5OH + Na2CO3, the equation is balanced.