To balance the equation C6H5OH + HNO3 = C6H2(NO2)3OH + 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.
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Step 1: Label Each Compound With a Variable
Label each compound (reactant or product) in the equation with a variable to tướng represent the unknown coefficients.
a C6H5OH + b HNO3 = c C6H2(NO2)3OH + d H2O
Step 2: Create a System of Equations
Create an equation for each element (C, H, O, N) where each term represents the number of atoms of the element in each reactant or product.
C: 6a + 0b = 6c + 0d H: 6a + 1b = 3c + 2d O: 1a + 3b = 7c + 1d N: 0a + 1b = 3c + 0d
Step 3: Solve For All Variables
Use substitution, Gaussian elimination, or a calculator to tướng solve for each variable.
- 6a - 6c = 0
- 6a + 1b - 3c - 2d = 0
- 1a + 3b - 7c - 1d = 0
- 1b - 3c = 0
Use your graphing calculator's rref() function (or an online rref calculator) to tướng convert the following matrix into reduced row-echelon-form:
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[ 6 0 -6 0 0] [ 6 1 -3 -2 0] [ 1 3 -7 -1 0] [ 0 1 -3 0 0]
The resulting matrix can be used to tướng determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.
Simplify the result to tướng get the lowest, whole integer values.
- a = 1 (C6H5OH)
- b = 3 (HNO3)
- c = 1 (C6H2(NO2)3OH)
- d = 3 (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.
C6H5OH + 3 HNO3 = C6H2(NO2)3OH + 3 H2O
| C | 6 | 6 | ✔️ |
|---|---|---|---|
| H | 9 | 9 | ✔️ |
| O | 10 | 10 | ✔️ |
| N | 3 | 3 | ✔️ |
Since there is an equal number of each element in the reactants and products of C6H5OH + 3HNO3 = C6H2(NO2)3OH + 3H2O, the equation is balanced.
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