Metamolecular, LLC Rich Apodaca Eliminating Guesswork with a Systematic Method How to Balance Any Chemical Equation
How to Balance Any Chemical Equation
Jun 23, 2015
A systematic method that eliminates guesswork. An app that does the calculation is available at: http://itunes.apple.com/us/app/reactionmate/id524541263
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Metamolecular, LLCRich Apodaca
Eliminating Guesswork with a Systematic Method
How to Balance Any Chemical Equation
“Balancing By Inspection”
2H2 + O2 → 2H2O
“Balancing By Inspection”
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O
“Balancing By Inspection”
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O
Not So Easy
P2I4 + P4 + H2O → PH4I + H3PO4
No More Guessing
Thorne Chem. Educator 2010, 15, 304
• Balance even the most difficult equations.
No More Guessing
Thorne Chem. Educator 2010, 15, 304
• Balance even the most difficult equations.
• Balance both redox and non-redox equations.
No More Guessing
Thorne Chem. Educator 2010, 15, 304
• Balance even the most difficult equations.
• Balance both redox and non-redox equations.
• Know when an equation is unbalanceable.
No More Guessing
Thorne Chem. Educator 2010, 15, 304
• Balance even the most difficult equations.
• Balance both redox and non-redox equations.
• Know when an equation is unbalanceable.
• An app can do this for you (details at the end).
No More Guessing
Thorne Chem. Educator 2010, 15, 304
Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation
Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O
H 2 0 2O 0 2 1
Composition Table
Unbalanced Equation
Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O
H 2 0 2O 0 2 1
Composition Table
Unbalanced Equation
2 0 20 2 1
Composition Matrixb
aa=a/2
b=b/2
Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O
H 2 0 2O 0 2 1
Composition Table
Unbalanced Equation
1 0 10 1 1/2
Reduced Row EchelonForm
2 0 20 2 1
Composition Matrixb
aa=a/2
b=b/2
Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O
H 2 0 2O 0 2 1
Composition Table
Unbalanced Equation
1 0 10 1 1/2
Reduced Row EchelonForm
1 0 10 1 1/20 0 1
Augmented Matrix(Add enough rows tomake a square.)
2 0 20 2 1
Composition Matrixb
aa=a/2
b=b/2
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 0 1 0 -10 1 0 0 1 -1/20 0 1 0 0 1
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
1 0 0 1 0 -10 1 0 0 1 -1/20 0 1 0 0 1
1 0 1 1 0 00 1 1/2 0 1 00 0 1 0 0 1 c
b
a
b=b-1/2c
a=a-c
1 0 -10 1 -1/20 0 1
Matrix Inverse
Step 3: Assign Coefficients
H2 + O2 → H2O
1 0 -10 1 -1/20 0 1
Matrix Inverse
Step 3: Assign Coefficients
H2 + O2 → H2O
[-1 -1/2 1] Null Space Vector Transpose
1 0 -10 1 -1/20 0 1
Matrix Inverse
Step 3: Assign Coefficients
H2 + O2 → H2O
[-1 -1/2 1] Null Space Vector Transpose
1 0 -10 1 -1/20 0 1
Matrix Inverse
[-2 -1 2] Scaled Transpose
Negative: reactantPositive: product
x2 removes fractions
Step 3: Assign Coefficients
H2 + O2 → H2O
[-1 -1/2 1] Null Space Vector Transpose
1 0 -10 1 -1/20 0 1
Matrix Inverse
2H2 + 1O2 → 2H2O
[-2 -1 2] Scaled Transpose
Negative: reactantPositive: product
x2 removes fractions
Matrix Operations
• Multiply row by number
Matrix Operations
8 4 2 11 0 1 10 1 0 15 2 1 3
1 1/2 1/4 1/81 0 1 10 1 0 15 2 1 3
1/8x
Matrix Operations
8 4 2 11 0 1 10 1 0 15 2 1 3
8 4 2 10 -32 -15 -70 1 0 15 2 1 3
-8x
• Multiply row by number
• Add multiple of one row to another
Matrix Operations
8 4 2 11 0 1 10 1 0 15 2 1 3
8 4 2 10 1 0 11 0 1 15 2 1 3
• Multiply row by number
• Add multiple of one row to another
• Exchange two rows
A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl
A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl
K 1 1 0 0 0 1I 1 0 0 2 0 0O 0 3 0 0 1 0H 0 0 1 0 2 0Cl 0 1 1 0 0 1
A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl
K 1 1 0 0 0 1I 1 0 0 2 0 0O 0 3 0 0 1 0H 0 0 1 0 2 0Cl 0 1 1 0 0 1
1 1 0 0 0 11 0 0 2 0 00 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
e
1 1 0 0 0 11 0 0 2 0 00 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
b=b-a, b=-b
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
e
1 1 0 0 0 11 0 0 2 0 00 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
b=b-a, b=-b
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
e
1 1 0 0 0 11 0 0 2 0 00 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
b=b-a, b=-b
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
e
1 1 0 0 0 11 0 0 2 0 00 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
b=b-a, b=-b
1 1 0 0 0 10 1 0 -2 0 10 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
e
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
e
d⇔c
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 3 0 0 1 00 0 1 0 2 00 1 1 0 0 1
a
b
c
d
e
d⇔c
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 3 0 0 1 00 1 1 0 0 1
a
b
c
d
e
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 3 0 0 1 00 1 1 0 0 1
a
b
c
d
e
d=d-3b, d=d/6
e=e-c, e=e/2
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 1 -1 0
a
b
c
d
e
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 3 0 0 1 00 1 1 0 0 1
a
b
c
d
e
d=d-3b, d=d/6
e=e-c, e=e/2
Reduce the Composition Matrix...
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 1 -1 0
a
b
c
d
ee=e-f, e = -(6/7)e
KI + KClO3 + HCl → I2 + H2O + KCl
Reduce the Composition Matrix...
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 1 -1 0
a
b
c
d
ee=e-f, e = -(6/7)e
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 0 1 -3/7
a
b
c
d
e
Reduce the Composition Matrix...
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 1 -1 0
a
b
c
d
ee=e-f, e = -(6/7)e
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 0 1 -3/7
a
b
c
d
e
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 0 1 -3/7
a
b
c
d
e
d=d-e/6
c=d-2e
b=b+2d
a=a-b
Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 10 1 0 -2 0 10 0 1 0 2 00 0 0 1 1/6 -1/20 0 0 0 1 -3/7
a
b
c
d
e
d=d-e/6
c=d-2e
b=b+2d
a=a-b
1 0 0 0 0 6/70 1 0 0 0 1/70 0 1 0 0 6/70 0 0 1 0 -3/70 0 0 0 1 -3/7
a
b
c
d
e
Augment and Invert the Matrix
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/7 1 0 0 0 0 00 1 0 0 0 1/7 0 1 0 0 0 00 0 1 0 0 6/7 0 0 1 0 0 00 0 0 1 0 -3/7 0 0 0 1 0 00 0 0 0 1 -3/7 0 0 0 0 1 00 0 0 0 0 1 0 0 0 0 0 1
a
b
c
d
ee=e+3f/7
d=d+3f/7
c=c-6e/7
f
b=b-e/7
a=a-6e/7
Augment and Invert the Matrix
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/7 1 0 0 0 0 00 1 0 0 0 1/7 0 1 0 0 0 00 0 1 0 0 6/7 0 0 1 0 0 00 0 0 1 0 -3/7 0 0 0 1 0 00 0 0 0 1 -3/7 0 0 0 0 1 00 0 0 0 0 1 0 0 0 0 0 1
a
b
c
d
ee=e+3f/7
d=d+3f/7
c=c-6e/7
f
b=b-e/7
a=a-6e/7
1 0 0 0 0 0 1 0 0 0 0 -6/70 1 0 0 0 0 0 1 0 0 0 -1/70 0 1 0 0 0 0 0 1 0 0 -6/70 0 0 1 0 0 0 0 0 1 0 3/70 0 0 0 1 0 0 0 0 0 1 3/70 0 0 0 0 1 0 0 0 0 0 1
a
b
c
d
e
f
Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/70 1 0 0 0 -1/70 0 1 0 0 -6/70 0 0 1 0 3/70 0 0 0 1 3/70 0 0 0 0 1
Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/70 1 0 0 0 -1/70 0 1 0 0 -6/70 0 0 1 0 3/70 0 0 0 1 3/70 0 0 0 0 1
[-6/7 -1/7 -6/7 3/7 3/7 1][-6 -1 -6 3 3 7]
x7
Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/70 1 0 0 0 -1/70 0 1 0 0 -6/70 0 0 1 0 3/70 0 0 0 1 3/70 0 0 0 0 1
6KI + 1KClO3 + 6HCl → 3I2 + 3H2O + 7KCl
[-6/7 -1/7 -6/7 3/7 3/7 1][-6 -1 -6 3 3 7]
x7
A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/70 1 0 0 0 1/70 0 1 0 0 6/70 0 0 1 0 -3/70 0 0 0 1 -3/70 0 0 0 0 1
Reduced AugmentedMatrix
A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/70 1 0 0 0 1/70 0 1 0 0 6/70 0 0 1 0 -3/70 0 0 0 1 -3/70 0 0 0 0 1
Reduced AugmentedMatrix
[6/7 1/7 6/7 -3/7 -3/7 1][6 1 6 3 3 7]x7
A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
6KI + 1KClO3 + 6HCl → 3I2 + 3H2O + 7KCl
1 0 0 0 0 6/70 1 0 0 0 1/70 0 1 0 0 6/70 0 0 1 0 -3/70 0 0 0 1 -3/70 0 0 0 0 1
Reduced AugmentedMatrix
[6/7 1/7 6/7 -3/7 -3/7 1][6 1 6 3 3 7]x7
Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0Cl 0 2 0 1
charge 0 0 3 -1
Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0Cl 0 2 0 1
charge 0 0 3 -1
1 0 3 00 2 0 10 0 1 -1
1 0 0 -1/30 1 0 -1/20 0 1 1/30 0 0 1
Reduced, augmented composition matrix inverseComposition matrix
Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0Cl 0 2 0 1
charge 0 0 3 -1
1 0 3 00 2 0 10 0 1 -1
1 0 0 -1/30 1 0 -1/20 0 1 1/30 0 0 1
Reduced, augmented composition matrix inverseComposition matrix
2Fe + 3Cl2 → 2Fe3+ + 6Cl-
[-1/3 -1/2 1/3 1] → [-2 -3 2 6]x7
Know When an Equation is Unbalanceable
FeS2 +HNO3 → Fe2(SO4)3 + NO + H2SO4
Fe 1 0 2 0 0S 2 0 3 0 1H 0 1 0 0 2N 0 1 0 1 0O 0 3 12 1 4
1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1
Reduced product matrix is square diagonal. No solution is possible.
Balance Chemical Equations Systematically
• Build the Composition Matrix
Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix
Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix
• Augment and Invert the Reduced Composition Matrix
Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix
• Augment and Invert the Reduced Composition Matrix
• Assign Coefficients
Balance Chemical Equations Systematically
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