The document outlines a systematic method for balancing chemical equations, emphasizing a step-by-step approach that eliminates guesswork. It describes techniques such as obtaining an augmented composition matrix, inverting it, and assigning coefficients, applicable to both redox and non-redox reactions. The content includes examples and mentions an app that can assist in this balancing process.

1. How to Balance Any
Chemical Equation
Eliminating Guesswork with a Systematic Method
Rich Apodaca Metamolecular, LLC

2. “Balancing By Inspection”
2H2 + O2 → 2H2O

3. “Balancing By Inspection”
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O

4. “Balancing By Inspection”
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O
2H2 + O2 → 2H2O

5. Not So Easy
P2I4 + P4 + H2O → PH4I + H3PO4

6. No More Guessing
Thorne Chem. Educator 2010, 15, 304

7. No More Guessing
• Balance even the most difficult
equations.
Thorne Chem. Educator 2010, 15, 304

8. No More Guessing
• Balance even the most difficult
equations.
• Balance both redox and non-
redox equations.
Thorne Chem. Educator 2010, 15, 304

9. No More Guessing
• Balance even the most difficult
equations.
• Balance both redox and non-
redox equations.
• Know when an equation is
unbalanceable.
Thorne Chem. Educator 2010, 15, 304

10. No More Guessing
• 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).
Thorne Chem. Educator 2010, 15, 304

11. Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation

12. Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation
H 2 0 2 Composition Table
O 0 2 1

13. Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation
H 2 0 2 Composition Table
O 0 2 1
a=a/2
2 0 2 a
Composition Matrix
b=b/2
0 2 1 b

14. Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation
H 2 0 2 Composition Table
O 0 2 1
a=a/2
2 0 2 a
Composition Matrix
b=b/2
0 2 1 b
1 0 1 Reduced Row Echelon
Form
0 1 1/2

15. Step 1: Find Augmented Composition Matrix
2H2 + O2 → 2H2O Unbalanced Equation
H 2 0 2 Composition Table
O 0 2 1
a=a/2
2 0 2 a
Composition Matrix
b=b/2
0 2 1 b
1 0 1 Reduced Row Echelon
Form
0 1 1/2
1 0 1 Augmented Matrix
(Add enough rows to
0 1 1/2 make a square.)
0 0 1

16. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c

17. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c

18. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c

19. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c

20. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c

21. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c
1 0 0 1 0 -1
0 1 0 0 1 -1/2
0 0 1 0 0 1

22. Step 2: Invert the Augmented Matrix
H2 + O2 → H2O
a=a-c
1 0 1 1 0 0 a
b=b-1/2c
0 1 1/2 0 1 0 b
0 0 1 0 0 1 c
1 0 0 1 0 -1
0 1 0 0 1 -1/2
0 0 1 0 0 1
1 0 -1
0 1 -1/2 Matrix Inverse
0 0 1

23. Step 3: Assign Coefficients
H2 + O2 → H2O
1 0 -1
0 1 -1/2 Matrix Inverse
0 0 1

24. Step 3: Assign Coefficients
H2 + O2 → H2O
1 0 -1
0 1 -1/2 Matrix Inverse
0 0 1
[-1 -1/2 1] Null Space Vector Transpose

25. Step 3: Assign Coefficients
H2 + O2 → H2O
1 0 -1
0 1 -1/2 Matrix Inverse
0 0 1
[-1 -1/2 1] Null Space Vector Transpose
x2 removes fractions [-2 -1 2] Scaled Transpose
Negative: reactant
Positive: product

26. Step 3: Assign Coefficients
H2 + O2 → H2O
1 0 -1
0 1 -1/2 Matrix Inverse
0 0 1
[-1 -1/2 1] Null Space Vector Transpose
x2 removes fractions [-2 -1 2] Scaled Transpose
Negative: reactant
Positive: product
2H2 + 1O2 → 2H2O

27. Matrix Operations

28. Matrix Operations
1/8x
8 4 2 1 1 1/2 1/4 1/8
1 0 1 1 1 0 1 1
0 1 0 1 0 1 0 1
5 2 1 3 5 2 1 3
• Multiply row by number

29. Matrix Operations
-8x
8 4 2 1 8 4 2 1
1 0 1 1 0 -32 -15 -7
0 1 0 1 0 1 0 1
5 2 1 3 5 2 1 3
• Multiply row by number
• Add multiple of one row to another

30. Matrix Operations
8 4 2 1 8 4 2 1
1 0 1 1 0 1 0 1
0 1 0 1 1 0 1 1
5 2 1 3 5 2 1 3
• Multiply row by number
• Add multiple of one row to another
• Exchange two rows

31. A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl

32. A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl
K 1 1 0 0 0 1
I 1 0 0 2 0 0
O 0 3 0 0 1 0
H 0 0 1 0 2 0
Cl 0 1 1 0 0 1

33. A More Complicated Example
KI + KClO3 + HCl → I2 + H2O + KCl
K 1 1 0 0 0 1
I 1 0 0 2 0 0
O 0 3 0 0 1 0
H 0 0 1 0 2 0
Cl 0 1 1 0 0 1
1 1 0 0 0 1
1 0 0 2 0 0
0 3 0 0 1 0
0 0 1 0 2 0
0 1 1 0 0 1

34. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
b=b-a, b=-b
1 0 0 2 0 0 b
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e

35. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
b=b-a, b=-b
1 0 0 2 0 0 b
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e

36. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
b=b-a, b=-b
1 0 0 2 0 0 b
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e

37. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
b=b-a, b=-b
1 0 0 2 0 0 b
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e

38. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
d c
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e

39. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
d c
0 3 0 0 1 0 c
0 0 1 0 2 0 d
0 1 1 0 0 1 e
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 3 0 0 1 0 d
0 1 1 0 0 1 e

40. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
d=d-3b, d=d/6
0 3 0 0 1 0 d
e=e-c, e=e/2
0 1 1 0 0 1 e

41. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
d=d-3b, d=d/6
0 3 0 0 1 0 d
e=e-c, e=e/2
0 1 1 0 0 1 e
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
0 0 0 1 -1 0 e

42. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
e=e-f, e = -(6/7)e
0 0 0 1 -1 0 e

43. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
e=e-f, e = -(6/7)e
0 0 0 1 -1 0 e
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
0 0 0 0 1 -3/7 e

44. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
e=e-f, e = -(6/7)e
0 0 0 1 -1 0 e
1 1 0 0 0 1 a
0 1 0 -2 0 1 b
0 0 1 0 2 0 c
0 0 0 1 1/6 -1/2 d
0 0 0 0 1 -3/7 e

45. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
a=a-b
1 1 0 0 0 1 a
b=b+2d
0 1 0 -2 0 1 b
c=d-2e
0 0 1 0 2 0 c
d=d-e/6
0 0 0 1 1/6 -1/2 d
0 0 0 0 1 -3/7 e

46. Reduce the Composition Matrix...
KI + KClO3 + HCl → I2 + H2O + KCl
a=a-b
1 1 0 0 0 1 a
b=b+2d
0 1 0 -2 0 1 b
c=d-2e
0 0 1 0 2 0 c
d=d-e/6
0 0 0 1 1/6 -1/2 d
0 0 0 0 1 -3/7 e
1 0 0 0 0 6/7 a
0 1 0 0 0 1/7 b
0 0 1 0 0 6/7 c
0 0 0 1 0 -3/7 d
0 0 0 0 1 -3/7 e

47. Augment and Invert the Matrix
KI + KClO3 + HCl → I2 + H2O + KCl
a=a-6e/7 1 0 0 0 0 6/7 1 0 0 0 0 0 a
b=b-e/7 0 1 0 0 0 1/7 0 1 0 0 0 0 b
c=c-6e/7 0 0 1 0 0 6/7 0 0 1 0 0 0 c
d=d+3f/7 0 0 0 1 0 -3/7 0 0 0 1 0 0 d
e=e+3f/7 0 0 0 0 1 -3/7 0 0 0 0 1 0 e
0 0 0 0 0 1 0 0 0 0 0 1 f

48. Augment and Invert the Matrix
KI + KClO3 + HCl → I2 + H2O + KCl
a=a-6e/7 1 0 0 0 0 6/7 1 0 0 0 0 0 a
b=b-e/7 0 1 0 0 0 1/7 0 1 0 0 0 0 b
c=c-6e/7 0 0 1 0 0 6/7 0 0 1 0 0 0 c
d=d+3f/7 0 0 0 1 0 -3/7 0 0 0 1 0 0 d
e=e+3f/7 0 0 0 0 1 -3/7 0 0 0 0 1 0 e
0 0 0 0 0 1 0 0 0 0 0 1 f
1 0 0 0 0 0 1 0 0 0 0 -6/7 a
0 1 0 0 0 0 0 1 0 0 0 -1/7 b
0 0 1 0 0 0 0 0 1 0 0 -6/7 c
0 0 0 1 0 0 0 0 0 1 0 3/7 d
0 0 0 0 1 0 0 0 0 0 1 3/7 e
0 0 0 0 0 1 0 0 0 0 0 1 f

49. Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/7
0 1 0 0 0 -1/7
0 0 1 0 0 -6/7
0 0 0 1 0 3/7
0 0 0 0 1 3/7
0 0 0 0 0 1

50. Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/7
0 1 0 0 0 -1/7
0 0 1 0 0 -6/7
0 0 0 1 0 3/7
0 0 0 0 1 3/7
0 0 0 0 0 1
x7 [-6/7 -1/7 -6/7 3/7 3/7 1]
[-6 -1 -6 3 3 7]

51. Assign Coefficients
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 -6/7
0 1 0 0 0 -1/7
0 0 1 0 0 -6/7
0 0 0 1 0 3/7
0 0 0 0 1 3/7
0 0 0 0 0 1
x7 [-6/7 -1/7 -6/7 3/7 3/7 1]
[-6 -1 -6 3 3 7]
6KI + 1KClO3 + 6HCl → 3I2 + 3H2O + 7KCl

52. A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/7
0 1 0 0 0 1/7
0 0 1 0 0 6/7 Reduced Augmented
Matrix
0 0 0 1 0 -3/7
0 0 0 0 1 -3/7
0 0 0 0 0 1

53. A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/7
0 1 0 0 0 1/7
0 0 1 0 0 6/7 Reduced Augmented
Matrix
0 0 0 1 0 -3/7
0 0 0 0 1 -3/7
0 0 0 0 0 1
[6/7 1/7 6/7 -3/7 -3/7 1]
x7 [6 1 6 3 3 7]

54. A Shortcut
KI + KClO3 + HCl → I2 + H2O + KCl
1 0 0 0 0 6/7
0 1 0 0 0 1/7
0 0 1 0 0 6/7 Reduced Augmented
Matrix
0 0 0 1 0 -3/7
0 0 0 0 1 -3/7
0 0 0 0 0 1
[6/7 1/7 6/7 -3/7 -3/7 1]
x7 [6 1 6 3 3 7]
6KI + 1KClO3 + 6HCl → 3I2 + 3H2O + 7KCl

55. Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-

56. Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0
Cl 0 2 0 1
charge 0 0 3 -1

57. Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0
Cl 0 2 0 1
charge 0 0 3 -1
1 0 3 0 1 0 0 -1/3
0 2 0 1 0 1 0 -1/2
0 0 1 -1 0 0 1 1/3
0 0 0 1
Composition matrix Reduced, augmented composition matrix inverse

58. Balance Equations Containing Charge
Fe + Cl2 → Fe3+ + Cl-
Fe 1 0 3 0
Cl 0 2 0 1
charge 0 0 3 -1
1 0 3 0 1 0 0 -1/3
0 2 0 1 0 1 0 -1/2
0 0 1 -1 0 0 1 1/3
0 0 0 1
Composition matrix Reduced, augmented composition matrix inverse
x7 [-1/3 -1/2 1/3 1] → [-2 -3 2 6]
2Fe + 3Cl2 → 2Fe3+ + 6Cl-

59. Know When an Equation is Unbalanceable
FeS2 +HNO3 → Fe2(SO4)3 + NO + H2SO4
Fe 1 0 2 0 0
S 2 0 3 0 1
H 0 1 0 0 2
N 0 1 0 1 0
O 0 3 12 1 4
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
Reduced product matrix is square diagonal. No solution is possible.

60. Balance Chemical Equations Systematically

61. Balance Chemical Equations Systematically
• Build the Composition Matrix

62. Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix

63. Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix
• Augment and Invert the Reduced
Composition Matrix

64. Balance Chemical Equations Systematically
• Build the Composition Matrix
• Reduce the Composition Matrix
• Augment and Invert the Reduced
Composition Matrix
• Assign Coefficients

65. ReactionMate
TM
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