2.2 Balancing of Equations

Chemical equations must obey the law of conservation of mass, meaning the number of atoms of each element must be the same on both the reactant and product sides. While simple equations can be balanced by inspection (trial and error), redox reactions are often complex and require systematic methods. This section introduces the two primary methods used to balance redox equations.

Main Idea

• Redox Method (Oxidation Number Method)
• Ion-Electron Method (Half-Reaction Method)

Both methods are based on the fundamental principle that in a redox reaction, the total number of electrons lost by the substance being oxidized must equal the total number of electrons gained by the substance being reduced.

Core Concept

The balancing process is essentially a way to ensure that electrons are conserved. When an element is oxidized, it loses electrons. When another is reduced, it gains them. To write a balanced equation, you must find the correct coefficients that make the electron loss equal to the electron gain, and then balance the remaining atoms (like hydrogen and oxygen) by inspection.

Detailed Explanation: The Oxidation Number Method

The textbook uses a detailed example to explain the oxidation number method: P + HNO_3 + H_2O ⟶ H_3PO_4 + NO

• Elemental phosphorus (P) is 0.
• In HNO_3, H is +1, O is -2, so N must be +5.
• In H_3PO_4, H is +1, O is -2, so P must be +5.
• In NO, O is -2, so N must be +2.
• Water (H_2O) has H=+1 and O=-2.

• Phosphorus (P): Goes from 0 to +5 (a change of 5 electrons lost).
• Nitrogen (N): Goes from +5 to +2 (a change of 3 electrons gained).

Step 3: Draw a bridge and indicate the number of electrons gained or lost. (This is a visual tool to help track the electrons).

• To balance a 3-electron gain with a 5-electron loss, multiply the N-containing compound by 5 and the P-containing compound by 3.
• This makes the total electrons lost = 5 × 3 = 15, and the total electrons gained = 3 × 5 = 15.
• Apply these coefficients to the respective substances: 3P and 3H_3PO_4; 5HNO_3 and 5NO.

• Balance Hydrogen: On the right, there are 3 × 3 = 9 H atoms in H_3PO_4. On the left, there are 5 H atoms in HNO_3. We need 4 more, so we place a coefficient of 2 in front of H_2O.
• Balance Oxygen:
• Left side: 5 × 3 = 15 (from HNO_3) + 2 × 1 = 2 (from H_2O) = 17 O atoms.
• Right side: 3 × 4 = 12 (from H_3PO_4) + 5 × 1 = 5 (from NO) = 17 O atoms.
• The oxygen atoms are balanced.

The final balanced equation is: 3P + 5HNO_3 + 2H_2O ⟶ 3H_3PO_4 + 5NO

Intermediate Level

The oxidation number method is a systematic, step-by-step procedure. The most crucial step is correctly identifying the elements whose oxidation numbers change. You then calculate the magnitude of the change per atom. By using cross-multiplication (multiplying the oxidation change of one element by the coefficient of the other), you ensure that the total electron transfer is balanced. After that, you balance the equation by inspection, starting with elements that appear in only one compound on each side (usually hydrogen and oxygen). This method is especially useful for equations in molecular form.

Important Points

• - Principle: Total electrons lost = Total electrons gained.
• - Method 1: Oxidation Number Method:
• 1. Assign oxidation numbers.
• 2. Identify the elements that change.
• 3. Determine the change in oxidation number per atom.
• 4. Equalize the total change (electrons lost = electrons gained) using a multiplier.
• 5. Balance the remaining atoms (H and O) by inspection.
• - Method 2: Ion-Electron Method (Half-Reaction Method): This is used for reactions in aqueous solutions and will be covered in the next section. It involves splitting the reaction into two half-reactions and balancing them separately.

Remember

• - Always start by assigning oxidation numbers correctly.
• - The change in oxidation number is the number of electrons lost or gained per atom.
• - The multiplier you use is determined by the need to make the total electrons lost equal to the total electrons gained.
• - Once the redox part is balanced, balance H and O by adding H_2O or H^+ as needed. (In neutral/basic conditions, add OH^- and H_2O to balance).
• - Never change the subscripts in a formula to balance an equation; only change the coefficients.

Common Mistakes

• - Changing subscripts instead of coefficients. For example, changing H_2O to H_4O_2 is wrong. You can only place numbers in front of the formula (e.g., 2H_2O).
• - Not multiplying the entire formula. When you multiply HNO_3 by 5, you must apply the coefficient to the whole formula, not just the N atom.
• - Forgetting to balance other atoms after the redox step. Many students stop once the electron transfer is balanced, forgetting to balance hydrogen and oxygen by inspection.
• - Making arithmetic errors when calculating electron transfer. Always double-check your math.
• - Misidentifying the oxidation state changes. If you assign the wrong oxidation numbers, the entire balancing process will be flawed.

Quick Revision

To balance a redox equation using the oxidation number method, you first assign oxidation numbers to all elements to identify which ones change. You then calculate the number of electrons lost (increase in oxidation number) and gained (decrease in oxidation number) per atom. To balance the electrons, you use cross-multiplication to find the right coefficients, ensuring the total electrons lost equals the total electrons gained. Finally, you balance the remaining atoms, usually hydrogen and oxygen, by inspection.

60-Second Revision

The oxidation number method balances redox equations by ensuring that the total increase in oxidation number (electrons lost) equals the total decrease (electrons gained). After assigning oxidation numbers and identifying the changes, use cross-multiplication to balance the electron transfer. Finally, balance the other atoms (like H and O) by inspection, adjusting coefficients only, never subscripts.

Frequently Asked Questions (FAQs)

Q1. What is the underlying principle of the oxidation number method? Answer: The underlying principle is the conservation of electrons. In a redox reaction, the total number of electrons lost by the substance being oxidized must be exactly equal to the total number of electrons gained by the substance being reduced.

Q2. What is the first step in balancing an equation by the oxidation number method? Answer: The first step is to assign oxidation numbers to all the atoms involved in the equation. This allows you to identify which elements are undergoing oxidation and reduction.

Q3. How do you equalize the number of electrons lost and gained? Answer: You find the change in oxidation number for the element being oxidized and the element being reduced. Then, you determine the lowest common multiple of these two changes. The multipliers needed to reach this common number become the coefficients for the respective substances.

Q4. In the example P + HNO_3 → H_3PO_4 + NO, why is the coefficient of P 3 and not 5? Answer: Phosphorus loses 5 electrons (0 to +5), and nitrogen gains 3 electrons (+5 to +2). To balance the electrons, you cross-multiply: multiply the P-containing substances by 3 (to get 3 × 5 = 15 electrons lost) and the N-containing substances by 5 (to get 5 × 3 = 15 electrons gained). This equalizes the electron transfer.

Q5. Can you change the subscripts of a chemical formula to balance an equation? Answer: No, you can never change the subscripts in a chemical formula to balance an equation. Doing so would change the identity of the substance. You can only change the coefficients (the numbers placed in front of the formula).

Q6. Why do you need to balance hydrogen and oxygen by inspection after the redox step? Answer: The redox step only balances the elements that change oxidation numbers. It does not necessarily balance the other elements, like hydrogen and oxygen, which are often present in the reactants and products. These must be balanced by adding coefficients to molecules like H_2O or H^+ to ensure the law of conservation of mass is satisfied.

Q7. What is the difference between the oxidation number method and the ion-electron method? Answer: The oxidation number method works with the overall molecular equation and tracks changes in oxidation numbers. The ion-electron method (or half-reaction method) splits the overall reaction into two half-reactions (oxidation and reduction) and balances each one separately for atoms and charge before combining them.

Q8. What is a "bridge" in the oxidation number method? Answer: The "bridge" is a visual aid used to show the connection between the same element on the reactant and product sides of the equation. It helps you see the change in oxidation number and calculate the number of electrons lost or gained.

Multiple Choice Questions (MCQs)

• Q1. What is the fundamental principle behind balancing a redox equation?
• A. The number of atoms of each element must be equal on both sides.
• B. The total number of electrons lost equals the total number of electrons gained.
• C. The number of molecules must be equal on both sides.
• D. The total charge on both sides must be zero.
• Correct Answer: B
• Explanation: Balancing redox equations is based on the conservation of electrons; the electrons lost by oxidation must equal the electrons gained by reduction.

• Q2. In the oxidation number method, what is the first step?
• A. Balance the oxygen atoms.
• B. Assign oxidation numbers to all atoms.
• C. Equalize the electrons lost and gained.
• D. Balance the hydrogen atoms.
• Correct Answer: B
• Explanation: The first step is to assign oxidation numbers to identify which elements are being oxidized and reduced.

• Q3. In the reaction P + HNO_3 → H_3PO_4 + NO, what is the change in oxidation number for nitrogen?
• A. From +5 to +2 (a gain of 3 electrons)
• B. From +5 to +4 (a gain of 1 electron)
• C. From 0 to +2 (a loss of 2 electrons)
• D. From +2 to +5 (a loss of 3 electrons)
• Correct Answer: A
• Explanation: N in HNO_3 is +5 and in NO it is +2. This is a decrease of 3, which corresponds to a gain of 3 electrons.

• Q4. Why do we cross-multiply the changes in oxidation numbers?
• A. To balance the number of molecules.
• B. To balance the number of oxygen atoms.
• C. To ensure the total electrons lost equals the total electrons gained.
• D. To make the equation look balanced.
• Correct Answer: C
• Explanation: Cross-multiplying ensures that the total increase in oxidation number (electrons lost) is equal to the total decrease (electrons gained).

Q5. What is the correct coefficient for H_2O in the balanced equation: 3P + 5HNO_3 + ? H_2O → 3H_3PO_4 + 5NO? A. 1 B. 2 C. 3 D. 4 Correct Answer: B Explanation: Balancing the hydrogen atoms shows there are 9 on the right (in H_3PO_4) and 5 on the left (in HNO_3). To get 9, you need 4 more H atoms, which come from 2H_2O. So the coefficient is 2.

• Q6. In the oxidation number method, which atoms are usually balanced by inspection after the redox step?
• A. The atoms that changed oxidation number.
• B. Hydrogen and oxygen.
• C. All metal atoms.
• D. Only the atoms in the products.
• Correct Answer: B
• Explanation: After balancing the electron transfer, the remaining elements, typically hydrogen and oxygen, are balanced by inspection.

• Q7. What is the change in oxidation number for phosphorus in the reaction P → H_3PO_4?
• A. 0 to +3
• B. 0 to +5
• C. +5 to 0
• D. -3 to +5
• Correct Answer: B
• Explanation: Elemental P has an oxidation number of 0. In H_3PO_4, the oxidation number of P is +5. This is an increase of 5.

Q8. When balancing an equation, you should only change the: A. Subscripts of the chemical formulas. B. Coefficients in front of the chemical formulas. C. Chemical symbols. D. State symbols. Correct Answer: B Explanation: Only the coefficients (the numbers placed in front of formulas) can be changed when balancing an equation. Changing subscripts would change the chemical identity.

• Q9. Which of the following is NOT a step in the oxidation number method?
• A. Assign oxidation numbers.
• B. Identify elements that change oxidation number.
• C. Split the reaction into two half-reactions.
• D. Equalize electrons lost and gained.
• Correct Answer: C
• Explanation: Splitting the reaction into half-reactions is part of the ion-electron method, not the oxidation number method.

Q10. What does the arrow or "bridge" in the oxidation number method represent? A. The flow of electrons from one atom to another. B. The change in the oxidation state of an element. C. The formation of a new bond. D. The breaking of a bond. Correct Answer: B Explanation: The bridge visually connects the same element before and after the reaction, showing the change in its oxidation number and the number of electrons lost or gained.