How many moles of electrons are involved in balancing the following redox equations?
`H_2S + NO_3^(-) to S + NO`
`Mn (OH)_2 + H_2O_2 to MnO_2`

Answer

To determine the number of moles of electrons involved in balancing the given redox equations, we will analyze each reaction step by step.

### Step 1: Analyze the first reaction
**Reaction:**
\[ \text{H}_2\text{S} + \text{NO}_3^{-} \rightarrow \text{S} + \text{NO} \]

1. **Identify oxidation states:**
- In H₂S, sulfur (S) has an oxidation state of -2.
- In S (elemental sulfur), the oxidation state is 0.
- In NO₃⁻, nitrogen (N) has an oxidation state of +5.
- In NO, nitrogen (N) has an oxidation state of +2.

2. **Determine the changes in oxidation states:**
- Sulfur changes from -2 to 0:
Change = 0 - (-2) = +2
- Nitrogen changes from +5 to +2:
Change = +2 - (+5) = -3

3. **Calculate the number of moles of electrons transferred:**
- For sulfur: 1 mole of sulfur undergoes a change of +2, so it contributes 2 moles of electrons.
- For nitrogen: 1 mole of nitrogen undergoes a change of -3, so it requires 3 moles of electrons.

4. **Balance the reaction:**
- To balance the electrons, we need to multiply the sulfur change by 3 and the nitrogen change by 2:
- 3 moles of H₂S will produce 3 moles of S (2 moles of electrons each) = 6 moles of electrons.
- 2 moles of NO₃⁻ will consume 2 moles of NO (3 moles of electrons each) = 6 moles of electrons.

### **Total for the first reaction:**
**6 moles of electrons are involved.**

### Step 2: Analyze the second reaction
**Reaction:**
\[ \text{Mn(OH)}_2 + \text{H}_2\text{O}_2 \rightarrow \text{MnO}_2 \]

1. **Identify oxidation states:**
- In Mn(OH)₂, manganese (Mn) has an oxidation state of +2.
- In MnO₂, manganese (Mn) has an oxidation state of +4.
- In H₂O₂, oxygen (O) has an oxidation state of -1.
- In O₂ (in MnO₂), oxygen (O) has an oxidation state of -2.

2. **Determine the changes in oxidation states:**
- Manganese changes from +2 to +4:
Change = +4 - (+2) = +2
- Oxygen changes from -1 (in H₂O₂) to -2 (in MnO₂):
Change = -2 - (-1) = -1 (for each O, and there are 2 O in H₂O₂).

3. **Calculate the number of moles of electrons transferred:**
- For manganese: 1 mole of Mn undergoes a change of +2, contributing 2 moles of electrons.
- For hydrogen peroxide: 1 mole of H₂O₂ (2 O) undergoes a change of -2, requiring 2 moles of electrons.

4. **Balance the reaction:**
- The total number of moles of electrons transferred is:
- 1 mole of Mn contributes 2 moles of electrons.
- 1 mole of H₂O₂ requires 2 moles of electrons.

### **Total for the second reaction:**
**4 moles of electrons are involved.**

### Final Answers:
- For the first reaction: **6 moles of electrons**
- For the second reaction: **4 moles of electrons**