To balance the equation for the reaction between [tex]\( SO_2 \)[/tex] and [tex]\( Br_2 \)[/tex] under acidic conditions, we need to follow a series of systematic steps:
1. Assign Oxidation States:
- For [tex]\( SO_2 \)[/tex]:
- Oxidation state of S: +4
- For [tex]\( SO_4^{2-} \)[/tex]:
- Oxidation state of S: +6
- For [tex]\( Br_2 \)[/tex]:
- Oxidation state of Br: 0 (elemental form)
- For [tex]\( Br^- \)[/tex]:
- Oxidation state of Br: -1
2. Identify the Change in Oxidation States:
- The sulfur in [tex]\( SO_2 \)[/tex] is oxidized from +4 to +6, losing 2 electrons.
- Each bromine in [tex]\( Br_2 \)[/tex] is reduced from 0 to -1, each gaining 1 electron. Since there are two bromine atoms, the total gain is 2 electrons.
3. Balance the Atoms Undergoing Changes in Oxidation States:
- Write the half-reactions for the oxidation and reduction processes:
- Oxidation (Sulfur): [tex]\( SO_2 \rightarrow SO_4^{2-} \)[/tex]
- Reduction (Bromine): [tex]\( Br_2 \rightarrow 2Br^- \)[/tex]
4. Balance the Oxygen Atoms by Adding [tex]\( H_2O \)[/tex]:
- Balance the half-reaction for [tex]\( SO_2 \)[/tex] by adding [tex]\( H_2O \)[/tex]:
[tex]\[ SO_2 + 2H_2O \rightarrow SO_4^{2-} + 4H^+ + 2e^- \][/tex]
5. Balance the Hydrogen Atoms by Adding [tex]\( H^+ \)[/tex]:
- This step has already been included while balancing oxygens using [tex]\( H_2O \)[/tex]:
6. Combine the Half-Reactions Ensuring Electrons are Balanced:
- Write the reduction half-reaction:
[tex]\[ Br_2 + 2e^- \rightarrow 2Br^- \][/tex]
- Combine the oxidation half-reaction and the reduction half-reaction while ensuring the electrons are canceled out.
- Oxidation half-reaction: [tex]\( SO_2 + 2H_2O \rightarrow SO_4^{2-} + 4H^+ + 2e^- \)[/tex]
- Reduction half-reaction: [tex]\( Br_2 + 2e^- \rightarrow 2Br^- \)[/tex]
7. Add the Half-Reactions Together:
- Ensure that electrons cancel out:
[tex]\[ SO_2 + 2H_2O \rightarrow SO_4^{2-} + 4H^+ + 2e^- \][/tex]
[tex]\[ Br_2 + 2e^- \rightarrow 2Br^- \][/tex]
- Combining these, we get:
[tex]\[ SO_2 + Br_2 + 2H_2O \rightarrow SO_4^{2-} + 2Br^- + 4H^+ \][/tex]
Thus, the balanced equation for the reaction between [tex]\( SO_2 \)[/tex] and [tex]\( Br_2 \)[/tex] under acidic conditions is:
[tex]\[ SO_2 + Br_2 + 2H_2O \rightarrow SO_4^{2-} + 2Br^- + 4H^+ \][/tex]
