Sodium hydroxide is manufacture by the electrolysis of
brine solution. The reaction by `-` products are `:`

Options

A. `Cl_2` and `H_2`
B. `Cl_2` and `Na-Hg`
C. `Cl_2` and NaCl
D. `Cl_2` and `O_2`

Explanation

To solve the question regarding the by-products formed during the electrolysis of brine solution for the manufacture of sodium hydroxide (NaOH), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electrolysis Process**:
- Brine solution is a concentrated solution of sodium chloride (NaCl) in water (H2O). During electrolysis, brine undergoes a chemical reaction when an electric current is passed through it.

2. **Dissociation of Brine**:
- The brine solution dissociates into its ions:
\[ \text{NaCl} \rightarrow \text{Na}^+ + \text{Cl}^- \]
- Additionally, water also dissociates into:
\[ \text{H}_2\text{O} \rightarrow \text{H}^+ + \text{OH}^- \]

3. **Movement of Ions**:
- The sodium ions (\(\text{Na}^+\)) and hydrogen ions (\(\text{H}^+\)) migrate towards the cathode (negative electrode).
- The chloride ions (\(\text{Cl}^-\)) and hydroxide ions (\(\text{OH}^-\)) migrate towards the anode (positive electrode).

4. **Reactions at the Anode**:
- At the anode, chloride ions (\(\text{Cl}^-\)) are discharged because their discharge potential is lower than that of hydroxide ions (\(\text{OH}^-\)). The reaction at the anode is:
\[ 2\text{Cl}^- \rightarrow \text{Cl}_2 + 2\text{e}^- \]
- Chlorine gas (\(\text{Cl}_2\)) is evolved as a by-product.

5. **Reactions at the Cathode**:
- At the cathode, hydrogen ions (\(\text{H}^+\)) are discharged because their discharge potential is lower than that of sodium ions (\(\text{Na}^+\)). The reaction at the cathode is:
\[ 2\text{H}^+ + 2\text{e}^- \rightarrow \text{H}_2 \]
- Hydrogen gas (\(\text{H}_2\)) is evolved as another by-product.

6. **Formation of Sodium Hydroxide**:
- In the solution, sodium ions (\(\text{Na}^+\)) combine with hydroxide ions (\(\text{OH}^-\)) to form sodium hydroxide:
\[ \text{Na}^+ + \text{OH}^- \rightarrow \text{NaOH} \]

### Conclusion:
The by-products of the electrolysis of brine solution are chlorine gas (\(\text{Cl}_2\)) and hydrogen gas (\(\text{H}_2\)). Therefore, the answer is:
- **Chlorine (Cl2) and Hydrogen (H2)**.