The cell reaction occurring at anode in the electrolysis of aqueous \(NaCl\) solution is:

\(Cl (aq) \longrightarrow \frac{1}{2} Cl_2(g) + e^-,\,\ E_{cell} = 1.36\, \ V\)

The correct answer is option 2. \(Cl (aq) \longrightarrow \frac{1}{2} Cl_2(g) + e^-,\,\ E_{cell} = 1.36\, \ V\).

In the electrolysis of aqueous sodium chloride (\(NaCl\)), the ions present are \(Na^+\), \(Cl^-\), \(H^+\), and \(OH^-\).

At the anode (positive electrode), oxidation occurs. Since chloride ions (\(Cl^-\)) are negatively charged, they are attracted to the positively charged anode. The most common reaction that occurs at the anode during the electrolysis of aqueous sodium chloride is the oxidation of chloride ions (\(Cl^-\)) to chlorine gas (\(Cl_2\)).

Let us analyze each option:

1. \(H^+(aq) + e^- \longrightarrow \frac{1}{2}H_2(g), \, E_{\text{cell}} = 0.00 V\) - This reaction involves the reduction of hydrogen ions (\(H^+\)) to hydrogen gas (\(H_2\)), not oxidation at the anode. Additionally, the standard electrode potential (\(E_{\text{cell}}\)) given is not consistent with the process occurring at the anode during the electrolysis of aqueous sodium chloride.

2. \(Cl (aq) \longrightarrow \frac{1}{2} Cl_2(g) + e^-,\, E_{\text{cell}} = 1.36\, V\) - This reaction involves the oxidation of chloride ions (\(Cl^-\)) to chlorine gas (\(Cl_2\)). The standard electrode potential (\(E_{\text{cell}}\)) given is consistent with the process occurring at the anode during the electrolysis of aqueous sodium chloride. This option is a suitable representation of the reaction at the anode.

3. \(Na^+ (aq) + e^- \longrightarrow Na(s), \, E_{\text{cell}} = - 2.71\, V\) - This reaction involves the reduction of sodium ions (\(Na^+\)) to solid sodium (\(Na\)), not oxidation at the anode. Additionally, the standard electrode potential (\(E_{\text{cell}}\)) given is not consistent with the process occurring at the anode during the electrolysis of aqueous sodium chloride.

4. \(2H_2O(l) \longrightarrow O_2(g) + 4H^+(aq) + 4e^-,\, E_{\text{cell}} = - 1.23\, V\) - This reaction involves the oxidation of water to oxygen gas (\(O_2\)), not oxidation at the anode during the electrolysis of aqueous sodium chloride. Additionally, the standard electrode potential (\(E_{\text{cell}}\)) given is not consistent with the process occurring at the anode during the electrolysis of aqueous sodium chloride.

Therefore, the correct answer is option (2) \(Cl (aq) \longrightarrow \frac{1}{2} Cl_2(g) + e^-,\, E_{\text{cell}} = 1.36\, V\). This represents the oxidation of chloride ions (\(Cl^-\)) to chlorine gas (\(Cl_2\)) at the anode during the electrolysis of aqueous sodium chloride.