## 3 Results and Discussion
### Effect of Electrode Configuration
relative to the potential of the solution (\(\psi_{0}\))), and \(\psi_{0}\) is the potential of the solution, whose value is zero; therefore, Equation (5) is expressed as:  
\[\Delta_{\rm r}G_{\rm m}=\mu_{\rm aq}^{\emptyset}-RT\ln\left[\rm Cl_{\rm aq}^{-} \right]-\mu_{\rm ads}^{\emptyset}-RT\ln\left[\rm Cl_{\rm ads}^{-}\right]\ +\psi F \tag{6}\]  
and, finally  
\[\Delta_{\rm r}G_{\rm m}=\mu_{aq}^{\emptyset}-\mu_{ads}^{\emptyset}+RT\ln\frac{ \left[\rm Cl_{\rm aq}^{-}\right]}{\left[\rm Cl_{\rm ads}^{-}\right]}+\psi F \tag{7}\]  
In the desorption process, the standard Gibbs free energy change of uncharged particles transferred from a solid phase to a liquid phase is the same as the difference in the chemical potential between the solid phase and the liquid phase. Therefore, the standard Gibbs free energy change in the standard state can be expressed as:  
\[\Delta_{\rm r}G_{\rm m}^{\emptyset}=\mu_{aq}^{\emptyset}-\mu_{ads}^{\emptyset} \tag{8}\]  
The standard Gibbs free energy change is given by:  
\[\Delta_{\rm r}G_{\rm m}^{\emptyset}=-RT\ln K_{\rm d} \tag{9}\]  
where \(K_{\rm d}\) is the desorption equilibrium constant. According to Equations (8) and (9), we obtain  
\[\Delta_{\rm r}G_{\rm m}=-RT\ln K_{\rm d}+RT\ln\frac{\left[\rm Cl_{\rm aq}^{-} \right]}{\left[\rm Cl_{\rm ads}^{-}\right]}+\psi F \tag{10}\]  
If  
\[Q=\frac{\left[\rm Cl_{\rm aq}^{-}\right]}{\left[\rm Cl_{\rm ads}^{-}\right]} \tag{11}\]  
Inserting Equation (11) into Equation (10), we obtain  
\[\Delta_{\rm r}G_{\rm m}=-RT\ln K_{\rm d}+RT\ln Q+\psi F \tag{12}\]  
and, then  
\[\Delta_{\rm r}G_{\rm m}=RT\ln\frac{Q}{K_{\rm d}}+\psi F \tag{13}\]  
When desorption reaction reaches equilibrium, Equation (13) can be expressed as:  
\[RT\ln\frac{Q}{K_{\rm d}}=-\psi F \tag{14}\]  
If cathodic polarization occurs, then the value of parameter \(\psi\) is negative. \[\Delta_{\rm r}G_{\rm m}=\mu_{\rm aq}^{\emptyset}-RT\ln\left[\rm Cl_{\rm aq}^{-} \right]-\mu_{\rm ads}^{\emptyset}-RT\ln\left[\rm Cl_{\rm ads}^{-}\right]\ +\psi F \tag{6}\]0  
Thus, Equations (13) and (15) show that \(\rm Cl^{-}\) on the GAC surface is desorbed and entered the solution because the Gibbs free energy change exceeded 0 when the cathode was polarized. This is consistent with the experimental electrodesorption results at different cathode potentials (Section 3.2), implying that \(\rm Cl^{-}\) desorption is affected by cathodic polarization in the CDI process. However, as confirmed in the experimental electrodesorption results, the desorption efficiency cannot reach 100% when the cathode potential cannot reach \(-\)1.7 V; that is, the cathode potential distribution must be sufficient to completely desorb \(\rm Cl^{-}\). The Gibbs free energy of desorption depends on the cathode potential;specifically, as the range of cathode potential broadens, the desorption becomes easier. Based on the above analysis, we speculated that the incomplete desorption of the electrode in previous studies [18; 19; 41], even under cathodic polarization, is linked to insufficient cathode potential. The electrodeposition process has successfully predicted the complete desorption potential (\(\psi\)) that achieves complete Cl\({}^{-}\) desorption. The parameter [Cl\({}^{-}\)aq] represents the Cl\({}^{-}\) concentration in solution when the Cl\({}^{-}\) have completely desorbed from the GAC electrode, which can be derived from the equilibrium absorption capacity. The other parameter \(\left[\text{Cl}_{\text{ads}}^{-}\right]\) represents the amount of Cl\({}^{-}\) adsorbed on the GAC surface. Its value is nearly zero when the Cl\({}^{-}\) completely desorbs, but for a convenient calculation, we replaced the value of \(\left[\text{Cl}_{\text{ads}}^{-}\right]\) by the detection limit of the ion chromatograph. During the electrodeposition process, \(K_{\text{d}}\) was calculated by the method reported in our previous research [38]. By Equations (11) and (13), the parameter \(\psi\) (i.e., the potential of complete desorption) was determined as \(-1.64\) V, close to the experimental value.