# Electrochemical Ion Pumping Device for Blue Energy Recovery: Mixing Entropy Battery
## 2 Theoretical Background for Mixing Process
The theoretical \"voltage rise\" for this lithium device can be calculated from the Nernst equation, according to the following expression [22]:  
\[\Delta G_{mix}=-(n_{h}+n_{l})T\Delta S_{mix,m}-\left(-n_{h}T\Delta S_{mix,h}-n _{l}T\Delta S_{mix,l}\right) \tag{2}\]9  
where \(z\) is the number of electron transferred in the reaction, and subscripts \(h\) and \(l\) refer to concentrated (brine) and diluted (recovery) solutions respectively. The potential difference between the two electrodes is [23]:  
\[\Delta S_{mix}=-R\sum_{i}x_{i}lnx_{i} \tag{3}\]0  
where \(E^{0}_{Cell}\) is the standard cell voltage, \(C_{LiCl}\) is the concentration of \(LiCl\) and \(\gamma_{LiCl}\) is the mean activity coefficient of \(LiCl\). The dependence of \(\gamma_{LiCl}\) on \(C_{LiCl}\) its described by the Debye-Huckel law [18], which is as follows:  
\[\Delta S_{mix}=-R\sum_{i}x_{i}lnx_{i} \tag{3}\]1  
where \(A\) and \(B\) are the constants. A mixing entropy battery device consists of a reversible electrochemical system, where the electroactive ions present in the natural solution are pseudo capacitively stored in their respective electrodes. These electrodes work selectively by capturing dissolved ions, where cationic ions (\(M^{+}\)) such as the \(Na^{+}\) (1.02 A), \(Li^{+}\) (0.76 A) or \(Mg^{2+}\) (0.65 A) [24; 25] (or trivalent ions, such as \(Al^{3+}\) and polyvalent (\(B^{n+}\)) as well) interact with the cathode electrode and anionic ions (\(A^{-}\)), such as the ion interacting with the anodic electrode. The CAPMIX and the mixing entropy battery are also known as the Mix Accumulator or AccMix [23]. The mixing entropy battery proposed by [PERSON] operates under a process by which anionic and cationic electrodes, selectively interact with \(Cl^{-}\) and \(Na^{+}\) ions in several steps. In step 1, the battery is charged when \(M^{+}\) and \(A^{-}\) ions are removed from the respective electrodes in a low ionic strength solution. In step 2, the recovery solution is exchanged for the concentrated solution, generating an increase in cell voltage or open circuit potential; this phenomenon is known as \"Voltage Rise\" [1, 2, 4, 5, 7, 11, 19, 22, 26]. In step 3, the battery is discharged with this high potential, capturing the \(M^{+}\) ions and \(A^{-}\) from the concentrated solution, interspersing these ions within the crystalline structure of both electrodes at a constant current and in a closed circuit, generating energy. Finally, in step 4, the concentrated solution is exchanged for a recovery solution with low ionic force, leaving the device ready to start a new cycle. During steps 2 and 4, there is no power consumption or generation. Energy extraction, \(W\), is given by the following expression:  
\[\Delta S_{mix}=-R\sum_{i}x_{i}lnx_{i} \tag{3}\]2  
where \(I\) is the applied current, \(q\) is the charge, and \(t\) is the time. During the ion capture and release stage, energy extraction (\(W\)) is given by the sum of thermodynamic and kinetic contributions according the following expression:  
\[\Delta S_{mix}=-R\sum_{i}x_{i}lnx_{i} \tag{3}\]3  
where \(\Delta E_{eq}\) is the equilibrium potential of the electrochemical system, \(\eta\) is the overpotential of the device, and \(W_{th}\) and \(W_{k}\) are the energy consumption products of the thermodynamic and kinetic contribution, respectively. Kinetic effects are usually related to overpotentials and are essential in determining the amount of energy extracted, on the other hand, kinetic limitations come hand in hand with mass diffusion processes.