# Electrochemical Ion Pumping Device for Blue Energy Recovery: Mixing Entropy Battery
## 2 Theoretical Background for Mixing Process  
[PERSON], in 1954, was the first to propose the mixture between river water and seawater as an interesting alternative renewable energy source, with an electrochemical device called the Hydroelectric Pile [16]. Later, [PERSON] in 1976, proposed an electrochemical concentration cell design capable of recovering entropy energy from the mixture of solutions with different ionic strengths [17]. More recently, [PERSON] in 2009 proposed a novel method based on an electric double-layer (EDL) capacitor called CAPMIX [4]. The mixing entropy battery is a new concept of recovering chemical energy from salinity gradient based on charge storage as chemical energy inside the electrode materials. This process was described by [PERSON] et al. [5] in 2011 as electrochemical capacitors with a faradic charge transfer process [18] delimited by a Nernst potential associated to salt concentration [2; 5; 6; 19]. The \"blue energy\" that can be extracted from the mingling of fresh water and seawater is best illustrated by its reverse process, \" desalination\" energy that is required to extract fresh water from seawater. The theoretical non-expansion work that can be produced from mixing a relatively concentrated salt solution \(h\) (seawater) and a dilute salt solution \(l\) (river water), at constant pressure \(p\) and absolute temperature \(T\), to give a brackish solution \(m\), is defined by the Gibbs free energy of mixing \(\Delta G_{mix}\)[5; 20]. \[\Delta G_{mix}=\Delta G_{m}-(\Delta G_{h}+\Delta G_{l}) \tag{1}\]  
Assuming that solutions are ideal dilute, the Gibbs free energy of the mixing process can be calculated just from the charge in molar entropy (i.e., \(\Delta H_{mix}=0\)) [18; 20]:  
\[\Delta G_{mix}=-(n_{h}+n_{l})T\Delta S_{mix,m}-\left(-n_{h}T\Delta S_{mix,h}-n _{l}T\Delta S_{mix,l}\right) \tag{2}\]  
where \(n\) is the amount (moles), \(T\) is the temperature, and \(\Delta S_{mix}\) represents the contribution of the molar entropy of mixing to the total molar entropy of the mixing electrolyte solution (J/mol.K), according to:  
\[\Delta S_{mix}=-R\sum_{i}x_{i}lnx_{i} \tag{3}\]  
where \(R\) is the universal gas constant (8.314 J/mol.K) and \(x\) is the mole fraction of component \(i\left(i=Na^{+},\;Li^{+},\;Mg^{2+},\;Cl^{-},\;H_{2}O\right)\). The mixing entropy battery operates through the Gibbs free energy change, for transferring a species \(i\) from a system 1 to a system 2 (\(\Delta G_{i\,1,2}\)) in isothermal conditions, which can be obtained by adding the Gibbs free energy changes of the two systems separately. \[\Delta G_{i\,1,2}=\Delta G_{i,1}+\Delta G_{i,2}=G_{i,1_{F}}-G_{i,1_{I}}+G_{i,2 _{F}}-G_{i,2_{I}} \tag{4}\]In terms of composition it can be calculated as follows:  
\[\frac{\Delta G_{i\;1,2}}{RT}=n_{i,1_{F}}\;ln\left(C_{i,1_{F}}\right)-n_{i,1_{I}} \;ln\left(C_{i,1_{I}}\right)+n_{i,2_{F}}\;ln\left(C_{i,2_{F}}\right)-n_{i,2_{I} }\;ln\left(C_{i,2_{I}}\right) \tag{5}\]  
where \(F\) and \(I\) are final and initial conditions of each system, respectively. On the other hand, the conservation of mass for the transferred species can be represented by the following expression:  
\[n_{i,1_{F}}-n_{i,1_{I}}=n_{i,2_{F}}-n_{i,2_{I}}=\Delta n_{i} \tag{6}\]  
Substitution of (6) into (5) and carrying out some mathematical arrangements gives the following:  
\[\frac{\Delta G_{i\;1,2}}{RT}=n_{i,1_{F}}\;ln\left(\frac{C_{i,1_{F}}}{C_{i,1_{I }}}\right)+n_{i,2_{I}}\;ln\left(\frac{C_{i,2_{F}}}{C_{i,2_{I}}}\right)+\Delta n _{i}\;ln\left(\frac{C_{i,2_{F}}}{C_{i,1_{F}}}\right) \tag{7}\]  
In the present work system \"1\" represents the brine, and system \"2\" the recovery solution or fresh water. So, the following approximations can be made:  
\[n_{i,2_{F}}-n_{i,2_{I}}\;\cong\;n_{i,2_{F}} \tag{8}\]  
\[n_{i,1_{F}}-n_{i,1_{I}}\;\cong\;0 \tag{9}\]  
This simplifies Equations (7)-(10), used to calculate the Gibbs free energy change, for transferring a species \(i\) from brine to the recovery solution in isothermal conditions.