## 2 PEM Fuel Cell  
As shown in Figure 1, a PEMFC is composed of a catalyst layer (CL), a gas diffusion layer (GDL), and an electrolyte membrane. Each of these components is fabricated individually and then pressed to each other at high pressures and temperatures. The CL and GDL are placed on both anode and cathode sides. The electrolyte membrane permits only the appropriate ions (protons) to migrate toward the cathode. The PEMFC is supplied by pressurized hydrogen (H\({}_{2}\)) and oxygen (O\({}_{2}\)) as a fuel and generates electricity, water, and heat. The hydrogen atoms (H\({}_{2}\)) enter the PEMFC at the anode side, where the CL divides them into protons (H\({}^{+}\)) and electrons (\(e^{-}\)). The protons flow to the cathode via the electrolyte membrane, while the electrons flow through the external circuit to provide electric energy along the way. The oxygen atoms (O\({}_{2}\)) enter the PEMFC at the cathode side and react with electrons returning from the external circuit and with protons that have traveled through the membrane to produce water and heat [60]. The electrochemical reactions occurring on the electrodes can be described in Equations (1)-(3). The first and the second equations show the anode and the cathode side reactions, respectively, and the third equation shows the overall electrochemical reaction [61]. \[H_{2} \implies 2H^{+} + 2e^{-} \tag{1}\] \[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\] (2) \[H_{2} + \tfrac{1}{2}O_{2} \implies H_{2}O + \text{Energy.} \tag{3}\]  
The energy of Equation (3) is called the enthalpy of formation \(\triangle H\). It can be divided into two kinds of energies: the first one is the thermal energy represented by the specific entropy \(\triangle S\) multiplied by the temperature \(T\), and the second is the useful work \(\triangle G\). \(\triangle G\) is also called the negative thermodynamic potential (or Gibbs free energy). Therefore, the total energy as given in [62] is  
\[\triangle H=\triangle G+T\times\triangle S. \tag{4}\]  
\(\triangle G\) can be extracted as an electric work, defined by the charge \(Q\) across the potential \(E\). \(Q\) is the number of electrons (released from the anode), multiplied by the Faraday constant \(F\). Therefore, the useful work can be calculated by Equation (5):  
\[\triangle G=-Q\times E=-2\times F\times E. \tag{5}\]  
Figure 1: Fuel cell operation diagram. Using Equations (4) and (5), the PEMFC potential can be calculated by Equation (6), where \(\bigtriangleup G\), \(\bigtriangleup H\), and \(\bigtriangleup S\) are negative due to the exothermic reaction (yields energy). \[E=-\frac{\bigtriangleup G}{2\times F}=-\frac{\bigtriangleup H-T\times\bigtriangleup S }{2\times F}. \tag{6}\]  
The values of the useful work \(\bigtriangleup G\), which is given in Equation (5), also depend on the reactants. Therefore, it can also be calculated using Equation (7):  
\[\bigtriangleup G=\bigtriangleup G^{0}-RT[ln(P_{\mathrm{H_{2}}})+\frac{1}{2} ln(P_{\mathrm{O_{2}}})] \tag{7}\]  
where \(R\) is a universal constant, \(P_{\mathrm{H_{2}}}\) and \(P_{\mathrm{O_{2}}}\) are, respectively, the partial pressure of hydrogen and oxygen, and \(\bigtriangleup G^{0}\) is \(\bigtriangleup G\) at the standard condition. Therefore, by placing Equation (7) into Equation (6), the PEMFC potential can be given as  
\[E=-\frac{\Delta G}{2F}=-\frac{\Delta G^{0}}{2F}+\frac{RT}{2F}[ln(P_{\mathrm{H_{ 2}}})+\frac{1}{2}ln(P_{\mathrm{O_{2}}})] \tag{8}\]  
at the standard condition (\(T_{std}\) = 25 \({}^{\circ}\)C, \(1 atm\)).