## 3 Fuel Cell Performance Calculations  
The basic structure of a PEM-FC is two electrodes (anode and cathode) separated by a solid membrane that acts as an electrolyte. The membrane is coated on both sides with a thin catalyst layer (Figure 2). Hydrogen fuel moves within a network of channels into the anode, where it remains divided into protons (at the anode catalyst layer), which pass through the membrane to the cathode, and electrons are captured as an electrical current through an external circuit connecting the two electrodes. The oxidant (air) travels to the cathode through channels where oxygen reacts with electrons within the external circuit and protons passing into the membrane (at the cathode catalyst layer) to produce water. The following chemical reactions take place at the anode and cathode electrodes of a PEM-FC:  
Anode reaction:  
\[2H_{2}\to 4H^{+}+4e^{-} \tag{7}\]  
Cathode reaction:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]  
Overall:  
\[2H_{2}+O_{2}\to 2H_{2}O \tag{9}\]  
While these reactions can have multiple intermediate levels and some (unwanted) side reactions, they correctly reflect the key rules in an FC for the time being. The ultimate reaction is identical to the hydrogen combustion reaction. Combustion is an exothermic reaction, meaning that energy is emitted during the process. Water, DC electricity, and heat are the end products of this process. Total cell reaction:  
\[2H_{2}+O_{2}\to 2H_{2}O+electricity+heat \tag{10}\]  
A PEM-FC's behavior depends largely on the electrochemical reactions that happen with hydrogen and oxygen. It is considered that a system is in thermodynamic equilibrium when three conditions are simultaneously verified: chemical equilibrium, heat balance and mechanical balance. To explain this situation, we consider the basic thermodynamic functions of temperature, T, pressure, P, volume, V, internal energy, U, and entropy, S. In addition, we must also consider the thermodynamic state functions. These functions' main characteristic is that their value consists only of the system's current thermodynamic state, regardless of how the state was achieved. In this sense, several different state functions can be combined to establish another state function, known as Gibbs free energy, G.  
Thermodynamic Characteristics and Governing Equations  
The enthalpy, H, Gibbs free energy, G, internal energy, U, and entropy, S, are the basic thermodynamic functions. These are first-order homogeneous functions of the system's components, and they are comprehensive properties of a thermodynamic system. The system's intensive properties are pressure and temperature, which are zero-order homogeneous functions of the system's components. Figure 2: PEM-FC principles of operation. The FC's electrochemical effects change free energy differences linked by the chemical reaction toward electrical energy in a straight line. The Gibbs free energy difference (\(\Delta G\)) within a chemical reaction contains the highest network achievable chemical reaction [16]. \[\Delta G=\Delta H-T\Delta S \tag{11}\]  
where \(\Delta H\) and \(\Delta S\) show the enthalpy change and the entropy change. The potential free energy does equal to the lower enthalpy difference amount \(T\Delta S\), which describes the unavailable energy rising from the system's entropy difference. The quantity from the heat that does perform with an FC working reversibly is \(T\Delta S\). These functions can be seen in a graphical representation in Figure 3.  
The enthalpy (or heat) regarding the chemical reaction signifies the variation among the heats of the development of outcomes and reactants.