## 3 Fuel Cell Performance Calculations
\[\Delta H=\left(h_{f}\right)_{H_{2}O}-\left(h_{f}\right)_{H_{2}}-\
icefrac{{1}}{ {2}}\left(h_{f}\right)_{O_{2}} \tag{12}\]  
Likewise, as \(\Delta H\), concerning the reaction, signifies the variation within the heats of the production of outcomes and reactants, \(\Delta S\) equals the contrast among entropies regarding products and reactants:  
\[\Delta S=\left(s_{f}\right)_{H_{2}O}-\left(s_{f}\right)_{H_{2}}-\
icefrac{{1}}{ {2}}\left(s_{f}\right)_{O_{2}} \tag{13}\]  
Generally, electrical work yields the following results in terms of charge and potential:  
\[W_{el}=q\cdot E \tag{14}\]  
where \(W_{el}\) is the electrical work (J/mole), \(q\) is the charge (Coulombs/mole), and \(E\) is the potential (E.M.F. of the basic FC) (Volts). The entire charge conveyed into an FC reaction per mole of H\({}_{2}\) employed is equal to:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]0  
Electrical work occurs accordingly:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]1  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]2 Ideal operation of a PEM-FC. As specified earlier, the maximum measure concerning electrical energy produced within the FC resembles Gibbs free energy, \(\Delta G\):  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]3  
The FC's theoretical potential is thus:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]4  
Thus, \(\Delta H\) and \(\Delta S\) denote functions regarding temperature. The specific enthalpy toward any species is displayed through the reaction, which is provided by:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]5  
Through constant pressure, the particular entropy concerning temperature \(T\) exists as follows:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]6  
where \(C_{p}\) shows the specific heat (J/(moic\(\cdot\)K)), which is also a function of temperature of any gas. It is possible to adopt the following empirical relationship [17]:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]7  
where the empirical coefficients \(a\), \(b\), and \(c\), listed in Table 2, are O\({}_{2}\), H\({}_{2}\), and H\({}_{2}\)O [2]. Substituting Equation (21) into Equations (18) and (19), and applying to Equations (12) and (15), integrates yields:  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]8  
\[O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{8}\]9  
where the differences between the coefficients a, b, and c are \(\Delta a\), \(\Delta b\), and \(\Delta c\), respectively, for reactants and products. These coefficients can be calculated by:  
\[2H_{2}+O_{2}\to 2H_{2}O \tag{9}\]0  
\[2H_{2}+O_{2}\to 2H_{2}O \tag{9}\]1  
\[2H_{2}+O_{2}\to 2H_{2}O \tag{9}\]2  
The difference within Gibbs free energy (\(G\)) can be seen to be during an isothermal rule [18]:  
\[2H_{2}+O_{2}\to 2H_{2}O \tag{9}\]3  
where \(P\) is the pressure (Pa), and \(v\) is the molar volume (m\({}^{3}\)/mole). \begin{table}
\begin{tabular}{c c c c} \hline  & \(a\) & \(b\) & \(c\) \\ \hline O\({}_{2}\) & 25.84512 & 0.012987 & \(-3.9\times 10^{-6}\) \\ H\({}_{2}\) & 28.91404 & \(-0.0008400\) & \(2.01\times 10^{-6}\) \\ H\({}_{2}\)O\({}_{(\text{gas})}\) & 30.62644 & 0.009621 & \(1.18\times 10^{-6}\) \\ \hline \end{tabular}
\end{table} 
Table 2: The \(C_{p}\) ’s temperature dependency coefficients. For an ideal gas:  
\[P\cdot v=R\cdot T \tag{28}\]  
Therefore,  
\[dG=R\cdot T\cdot\frac{dP}{P} \tag{29}\]  
After integration:  
\[G=G_{o}+R\cdot T\cdot\ln\left(\frac{P}{P_{o}}\right) \tag{30}\]  
For any chemical reaction:  
\[jA+kB\to mC+rD \tag{31}\]  
The difference within Gibbs free energy exists in the difference among outcomes and reactants:  
\[\Delta G=mG_{c}+rG_{D}-jG_{A}-kG_{B} \tag{32}\]  
By substituting into Equation (30) [19],  
\[\Delta G=\Delta G_{o}+R\cdot T\cdot\ln\left[\frac{a_{C}^{m}a_{D}^{r}}{a_{A}^{ r}\cdot a_{B}^{k}}\right] \tag{33}\]  
where the \(a\) are the thermodynamic motion coefficients toward the reacting varieties.