## 3 Fuel Cell Performance Calculations
Substituting Equation (18) into Equation (33) gives the relation (the Nernst equation):  
\[E=E_{o}+\frac{R\cdot T}{n_{e}F}\cdot\ln\left[\frac{a_{A}^{m}a_{D}^{r}}{a_{A}^{ r}\cdot a_{B}^{k}}\right] \tag{34}\]  
Additionally, the reaction itself is not restricted through the outcome of water concentration on this surface. Therefore, the Nernst equation becomes:  
\[E=E_{o}+\frac{R\cdot T}{n_{e}F}\cdot\ln\left[\frac{\left(P_{H_{2}}/P_{o} \right)\cdot\left(P_{O_{2}}/P_{o}\right)^{0.5}}{\left(P_{H_{2}O}/P_{sat} \right)}\right] \tag{35}\]  
The molar fraction of the species (\(x_{i}\)) is merely the ratio of the partial pressure of the species of interest (\(P_{i}\)) and the total pressure (\(P_{total}\)):  
\[x_{i}=\frac{P_{i}}{P_{total}} \tag{36}\]  
Substituting Equation (36) in Equation (35), the Nernst Equation for Hydrogen Oxygen/Air FC becomes:  
\[P\cdot v=R\cdot T \tag{28}\]0  
The species chosen in the Nernst equation pressure term are only the species which participate in the overall electrochemical reaction. For example, the nitrogen in the cathode is not represented directly, since it does not participate in the electrochemical reaction. Inert species do have an effect on the Nernst voltage, through reduction of the mole fraction of the active species. Additionally, note that the activity values use the pressure of the electrode where the reacting species is located. That is, even though there is water at the anode, it is inert in terms of the hydrogen oxidation reaction and only a participant in the electrochemical reaction at the cathode [20]. The thermodynamic equilibrium condition regarding the electrochemical mode corresponds to the reversible potential obtained from the Nernst equation. Nonetheless, as the current begins to pass within the cell, the cell potential falls below the reversible potential because of the different forms of overpotential, such as activation, ohmic, and concentration overpotential. Additionally, at open circuits, PEM-FCs endure other losses--for instance, internal flows and fuel crossover and the usual electrochemical overpotentials. A standard polarization trajectory of a PEM-FC is displayed in \[P\cdot v=R\cdot T \tag{28}\]1 The standard figure of merit for measuring FC efficiency is the polarization curve, reflecting the cell voltage-current relationship. Voltage versus current density, scaled by geometric electrode area, is widely given so that the results can be increased to various cell sizes. Other losses control the curve, which can be divided into four regions. When the outer circuit in Region 1 is disconnected, fuel crossover loss occurs. At low current densities in Region 2, activation loss dominates. The ohmic loss governs Region 3, and the diffusion overpotential is responsible for Region 4, so the polarization curve falls downwards [21]. Valuable output (electrical energy) remains a concern for an FC simply when a current is drawn, though the exact cell potential (\(E_{cell}\)) is reduced to its equilibrium thermodynamic potential (\(E\)) due to irreversible losses. A deviation of the thermodynamic potential happens to correspond with the cell's electrical output at a current flow. The variation in equilibrium rate that exists is described as the overpotential and has been assigned the symbol (\(\eta\)). The overpotentials originate prime of activation overpotential (\(\eta_{act}\)), ohmic overpotential (\(\eta_{ohmic}\)), diffusion (or concentration) overpotential (\(\eta_{diff}\)), and crossover of fuel and oxidizer through the electrolyte (\(\eta_{x}\)).