## 4 Selecting a Suitable Metal Oxide
### Removal Efficiency  
The reaction (Equation (1)) has a high equilibrium constant which means the forward reaction is favorable (i.e., removal of H\({}_{2}\)S). The equilibrium constant for reaction (Equation (1)) is defined as \(K=\left[\mathrm{H_{2}O}\right]/\left[\mathrm{H_{2}S}\right]\) and obeys the thermodynamic relation:  
\[K=\exp\left(-\frac{\Delta G}{RT}\right) \tag{11}\]  
where \(\Delta G\) is the standard Gibbs free energy change of reaction (Equation (1)), \(T\) is the temperature and \(R\) is the gas constant. Table 2 gives \(\Delta G\) for a few common metal oxides undergoing reaction (Equation (1)). The values given in Table 2 were calculated from the Gibbs free energy of formation of the reactants and the products, that can be found in [65]. Notice that reaction (Equation (1)) is exothermic. Hence, with increasing temperature the equilibrium constant decreases and the equilibrium becomes less favorable for the metal sulfide. In addition to the equilibrium, there always exist kinetic limitations that prevent reactions from reaching equilibrium in a given reactor. The intrinsic reaction rate (i.e., the reaction rate without transport limitations) of reaction (Equation (1)) is commonly expressed as:  
\[r=k_{s}C_{\mathrm{H_{2}S}}^{n} \tag{12}\]  
where \(r\) is the reaction rate defined as the rate of consumption of H\({}_{2}\)S per unit surface area of sorbent, \(k_{s}\) is the intrinsic rate function, \(C_{\mathrm{H_{2}S}}\) is the concentration of H\({}_{2}\)S at the surface of the sorbent and \(n\) is the reaction order. In order to decouple the effects of binders added to the sorbent some authors write the intrinsic reaction rate as [50; 66]:  
\[r=k_{s}^{\prime}C_{\mathrm{MO}}C_{\mathrm{H_{2}S}}^{n} \tag{13}\]  
where \(C_{\mathrm{MO}}\) is the molar concentration of the metal oxide in the solid sorbent. The reaction rate is typically measured by placing a small amount of sorbent in powder form in a gravimetric chamber whose atmosphere is kept at a constant H\({}_{2}\)S concentration, i.e., by continuously flushing the chamber with a gas stream containing a defined amount of H\({}_{2}\)S. The reaction rate is then deduced from the increase in solid weight over time. Using such an approach, most studies found the reaction (Equation (1)) to be of the first order with respect to H\({}_{2}\)S [48; 50; 51; 66; 67; 68; 69; 70; 71; 72]. An exception is copper-based sorbents that showed a reaction order smaller than one [72; 73]. For all sorbents, the rate function \(k_{s}\) (and \(k_{s}^{\prime}\)) are found to follow Arrhenius law:  
\[k_{s}=k_{0}\exp\left(-\frac{E_{a}}{RT}\right) \tag{14}\]  
where \(k_{0}\) is the kinetic prefactor and \(E_{a}\) is the activation energy. Table 3 gives an overview of the reaction rate for different metal oxides together with the temperature range over which the reaction rate was measured and the method that was used to determine the kinetic parameters. In cases where the reaction rate is reported in the form of Equation (13) we estimated \(C_{\mathrm{MO}}\) from the molar density of the solid sorbent [49] or from the weight fraction of MO \begin{table}
\begin{tabular}{c c} \hline \hline
**Desulfurization Reaction** & \(\Delta G\) **(kJ/mol)** \\ \hline ZnO + H\({}_{2}\)S \(\rightleftharpoons\) ZnS + H\({}_{2}\)O & \(-\)61.3 \\ MnO + H\({}_{2}\)S \(\rightleftharpoons\) MnS + H\({}_{2}\)O & \(-\)32.9 \\ CuO + H\({}_{2}\)S \(\rightleftharpoons\) CuS + H\({}_{2}\)O & \(-\)110.6 \\ CaO + H\({}_{2}\)S \(\rightleftharpoons\) CaS + H\({}_{2}\)O & \(-\)64.6 \\ FeO + H\({}_{2}\)S \(\rightleftharpoons\) FeS + H\({}_{2}\)O & \(-\)43.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Gibbs free energy change for the sulfidation of different metal oxides. and the true sorbent density [66]. Also, in cases where the experimental data was evaluated using the grain model (see Section 5) we estimated the grain radius as \(R_{g}=3/(S\rho)\), [51; 67] where \(S\) is the specific surface area and \(\rho\) is the sorbent density (see Equation (A1)).