# Direct Solar Thermal Water-Splitting Using Iron and Iron Oxides at High Temperatures: A Review
## 2 Theoretical Background
#### Kinetic Model
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Reaction**} & \multicolumn{1}{c}{**Gibbs Free Energy**} & \multicolumn{1}{c}{**Enthalpy**} & \multicolumn{1}{c}{**Temp. K**} & \multicolumn{1}{c}{**Refs.**} \\  & \(\Delta\)**G** **(kJ/mol)** & **AH** **(kJ/mol)** & **(pres. atm)** & **Refs.** \\ \hline \(0.75\,Fe+H_{2}O\ \to 0.25 Fe_{3}O_{4}+4H_{2}\) & \(-14.52\) & \(-32.09\) & \(600\) & [34] \\ \(Fe+2H_{2}O\ \to FeO+H_{2}\) & \(-199.3\) & - & \(1000\) & [35] \\ \(Fe+2H_{2}O\ \to Fe(OH)_{2(g)}+H_{2}\) & \(38.9+5\times 807\) (cal) & - & - & [28] \\ \(2 Fe+3H_{2}O\ \to Fe_{2}O_{3}+3H_{2}\) & \(48.9\) & - & \(973\)–\(1123\) & [36] \\ \(3 Fe+4H_{2}O\ \to Fe_{3}O_{4}+4H_{2}\) & \(11.0\) & - & \(973\)–\(1123\) & [36; 37] \\ \(3 FeO+H_{2}O\ \to Fe_{3}O_{4}+H_{2}\) & - & \(-33.6\) & \(873\) & [22; 38; 39] \\ \(Fe_{2}O_{3}+H_{2}O\ \to 2 Fe(OH)_{2}+2 FeO(OH)\) & \(10.39\) & \(-47.54\) & \(400\) & [34] \\ \(Fe_{3}O_{4}+2H_{2}O\ \to Fe(OH)_{2}+2 FeO(OH)\) & \(41.02\) & \(-79.23\) & \(400\) & [34] \\ \(Fe_{3}O_{4}+4H_{2}O\ \to Fe(OH)_{2}+2 Fe(OH)_{3}\) & \(228.22\) & \(-120.48\) & \(600\) & [34] \\ \(2 Fe_{3}O_{4}+H_{2}O\ \to 3 Fe_{2}O_{3}+H_{2}\) & \(40.5\) & - & - & [40] \\ \(2 Fe_{3}O_{4}+4H_{2}O\ \to 6 FeOOH+H_{2}\) & \(44.4\) & - & - & [40] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermodynamic properties of the reaction of iron or oxides in water Vapour. where \(\rm K_{w}\) is the parabolic rate constant in (g\({}^{2}\)/m\({}^{4}\)s), \(\rm F_{p}\) is the porosity factor of the inner layer, and \(\rm P_{H2O}\) and \(\rm P_{H2}\) are the partial pressures of \(\rm H_{2}O\) and \(\rm H_{2}\), respectively. [PERSON] et al. [46] studied the production of hydrogen using iron oxide and inert silica with \(\rm H_{2}O_{(g)}\) at temperatures ranging from 460 to 700 \({}^{\circ}\)C and vapour partial pressures between 0.002 and 0.02 MPa. Initially, Fe is oxidized to \(\rm Fe_{3}O_{\rm\AA}\); this is followed by the gradual oxidation of \(\rm Fe_{3}O_{\rm\AA}\) to \(\rm Fe_{2}O_{\rm\AA}\) in the second step. The researchers determined the equilibrium state of \(\rm H_{2}O_{(g)}\) adsorption and found that the rate-determining step of the reaction is the desorption of hydrogen from the active sites. The proposed initial oxidation rate when the partial pressure of water is fixed is represented by the following expression:  
\[\left(\frac{dX_{B}}{dt}\right)_{t=0}=\left(\frac{bk}{1+K_{e\_ads}.P_{H_{2}O}} \right)P_{H_{2}O}-\frac{\left(\frac{bk}{1+K_{e\_ads}.P_{H_{2}O}}\right)}{K_{e}} P_{H_{2}} \tag{9}\]  
where b is the stoichiometric coefficient of the oxidation reaction from Fe to \(\rm Fe_{2}O_{\rm\AA}\), k is the overall rate constant based on unit solid volume, \(\rm K_{e}\) is the chemical equilibrium constant, \(\rm K_{e\_ads}\) is the equilibrium adsorption constant, and \(\rm P_{H2O}\) and \(\rm P_{H2}\) are the partial pressures of water and hydrogen, respectively. The heterogeneous kinetic models have been detailed extensively by [PERSON] et al. [47], [PERSON] et al. [48], and [PERSON] et al. [49] However, [PERSON] & [PERSON] et al. [50] proposed a kinetic model for the non-catalytic heterogeneous reaction in a solid-gas system, considering both heat and mass transfer as combined effects. [PERSON] & [PERSON] et al. [51] analysed multiple mathematical models and developed a comprehensive model for a multi-reactant system using porous granules within a sample of a mobile packed bed. Our model accounts for various factors, including external mass transfer, internal diffusion through the pores, chemical reactions, heat generation or consumption due to reactions, and heat transfer through effective conduction within the solid matrix. The three main types of growth laws that have been established experimentally, the linear, parabolic and logarithmic laws, are illustrated in Table 3.  
Figure 2: Step diagram of the non-catalytic superficial heterogeneous reaction of the electrochemical dissolution of iron in water vapour.