## 3 Results and Discussions
\[t_{O^{2-}}=\frac{\sigma_{O^{2-}}}{\sigma_{\rm tot}}=\frac{\sigma_{\rm Ar}}{\sigma_{ \rm air}} \tag{4}\]  
The oxygen-ionic transport numbers for both BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) and BaLaInO\({}_{4}\) compositions did not change under temperature variation at \(p\)O\({}_{2}\) = 0.21 atm and were around 20% for each composition, and both compositions demonstrate mixed ionic-electronic (hole) conductivity under dry air, as it was shown earlier for the composition BaLaInO\({}_{4}\)[45]. Returning to Figure 4a, it can be seen that the values of oxygen-ionic conductivity for BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) were higher than for BaLaInO\({}_{4}\). It should the noted that the increasing in the oxygen-ionic conductivity values correlates well with the decreasing in the values of energy activation of oxygen transport (from 0.87 eV for BaLaInO\({}_{4}\) to 0.81 eV for BaLaInO\({}_{4}\)). Because the increase in the oxygen-ionic conductivity correlates with the increase in the number of perovskite layers in the structure, we can suppose that the transport of oxygen ions includes the oxygen positions in the In-containing polyhedra. Figure 4: The temperature dependencies of conductivity of BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\) (blue symbols) and BaLaInO\({}_{4}\) (black symbols) [29] obtained dry Ar (open symbols) and dry air (closed symbols) (**a**), wet Ar (open symbols) and wet air (closed symbols) (**b**), wet Ar (open symbols), and dry Ar (closed symbols) (**c**). The comparison of the dependencies of conductivity vs. oxygen partial pressure under dry and wet conditions is presented in Figure 5b. At the \(p\)O\({}_{2}\) range from 10\({}^{-4}\) to 10\({}^{-16}\) atm, the conductivity increases significantly in a wet atmosphere as a result of the appearance of a proton contribution due to the dissolution of H\({}_{2}\)O in BaLa\({}_{2}\)In\({}_{2}\)O\({}_{7}\). Water molecules can be dissolved in the oxygen-deficient perovskite layers leading to the formation of proton carriers as follows:  
\[\rm V_{o}^{\bullet\bullet}+H_{2}O+O_{o}^{\times}\Leftrightarrow 2(OH)_{o}^{\bullet} \tag{5}\]  
where \(\rm V_{o}^{\bullet\bullet}\) is the oxygen vacancy; \(\rm O_{o}^{\times}\) is the oxygen atom in the regular position; \(\rm(OH)_{o}^{\bullet}\) is the hydroxyl group in the oxygen sublattice. At the same time, for layered structures containing rock-salt blocks, the second mechanism of the dissociative dissolution of water molecules is possible:  
\[\rm H_{2}O+O_{o}^{\times}\Leftrightarrow(OH)_{o}^{\bullet}+(OH)_{i}^{\prime} \tag{6}\]  
where \(\rm(OH)_{O}^{\bullet}\) is the hydroxyl group in the regular oxygen position; \(\rm(OH)_{i}^{\prime}\) is the hydroxyl group located in the rock-salt block. This process is accompanied by the increase in the coordination number of atoms in the rock-salt block. The increase in the conductivity in a wet atmosphere compared to a dry atmosphere is reduced with increasing temperature. In a wet atmosphere below 450 \({}^{\circ}\)C, the conductivity does not depend on \(p\)O\({}_{2}\), which reflects the dominant ionic (proton) nature of the conductivity. Differences in the conductivities in dry and wet atmospheres at high \(p\)O\({}_{2}\)\(>\) 10\({}^{-4}\) atm are not so significant compared to the plateau region (Figure 5b). This is due to the presence of holes, the concentration of which decreases in a wet atmosphere:  
\[h^{\bullet}+\frac{1}{2}H_{2}O\ +O_{i}^{\prime\prime}\ \leftrightarrow\frac{1}{4}O_{2}+(OH)_{i}^{\prime} \tag{7}\]  
where \(h^{\bullet}\) is the hole; \(\rm O_{i}^{\prime\prime}\) is the oxygen atom in the interstitial position; \(\rm(OH)_{i}^{\prime}\) is the oxygen-hydrogen group. In accordance with the explanations presented above and the obtained data on conductivity-\(p\)O\({}_{2}\), the measurements of conductivity in a wide temperature range can be carried out by comparing the atmosphere of air and argon at different levels of humidity.