## 3 Results and Discussion
### Transport Numbers in SZY Membranes
#### 3.2.2 Conductivity and Transport Numbers of Oxygen Ions as Functions of the Oxygen
For illustration, the temperature dependences of the total, oxygen-ion and electron-hole conductivities of SZY in air (\(p\)O\({}_{2}\approx 21\) kPa) calculated with the help of Equations (7)-(12) are shown in Figure 8. For comparison, the total conductivity of the bulk SZY sample in air obtained from the impedance measurements and the conductivity of a bulk sample of SrZr\({}_{0.9}\)Y\({}_{0.1}\)O\({}_{3-8}\) measured in wet oxidizing conditions (wet oxygen, \(p\)H\({}_{2}\)O \(\approx 3000\) Pa) reported in [42] are also presented in Figure 8. It can be seen that the calculated and measured values of conductivity are in good agreement. The calculated and experimentally obtained values of the total, oxygen-ion and hole conductivity of the bulk SZY sample in humid air (\(p\)H\({}_{2}\)O = 3365 Pa) at 550 \({}^{\circ}\)C and the corresponding activation energies are given in Table 1.  
The \(p\)O\({}_{2}\)-dependences of the total conductivity and transport numbers of oxygen ions, holes and electrons in SZY calculated using Equations (2) and (7) are shown in Figure 9. The calculated conductivity satisfactorily agrees with the experimentally obtained values of conductivity of SZY in wet air and wet hydrogen (\(p\)H\({}_{2}\)O = 3365 Pa) that indicates the reliability of the proposed approach. For comparison, the conductivity of undoped SrZrO\({}_{3}\) as a function of \(p\)O\({}_{2}\) at 900 \({}^{\circ}\)C reported in [43] is shown in Figure 9a; it can be seen that the dependence is similar to those obtained in our research. Figure 9b shows that SZY possesses almost pure ionic conduction in a wide range of \(p\)O\({}_{2}\) at intermediate temperatures (500-700 \({}^{\circ}\)C), however the electrolytic area of SZY decreases with increasing temperature. \[\overline{\sigma}(\text{air}/\text{H}_{2})=\sigma_{\text{O}^{2-}}+\frac{ \sigma_{\text{ho}}\int\limits_{p\text{O}_{2}\text{,}\text{Air}}^{p\text{O}_{ 2}\text{,}\text{air}}p\text{O}_{2}^{1/4}\text{d}p+\sigma_{\text{eo}}\int \limits_{p\text{O}_{2}\text{,}\text{H}_{2}}^{p\text{O}_{2}\text{,}\text{air}} p\text{O}_{2}^{-1/4}\text{d}p}{p\text{O}_{2}(\text{air})-p\text{O}_{2}(\text{H}_{2})} \tag{8}\]0 The calculated total, oxygen–ion and hole conductivities and the measured total conductivity of the bulk SZY sample in humid air (\(p\)H\({}_{2}\)O = 3365 Pa) as functions of inverse temperature. The solid line presents the total conductivity of SrZr\({}_{0.9}\)Y\({}_{0.1}\)O\({}_{3-8}\) reported in [42]. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline  & \(\sigma\), & \(\sigma_{\text{O}^{2}\text{-}^{*}}\) & \(\sigma_{\text{h}_{\text{t}}}\) & \(\sigma_{\text{exp}_{\text{t}}}\) & \(\mathbf{E_{z}(\omega)}\), \(\mathbf{eV}\) & \(\mathbf{E_{z}(\omega_{\text{O}^{-}})}\), \(\mathbf{eV}\) & \(\mathbf{E_{z}(\omega_{\text{O}})}\), \(\mathbf{eV}\) & \(\mathbf{E_{z}(\omega_{\text{exp}})}\), \(\mathbf{eV}\) & \(\mathbf{eV}\) \\ \hline Our data & \(9.1\times 10^{-5}\) & \(7.6\times 10^{-5}\) & \(5.4\times 10^{-6}\) & \(1.2\times 10^{-4}\) & 1.03 & 0.94 & 1.58 & 0.92 \\ \hline Ref. [42]\({}^{1}\) & \(7.7\times 10^{-5}\) & & & 1.09 & & & \\ \hline \end{tabular}  
\({}^{1}\) Conductivity of a bulk SrZr\({}_{0.5}\)Y\({}_{0.1}\)O\({}_{3-.5}\) sample in wet oxygen (\(p\)H\({}_{2}\)O \(\approx\) 3000 Pa).  
\end{table} 
Table 1: The calculated values of the total, oxygen–ion and hole conductivity and the measured total conductivity of bulk SZY at 550 \({}^{\circ}\)C in humid air (\(p\)H\({}_{2}\)O = 3365 Pa) and the corresponding activation energies. Figure 9: Calculated (**a**) total conductivity and (**b**) transport numbers of oxygen ions, electrons and holes in SZY as functions of \(p\)O\({}_{2}\). Closed squares and triangles show the total conductivity of the bulk SZY sample in wet hydrogen and wet air, respectively, obtained from impedance data. Open squares show the conductivity data for SrZrO\({}_{3}\) reported in [43].