## 3 Results and Discussion
### Transport Numbers in SZY Membranes
#### 3.2.2 Conductivity and Transport Numbers of Oxygen Ions as Functions of the Oxygen  
Partial Pressure  
Total conductivity of a typical acceptor doped oxide includes ionic, hole and electron contributions and can be written as:  
\[\sigma=\sigma_{\text{O}^{2-}}+\sigma_{\text{ho}}p\text{O}_{2}^{1/4}+\sigma_{ \text{eo}}p\text{O}_{2}^{-1/4} \tag{7}\]  
where \(\sigma_{\text{ho}}\) and \(\sigma_{\text{eo}}\) denote the conductivity of electron holes and electrons, respectively, at \(p\text{O}_{2}=1\) atm, \(\sigma_{\text{O}^{2-}}\) is the oxygen ion conductivity which remains invariable with \(p\text{O}_{2}\) change. Three parameters, \(\sigma_{\text{O}^{2-}}\), \(\sigma_{\text{ho}}\) and \(\sigma_{\text{eo}}\), are required to calculate the total conductivity and therefore the transport number of oxygen ions as functions of \(p\text{O}_{2}\) using Equation (7). These parameters can be determined from the measurements on the gas concentration cells as follows. The effective total conductivity of the mixed-conducting membrane separating air and hydrogen atmospheres can be obtained by integrating Equation (7) over the corresponding \(p\text{O}_{2}\) range:  
\[\overline{\sigma}(\text{air}/\text{H}_{2})=\sigma_{\text{O}^{2-}}+\frac{ \sigma_{\text{ho}}\int\limits_{p\text{O}_{2}\text{,}\text{Air}}^{p\text{O}_{ 2}\text{,}\text{air}}p\text{O}_{2}^{1/4}\text{d}p+\sigma_{\text{eo}}\int \limits_{p\text{O}_{2}\text{,}\text{H}_{2}}^{p\text{O}_{2}\text{,}\text{air}} p\text{O}_{2}^{-1/4}\text{d}p}{p\text{O}_{2}(\text{air})-p\text{O}_{2}(\text{H}_{2})} \tag{8}\]  
To simplify the task, consider the membrane exposed to the \(p\text{O}_{2}\) gradient in oxidizing atmospheres, for example in the air/argon gradient. In this case, Equation (7) can be simplified by neglecting the last contribution, so that Equation (8) can be transformed to:  
\[\overline{\sigma}(\text{air}/\text{Ar})=\sigma_{\text{O}^{2-}}+\frac{\sigma_ {\text{ho}}\int\limits_{p\text{O}_{2}\text{,}\text{Ar}}^{p\text{O}_{2}\text{,}\text{air}}p\text{O}_{2}^{1/4}\text{d}p}{p\text{O}_{2}(\text{air})-p\text{O }_{2}(\text{Ar})} \tag{9}\]  
The value of \(\sigma_{\text{O}^{2-}}\) can be obtained using the following expression:  
\[\overline{t}_{\sigma^{2-}}(\text{air}/\text{Ar})=\frac{\sigma_{\sigma^{2-}}} {\overline{\sigma}(\text{air}/\text{Ar})} \tag{10}\]  
relating the oxygen-ion conductivity to the effective transport number and the effective conductivity in the membrane exposed to the air/argon gradient, which was obtained experimentally. Now the value of \(\sigma_{\text{ho}}\) can be easily derived from Equation (9):  
\[\sigma_{\text{ho}}=\frac{(\overline{\sigma}(\text{air}/\text{H}_{2})-\sigma_ {\text{O}^{2-}})\Delta p\text{O}_{2}}{\int\limits_{p\text{O}_{2}\text{,} \text{Air}}^{p\text{O}_{2}/4}\text{d}p} \tag{11}\]  
where \(\Delta p\text{O}_{2}=p\text{O}_{2}\) (air) \(-p\text{O}_{2}\) (Ar) = 21 kPa \(-\) 10 Pa \(\approx\) 21 kPa. Finally, the factor \(\sigma_{\rm eo}\) can be calculated from Equation (8) as follows:  
\[\sigma_{\rm eo}=\frac{(\vec{\sigma}(\rm air/H_{2})-\sigma_{\rm O^{2-}})\Delta pO _{2}-\sigma_{\rm ho}\int\limits_{pO_{2},H_{2}}^{pO_{2},\rm air}pO_{2}^{1/4} \mathrm{d}p}{\int\limits_{pO_{2},H_{2}}^{pO_{2},\rm air}pO_{2}^{-1/4}\mathrm{d}p} \tag{12}\]  
Using the obtained parameters of \(\sigma_{\rm O^{2-}}\), \(\sigma_{\rm ho}\) and \(\sigma_{\rm eo}\), the total, oxygen-ion, electron and hole conductivities, \(\sigma\), \(\sigma_{\rm O^{2-}}\), \(\sigma_{\rm e}\) and \(\sigma_{\rm h}\), as well as the transport numbers of oxygen ions, electrons and holes, \(t_{\rm O^{2-}}\), \(t_{\rm e}\) and \(t_{\rm h}\), can be calculated at any \(p\)O\({}_{2}\) value.