## 3 Results and Discussion
### Propane Analysis
The first one comprises the mechanisms of Fermi level control, grain boundary barriers control, and electron-depletion and hole-accumulation layers. Additionally, the changes in electrical properties are accompanied by changes in physical properties such as energy bands and work functions. The macroscopic category includes mechanisms of adsorption-desorption patterns, bulk resistance control, and gas diffusion control. In the case of bulk resistance control, the change in the electrical properties of the sensors can be caused by phase transformations, which occur in certain types of materials. The gas diffusion control mechanism dictates that the most important factor in the diffusion process is related to the microstructure of the material, which, in turn, affects the surface chemical reactions and the sensing properties [61; 62]. The adsorption/desorption model is the current conventional mechanism, which states that the electron capture by adsorbed species, and their subsequent desorption, is responsible for changes in the sensor's electrical properties. This model includes oxygen adsorption, chemical adsorption/desorption, and physical adsorption/desorption. The theory of electron depletion and hole accumulation layers is an extension of the oxygen adsorption model. As mentioned in the discussion section, the material's sensitivity is primarily related to the oxygen chemisorption, where the formed ionic or molecular species are temperature dependent. Chemical adsorption/desorption also extends to the material's exposure to different gases [11; 20; 61; 62]. Various theories have been developed on the adsorption of gases. For example, [PERSON] and [PERSON]'s mathematical modeling of an isotherm for gas adsorbates, [PERSON]'s isotherm theory of adsorption of the unimolecular layer, the BET theory of multilayer adsorption, and dynamic adsorption and desorption processes based on [PERSON]'s theory [63]. Additionally, the adsorption of toxic small molecules has been theoretically investigated through adsorption configurations, adsorption energy, charge transfers, and electronic properties. Eight possible orientations of CO adsorption are documented for pentagraphene-based materials, where chemical adsorption is favored more than physical adsorption [64]. Likewise, in CO sensing using GaN, four stable gas adsorption configurations have been reported [65]. Theoretical studies on propane adsorption have also been conducted, where it did not have a significant site preference on the studied surfaces [66]. For the design of our 600-ppm propane detector, changes in \(|\mathbf{Z}|\) at 1 kHz and 360 \({}^{\circ}\)C were considered. The calculations shown in Figure 10c-e were conducted with the following formula:  
\[\mathbf{Z}(\mathbf{t})=|\mathbf{Z}(\mathbf{t})|e^{\mathbf{\theta}(\mathbf{t})}=\mathbf{r}(\mathbf{t})+i \mathbf{X}_{\mathbf{c}}=\frac{1}{2\pi\mathbf{f}\mathbf{C}} \tag{4}\]  
where \(\mathbf{i}\) is an imaginary operator, \(\mathbf{r}(\mathbf{t})=|\mathbf{Z}(\mathbf{t})|\cos\mathbf{\theta}(\mathbf{t})\) is the resistance (Figure 10c), \(\mathbf{X}_{\mathbf{c}}=|\mathbf{Z}(\mathbf{t})|\mathrm{sen}\mathbf{\theta}(\mathbf{t})\) is the reactance (Figure 10d), and \(\mathbf{C}(\mathbf{t})=\frac{1}{2\pi\mathbf{f}|\mathbf{Z}(\mathbf{t})|\mathrm{sen}\mathbf{\theta} (\mathbf{t})}\) is the capacitance in faradays (Figure 10e). Considering the values in Figure 10c-e, we developed our electrical circuit using the following equation:  
\[\mathbf{z}_{\mathbf{s}}(\mathbf{t})=\mathbf{z}_{\mathbf{c}}+\mathbf{z}_{\mathbf{d}}(\mathbf{t})=\mathbf{z}_{\mathbf{c }}+\frac{\mathbf{r}_{\mathbf{d}}(\mathbf{t})}{\mathbf{r}_{\mathbf{d}}(\mathbf{t})+\mathbf{X}_{\mathbf{d}}(\bm {t})} \tag{5}\]  
where \(\mathbf{z}_{\mathbf{s}}(\mathbf{t})\) is the sensor's impedance, \(\mathbf{z}_{\mathbf{c}}\) is the constant impedance, and \(\mathbf{z}_{\mathbf{d}}(\mathbf{t})\) is the sensor's dynamic impedance in the test gas.