## 3 Results
### Mathematical Data Processing and Acetic Acid Concentration Determination  
As reported in Section 3.3, sensors BIOS1 and BIOS2 respond both to acetic acid and butanol. It can be supposed that the resistance measured for each sensor, when they are exposed to an atmosphere containing both volatiles, will be a linear combination of the resistance variation for each gas. Thus, from a mathematical standpoint this can be expressed as:  
\[\left\{ \begin{array}{l}R_{spcr}^{Bis1}=\alpha(R_{AC}^{Bis1}-R_{but}^{Bis1})+R _{but}^{Bis1}\\ R_{spcr}^{Bis2}=\alpha(R_{AC}^{Bis2}-R_{but}^{Bis2})+R_{but}^{Bis2}\\ \end{array} \right. \tag{1}\]  
Figure 4: Response of BIOS1 and BIOS2 MOS sensors in terms of absolute resistance values when exposed to (**A**) acetic acid; (**C**) butanol. Obtained values are also reported in terms of percentage reduction compared to baseline when sensors are exposed to (**B**) acetic acid; (**D**) butanol. Each reported value is calculated as an average of three independent measures. where \(R_{sper}^{Biosi}\), and \(R_{sper}^{Biosi}\) are the resistance values detected by the two sensors, \(\alpha\) is the volume fraction of the acetic acid. \(R_{AC}^{Biosi},R_{but}^{Biosi},R_{AC}^{Biosi2},R_{but}^{Biosi2}\) are the calibration curves of BIOS1 and BIOS2, respectively, when exposed to Acetic acid (AC) or butanol (but). The calibrations curves obtained in Section 3.3 are approximated well by:  
\[\left\{ \begin{array}{l}R_{AC}^{Biosi1}=597,373\cdot x_{AC}{}^{-0.592}\\ R_{but}^{Biosi}=40,206\cdot x_{but}{}^{-0.442}\\ R_{AC}^{Biosi2}=411,714\cdot x_{AC}{}^{-0.484}\\ R_{but}^{Biosi2}=56,128\cdot x_{but}{}^{-0.322}\end{array} \right. \tag{2}\]  
where \(x_{AC}\) and \(x_{but}\) are the concentration of each volatile species. Due to the complexity and non-linearity of the system obtained, machine-assisted numeric resolution is needed to solve it and accurately calculate the acetic acid concentration in a multiple volatiles mixture. To solve this system, a software algorithm has been developed using an open-source mathematics-oriented syntax software called GNU Octave. The developed software takes as inputs the resistance values provided by the sensors and returns the total concentration of gases and the molar fraction of acetic acid, from which the acetic acid concentration can be calculated. To solve the system illustrated above, the developed algorithm uses the GNU Octave function _vpasolve_, which allows the resolution of the non-linear equation using a Newton-type algorithm. By rearranging the system as shown below in Equation (3):  
\[\left\{ \begin{array}{c}\left(R_{sper}^{Biosi}-R_{but}^{Biosi}\right)/\left(R _{AC}^{Biosi}-R_{but}^{Biosi}\right)=\left(R_{sper}^{Biosi2}-R_{but}^{Biosi2} \right)/\left(R_{AC}^{Biosi2}-R_{but}^{Biosi2}\right)\\ \alpha=\left(R_{sper}^{Biosi2}-R_{but}^{Biosi2}\right)/\left(R_{AC}^{Biosi2}-R_ {but}^{Biosi2}\right)\end{array} \right. \tag{3}\]  
Substituting the calibration curves obtained, shown in Equation (2), for the two sensors against each target volatile, we can write Equation (4):  
\[\left\{ \begin{array}{c}\left(R_{sper}^{Biosi}-40,206\cdot x^{-0.442}\right)/ \left(597,373\cdot x^{-0.592}-40,206\cdot x^{-0.442}\right)=\left(R_{sper}^{ Biosi2}-56,128\cdot x^{-0.322}\right)/\left(411,714\cdot x^{-0.484}-56,128 \cdot x^{-0.322}\right)\\ \alpha=\left(R_{sper}^{Biosi2}-56,128\cdot x^{-0.322}\right)/\left(411,714\cdot x ^{-0.484}-56,128\cdot x^{-0.322}\right)\end{array} \right. \tag{4}\]  
The numerical solution of Equation (4) gives as output the total concentration of released volatiles and the volume fraction of acetic acid \(\alpha\). It was previously seen that the two sensors only respond to acetic acid and butanol, and the two values \(x\) and \(\alpha\) can be put in a mathematical relationship, as shown in Equation (5):  
\[\left\{ \begin{array}{c}x=x_{AC}+x_{but}\\ \alpha=\frac{x_{AC}}{x}\end{array} \right. \tag{5}\]  
The algorithm can thus, using Equations (4) and (5), calculate the concentration of acetic acid (\(x_{AC}\)) and butanol (\(x_{but}\)).