## 3 Theory  
A typical segment of a chromatogram, as depicted in Figure 2, illustrates sample peaks corresponding to ethanol obtained from the alcoholic fermentation of dried fig syrup (12.5 \({}^{\circ}\)Be) at 18 \({}^{\circ}\)C, featuring the involvement of a wild type of _S. cerevisiae_.  
The observed sample peaks' height, denoted as \(H\), pertinent to the concentration of the ethanol, \(c\), near the intersection of the sample column and the diffusion column (\(x=l^{\prime}\)), at the time of each flow perturbation (\(t_{0}\)), is approximated by  
\[H\cong 2\;c\;\big{[}l^{\prime},\;t_{0}\big{]} \tag{1}\]  
where \(c\;\big{(}l^{\prime},\;t_{0}\big{)}\) is the solute concentration at \(x=l^{\prime}\) and time \(t_{0}\).  
The gaseous substance that comes out from a liquid phase, which is positioned near the bottom of the diffusion column, is transported until the point \(x=l^{\prime}\) of the sampling column. The only procedure of this mass transport is the diffusion phenomenon and is described by Fick's second law:  
\[\frac{\partial c}{\partial t_{0}}=D\;\frac{\partial^{2}c}{\partial z^{2}} \tag{2}\]  
Figure 1: Experimental configuration of reversed-flow gas chromatography (R.F.G.C.).  
Utilizing Fick's second law and solving this equation under the defined initial condition \(c(z,\,0)=0\), the boundary conditions in the diffusion column at \(z=0\), \(c(0,\,t_{0})=c(l,\,t_{0})\) and \(-D(\partial c/\partial z)_{z=0}=uc(l,\,t_{0})\) and the boundary conditions at \(z=L\), \(D(\partial c/\partial z)_{z=L}=k_{c}(c_{0}-c_{z_{L}})\), the expression of the substance concentration at the junction point of the diffusion and the sample column at various times is derived [47].  
\[c\left[l^{\prime},t_{0}\right]=\frac{k_{c}Dc_{0}}{v(k_{c}L+D)}\bigg{\{}1-\exp \left[-\frac{2(k_{c}L+D)t_{0}}{L^{2}}\right]\bigg{\}} \tag{3}\]  
where \(c_{0}\) is the concentration of the substance at the gas-liquid interphase in equilibrium with the bulk liquid phase, \(c_{z_{L}}\) is the concentration of the substance at the boundary layer of the gas-liquid interphase in equilibrium with \(c_{0},\,k_{c}\) is the mass transfer coefficient for the evaporation process, \(D\) is the diffusion coefficient of the substance into the carrier gas, \(v\) is the linear velocity of the carrier gas, and \(L\) is the length of the diffusion column in the R.F.G.C. system.  
The combination of Equations (1) and (3) yields  
\[H=2\frac{k_{c}Dc_{0}}{v\left(k_{c}L+D\right)}\,\left\{1-\exp\left[-\frac{2(k_{ c}L+D)t_{0}}{L^{2}}\right]\right\} \tag{4}\]  
Equation (4) predicts the elution curve's form. Moreover, an infinity value of \(H_{\text{max}}\) can be deduced for extended time periods [37]:  
\[H_{\text{max}}=\ \frac{2k_{c}Dc_{0}}{v\left(k_{c}L+D\right)} \tag{5}\]  
Here, \(H_{\text{max}}\) signifies the sample peak height at the plateau region of the elution curve. Notably, this expression indicates that peak height is inversely proportional to the concentration of the substance at the gas-liquid interphase, a quantity analogous to the substance's bulk concentration.  
The measured values of \(H_{\text{max}}\) can be utilized in a suitable mathematical expression for the kinetic phenomenon under investigation. For alcoholic fermentation, a first-order reaction rate expression aligns well with experimental results. The modified model, incorporating experimental peak height values, is expressed as follows [48]:  
\[\ln(H_{\text{maxo}}-H_{\text{max}})=\ln(H_{\infty})-k\ t_{0} \tag{6}\]  
Here, \(H_{\text{maxo}}\) denotes the sample peak value when the fermentation process concludes at extended times, and \(k\) represents the rate constant for the first-order alcohol-producing reaction.  
Figure 2: Reversed-flow gas chromatograph showing sample peaks of ethanol at 291.15 K.