Mass transfert in process chemistry engineering, two film theory

I'm having troubles to understand a part of my lesson. I have no idea where to ask my question, but it is consider as chemistry so here we go.



> 
> Let me introduce some notations :
> 
> 
> $x$ and $y$ are the mole fractions
> of the same compound but respectively in liquid phase and in gas
> phase.
> 
> 
> $k\_{c\_l}$ and $k\_{c\_g}$ are the coefficient of exchange of matter in
> $\ce{m.s^{-1}}$
> 
> 
> $N\_{1n}$ is the molar flux density of the compound $1$ in the normal position
> 
> 
> 


Let now me introduce (translating from french to english) you the part of the lessons in which I have troubles :


No phenomenas of production or consommation of the compound $1$ occur in the interface $I$, and we are in stationnary state then the molar flux density is the same in both parts and we have $$N\_{1n}=c\_lk\_{c\_l}\left(x\_1^{I}-x\_1\right)=c\_gk\_{c\_g}\left(y\_1-y\_1^{I}\right)$$


The fluids at the interface are supposed to be in thermodynamical equilibrum then we have $$y\_1^I=K\_1^{eq}x\_1^I$$


[![two film theory](https://i.stack.imgur.com/g4EHV.png)](https://i.stack.imgur.com/g4EHV.png)


We can for example define a fictive gas phase which will be in equilibrium with the liquid phase (or the contrary). The mole fraction of the compound $1$ is by definition $$y\_1^\*=K\_1^{eq}x\_1$$


The total driving force is then $$\begin{align} (y\_1-y\_1^\*)&=(y\_1-y\_1^I)+(y\_1^I-y\_1^\*)\\ &=(y\_1-y\_1^I)+ K\_1^{eq}\left(x\_1^I-x\_1\right)\end{align}$$ if we suppose to have a unique average equilibrum constant in the liquid film.


**Now I'm having troubles their :**


We can then define a global transfer constant in the ralation to the driving force in gas phase : $$\frac{N\_{1n}}{K\_{c\_g}c\_g}=\frac{N\_{1n}}{k\_{c\_g}c\_g}+\frac{N\_{1n}K\_1^{eq}}{k\_{c\_l}c\_l}$$


I really don't understand this last relation, if someone can help me, I would be glad. I apologize in advance for the troubles you can have too, to understand what I tried to translate here.


Thank you


PS : I have really no idea for the tags, feel free to edit them




You already derived this relation (eq 1),  

\begin{align} (y\_1-y\_1^\*)&=(y\_1-y\_1^I)+(y\_1^I-y\_1^\*)\\ &=(y\_1-y\_1^I)+ K\_1^{eq}\left(x\_1^I-x\_1\right)\end{align}


Now, to keep the molar flux density between the gas phase concentration $y\_1$ and fictitious conc. on the other side $y\_1^\*$ we assumed a fictitious overall transfer coefficient $K\_{cg}$. So, now we have another relation for molar flux density  

$$N\_{1n}=K\_{cg}c\_g\times(y\_1-y\_1^\*)$$
that follows,
$$y\_1-y\_1^\*=\frac{N\_{1n}}{K\_{cg}c\_g}$$
Similarly,
$$y\_1-y\_1^I=\frac{N\_{1n}}{k\_{cI}c\_I}$$ 
and
$$x\_1^I-x\_1=\frac{N\_{1n}}{k\_{cg}c\_g}$$ 
Plug these above three relations into eq 1 and you will get your last equation.


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