$(35$p$)$ Q$1.$ Imagine a tank containing liquid A$.$ The liquid is in contact with soluble gas B$.$ As the gas is soluble, it dissolves at the free surface of the liquid and then diffuses into it$.$ While it diffuses, it also undergoes a first order chemical reaction. The transport of the gas within the liquid is purely diffusional i$.$e$.,$ no convection.

Prepare the framework of a mathematical model to determine the unsteady$-$state concentration distribution of the soluble gas $B$ inside liquid $A.$ Keep in mind that liquid A does not contain any soluble gas $B$ in the beginning.

Using these variables as nomenclature:

$D=$ Diffusivity

$k-$ first order reaction rate constant

$C^{**}=$ physical solubility of the gas in the liquid i$.$e$.$ solute gas concentration at the interface

Answer the following questions:

$($a$)$ Prepare a neat sketch to explain the process.

$($b$)$ One of the assumptions of this model is interfacial equilibrium. What are the other assumptions of this model?

$($c$)$ Write the governing partial differential equation $($PDE$)$ of the model.

$($d$)$ Write the initial condition $($IC$)$ and boundary condition $($BC$)$ of the model.

$($e$)$ Introduce a dimensionless concentration variable $($call it $\frac{?}{\text{b}\text{a}\text{r}}(C)).$ Write the new PDE, IC and BC of the model based on the $\frac{?}{\text{b}\text{a}\text{r}}(C).$

$($f$)$ Considering a new variable hat$(C)=\frac{e^k}{b}ar(C)$ write the new governing PDE, IC and BC of the model based on the hat$(C).$

$($g$)$ On what variables would $\frac{?}{\text{b}\text{a}\text{r}}(C)$ depend on in the final equation?