Need help with Matlab: Adsorption isotherms are phase coexistence relationships that relate the partial pressure of a gas phase substance to the amount of gas molecules adsorbed inside a porous material $($i$.$e$.,$ loading$).$ In thermodynamics, we have learned how to predict the properties of two coexisting phases, such as vapor$-$liquid equilibria $($VLE$).$ If we replace the liquid phase with an adsorbed phase and treat the adsorbed phase using a formalism similar to Raoult$$s law,

we have the ideal adsorbed solution theory $($IAST$).$ This assignment and the next will walk you through a few important coding concepts. At its completion, you will be able to make your own IAST predictions.

Complete Part $(1)$ Part $(5)$ for this assignment.

Use water$\_$data, ehtanol$\_$data, and binary$\_$data when referencing data used

$1.$ Inspect the provided dataset by plotting the single$-$component adsorption isotherms,

Ni$($pi$),$ using the logarithmic scale for the horizontal axis. Use empty blue circles for

water $($i $=1)$ and empty red up triangles for ethanol $($i $=2).$

Note: For this part, you need to search for the plotting command and read its docu$-$

mentation to learn how to use it$.$

Interpretation of the plot: If we flow water vapor at a given pressure, say p$1=$

$11302\mathrm{.}26$ Pa$,$ over silicalite, $1\mathrm{.}367$ water molecules will be adsorbed per unit cell of

the zeolite sorbent at equilibrium.

$2.$ For each of the two unary isotherms, write a function to implement Ni$($ln pi$)$ by fitting

to the given data points. You can either use a file or the anonymous function syntax

@$($x$)().$ A user needs to be able to call your function via, e$.$g$.,$ isotherm$1($t$)$ with

the independent variable t $=$ ln pi$.$ pi can be any pressure not necessarily from the

provided data set.

Note: You may want to read the documentation of F $=$ griddedInterpolant$($x$,$ y$,$

$$linear$).$ To use the returned object with fsolve in $(3),$ you can create a true

function isotherm$1($t$)=$@$($t$)$F$($t$).$

$3.$ Find the roots ln p$1,0$ and ln p$2,0$ for the two isotherm functions using fsolve. In the

same figure of $(1),$ plot your fitted functions from ln pi$,0$ to $2$ ln pi$,$max, where pi$,$max is

the largest pressure from the provided dataset.

In a binary adsorption experiment, we put the water$/$ethanol mixture in contact with the

sorbent, and both molecules will adsorb. Of course, they will do so in different quantities,

resulting in preferential adsorption that can be used to separate the mixture. For a mixture

at given T and p$,$ the specification of its composition $($e$.$g$.,$ expressed in terms of the mole

fraction of ethanol, x$2)$ then completely fixes the chemical potentials of water and ethanol,

which can be converted to equivalent partial pressures. The first two columns of the mixture

isotherm data give the partial pressures, p$1$ and p$2.$ The last two columns give the measured

loadings for water $($N$1)$ and ethanol $($N$2).$

$4.$ In a new figure, plot both water and ethanol loadings as a function of p$2,$ using the

same symbol conventions as in $(1).$

$1$

The IAST equations for a C$-$component mixture are as follows:

p$0$

i zi$\backslash$gamma i $=$ fi $$ pi$,$ i $=1...$ C $(1)$

$\backslash$Psi $=\backslash$pi A

RT $=$

Z ln p$0$

$1$

ln p$1,0$

N$1($ln p$1)$ d ln p$1=$

Z ln p$0$

i

ln pi$,0$

Ni$($ln pi$)$ d ln pi$,$ i $=2...$ C $(2)$

where A is the $($unkwown$)$ surface area of the sorbent material, R is the gas constant, and

T is temperature. Eqn. $2$ also defines spreading pressure $\backslash$pi and adsorption potential $\backslash$Psi $.$

When using IAST to predict binary adsorption $($C $=2)$ from single$-$component isotherms

Ni$($ln pi$),$ the partial pressures $($pi$)$ are specified, and adsorbed$-$phase activity coefficients $(\backslash$gamma i$)$

are assumed to be unity. Using the three independent equations $($two of Eqn. $1$ and one of

Eqn. $2),$ one can solve for the adsorbed$-$phase composition, z$1,$ and the two pressures, ln p$0$

i $.$

$5.$ Using the two fitted isotherm functions, Ni$($ln pi$),$ write another two functions $($one for

water and the other for ethanol$)$ to compute the adsorption potential as a function of

ln p$0$

i : $\backslash$Psi i$($ln p$0$

i $)=$ R ln p$0$

i

ln pi$,0$ Ni$($ln pi$)$ d ln pi$,$ where the integration lower limit is what you

found in $(3).$ That is$,$ a user needs to be able to call ads pot$($t$),$ where t $=$ ln p$0$

i $,$ to

obtain the value of the integral.