The steam methane reforming $($SMR$)$ reaction is commonly used to produce hydrogen:

CH$4+2$H$2$O $$ CO$2+4$H$2$

The reaction occurs at high temperatures, so you can assume that all species are in the gas phase. The real reaction typically takes place over a Ni$/$Al$2$O$3$ catalyst and is extremely complex to

model$1,$ but here we will greatly simplify it while using numbers from the real reaction wherever possible. In this problem, you will assume the reaction is in a batch reactor and use a

combination of materials balances, rate laws, and numerical integration of differential equations

to evaluate the conversion water to hydrogen via the water$-$gas shift reaction. You will work with partial pressures and can assume that the total pressure has been included in the rate constants.

$1.$ Assume that the reaction occurs reversibly, with a forward rate constant of kf $=0\mathrm{.}32$ min$-1,$ and a reverse rate constant of kr $=0\mathrm{.}001$ min$-1.$ Assume that both reaction orders follow the law of mass action and use initial partial pressures of pCH$4=$ pH$2$O $=0\mathrm{.}5,$ and pCO$2=$pH$2=0.$ Derive all relevant equations and numerically solve the resulting initial value problem to plot the pressures of H$2$O and H$2$ from t $=0$ to $60$ minutes. You should use the most accurate built$-$in solver available $($e$.$g$.$ ODE$45$ in Matlab, solve$\_$ivp in Python$)$

for this part of problem.

a$.$ Setup: Derive equations, place them in standard form, and indicate initial conditions