$C_1=$exp$(-13\mathrm{.}4+\frac{9380}{T})$

$C_2=$exp$(-6\mathrm{.}2+\frac{4883}{T})$

$C_3=$exp$(-74\mathrm{.}6-\frac{39060}{T})$

$C_4=$exp$(-27\mathrm{.}8+\frac{15948}{T})$

$C_5=$exp$(-13\mathrm{.}7+\frac{8229}{T})$

To solve the problem, you will need to know the Gibbs free energy of reaction, $G_{rxn},$ for both reactions. In

thermodynamics you will learn how to calculate these quantities from reference tables like the NIST Webbook,

but for this case you can assume that $H_{RWGS}=41\mathrm{.}15k\frac{J}{m}ol,S_{RWGS}=41\mathrm{.}97\frac{J}{\text{m}\text{o}\text{l}\text{K}},H_{\text{M}\text{e}\text{O}\text{H}}=-94\mathrm{.}47$

$k\frac{J}{m}ol$ and $S_{\text{M}\text{e}\text{O}\text{H}}=-219\mathrm{.}06\frac{J}{\text{m}\text{o}\text{l}\text{K}}$ and that they are independent of temperature. Furthermore, recall that:

$G_{rxn}=H_{rxn}-T{S}_{rxn}$

and pay careful attention to the different units of enthalpy and entropy!

Set up all required mass balances to derive ODEs that govern the system of both reactions using the

extent of reaction approach. Note that in this case, the extent of reaction will have units of bar$/$s$,$

which, via the ideal gas law, is directly proportional to $mo\frac{l}{v}$olume. Use the resulting equations to

solve an initial value problem assuming initial concentrations $1$ bar $C{O}_2,3$ bar of $H_2,$ and $0$ bar of all

other species at a temperature of $440K$ for $t=0$ to $36000s.$ Use the results to plot the extent of reaction

for both reactions as a function of time from $t=0$ to $36000s$ and $t=0$ to $600s.$ Assume that the reaction

is at equilibrium when the rate of change of the extent of reaction for MeOH synthesis is below $0\mathrm{.}01$

bar$/$s $($i$.$e

$r_i=\frac{d{}^{**}}{dt}$;$,p_i=p_i^0+v_i{}^{**}$

a$.$ Setup: Derive equations in terms of extent of reaction, place them in standard form, and

indicate initial conditions. Hint: there should be $2$ equations.

b$.$ Results: Two plots with two curves.

c$.$ Analysis: Approximately how long does it take for the RWGS reaction to equilibrate?