$2\mathrm{.}7$ Consider the following reaction of synthesis gas,

$CO+2H_{2}harrC{H}_{3}OH$

in a packed$-$bed reactor. The reactor is well insulated and may

be assumed to operate at steady state. The feed enters the

catalyst bed at $275C$ and the effluent leaves at $429C.$ The

reaction takes place at $300$atm. Unfortunately, the analytical

chromatograph has suffered a short, so you do not know the

effluent composition. You do know that the feed consists of

a mixture of $CO$ and $H_2$ in the mole ratio $1$:$2.$ The flow rate

through the reactor is sufficiently low that you believe that

reaction equilibrium is achieved. The standard heat of reac$-$

tion is given by

$H^0=-17,539-18\mathrm{.}19T+0\mathrm{.}0141T^2$

for $T$ in $K$ and $H^0$ in cal$/$g$-$mol. You may ignore the varia$-$

tion of enthalpy with pressure. For purposes of this problem

you may employ the following heat capacity values as being

independent of temperature and pressure:

$C_{p_{H_2}}=C_{p_{CO}}=7\mathrm{.}0ca\frac{l}{\text{m}\text{o}\text{l}*K}$

$C_{p_{C{H}_{3}OH}}=21\mathrm{.}0ca\frac{l}{\text{m}\text{o}\text{l}*K}$

$($a$)$ What are your best estimates of the effluent composition

and the equilibrium constant $K_a$ for this reaction? Use

only the information above, the assumption of ideal solu$-$

tion behavior, and the fact that the fugacity coefficients

$(\frac{f}{P})$ for $CO,H_2,$ and $C{H}_{3}OH$ at the temperature and

total pressure in question are $1\mathrm{.}178,1\mathrm{.}068,$ and $0\mathrm{.}762,$

respectively. Calculate $K_a$ relative to standard states

of unit fugacity for all species. Clearly state any other

assumptions that you make.

$($b$)$ Note: You may not use the information contained in part

$($b$)$ to solve part $($a$).$ Results of a previous study indicate

that at $390C$ and $300$atm, the standard Gibbs free energy

of reaction relative to standard states of unit fugacity is

$14,700ca\frac{l}{m}ol.$ Are the results you obtain in part $($a$)$ rea$-$

sonably consistent with this value? For your calculations

you may neglect the variation of $H^0$ with temperature

over the range $390$ to $429C$ by employing an average

value $($i$.$e$.,$ evaluate $H^0$ at $410C$ and presume it to be a

constant$).$