CO hydrogenation to form CH$4$:

CO $+3$H$2->$ CH$4+$ H$2$O

occurs on a Ru$/$SiO$2$ catalyst and is used to remove traces of CO from H$2$ streams used in fuel cell applications where ultra$-$high purity H$2$ is required $($even ppm levels of CO can poison fuel cells$).$

This reaction occurs through the following steps:

$(1)$ CO $+*$ CO$*$

$(2)$ H$2+2*2$ H$*$

$(3)$ CO$*+*$ C$*+$ O$*$

$(4)$ C$*+$ H$*->$ CH$*+*$

$(5)$ CH$*+$ H$*->$ CH$2*+*$

$(6)$ CH$2*+$ H$*->$ CH$3*+*$

$(7)$ CH$3*+$ H$*->$ CH$4+2*$

$(8)$ O$*+$ H$*$ OH$*+*$

$(9)$ OH$*+$ H$*$ H$2$O $+2*$

where Steps $1,2,3,8,$ and $9$ can all be considered quasi$-$equilibrated.

a$)$ What is the stoichiometric number of each step?

b$)$ Derive a steady$-$state rate expression $($turnover rate$)$ in terms of gas$-$phase pressures and rate and equilibrium constants. Assume that the MASI are $*$ and CO$*.$

c$)$ An undesired side$-$reaction forms CO$2$:

$(10)$ CO$*+$ O$*->$ CO$2+2*$

derive the rate equation $($turnover rate$)$ for CO$2$ formation, and then describe how the ratio of rates of formation between CH$4$ and CO$2$ vary with reaction conditions. $($Same MASI as in part b$).$