Partial oxidation of methane over a suitable catalyst can be employed to generate hydrogen for fuel cell applications through the reaction CH$4+0\mathrm{.}5$O$2$ CO $+2$H$2$ and the rate limiting step is activation of methane $($i$.$e$.$ CH$4$ CH$3+$ H$).$ Suppose you have Ni nanocatalyst with exposed $(111)$ and $(112)$ facets. Use the d$-$band center data from the lecture notes to estimate the ratio of the hydrogen production rate on each crystal facet assuming an operating temperature of $700$oC$.$